Abstract

Holonomies, arising from non-Abelian geometric transformations of quantum states in Hilbert space, offer a promising way for quantum computation. These holonomies are not commutable and thus can be used for the realization of a universal set of quantum logic gates, where the global geometric feature may result in some noise-resilient advantages. Here we report, to our knowledge, the first on-chip realization of a non-Abelian geometric controlled-NOT gate in a superconducting circuit, which is a building block for constructing a holonomic quantum computer. The conditional dynamics is achieved in an all-to-all connected architecture involving multiple frequency-tunable superconducting qubits controllably coupled to a resonator; a holonomic gate between any two qubits can be implemented by tuning their frequencies on-resonance with the resonator and applying a two-tone drive to one of them. This gate represents an important step towards the all-geometric realization of scalable quantum computation on a superconducting platform.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

When a nondegenerate quantum system makes a cyclic evolution in the Hilbert space, it will pick up a phase, which, in general, is contributed by both the dynamical and geometric effects. The dynamical part is the time integral of the energy, while the geometric one depends upon the area enclosed by the loop that the quantum state traverses in the Hilbert space. This effect, discovered by Berry in cyclic and adiabatic evolutions [1], has been generalized to nonadiabatic [2] and degenerate [3] cases. If a system has degenerate energy levels, the cyclic evolution of the corresponding degenerate subspaces will produce a matrix-valued quantum state transformation that is path dependent and referred to as non-Abelian geometric phase or holonomy [3]. The Berry phase and holonony depend upon the global geometry of the associated loops and have intrinsic resistance to certain kinds of small errors, suggesting that quantum gates based on geometric operations have practical advantages as compared to dynamical gates [47]. In particular, it was shown that all of the elementary one- and two-qubit gates needed for accomplishing any quantum computation task could be achieved with Berry phase and holonomic transformations, offering a possibility for implementations of geometric quantum computation [8,9].

The conditional Berry phase was first observed in nuclear magnetic resonance systems [10]. However, the relatively long operation time associated with an adiabatic evolution represents an unfavorable condition for the implementation of geometric quantum computation with such controlled phase gates. As such, geometric effects without the adiabatic restriction are highly desirable for the implementation of quantum logic gates that are robust against noises [1116]. So far, nonadiabatic geometric controlled-phase gates have been realized in ion traps [1720] and superconducting circuits [2123]. On the other hand, Sjöqvist et al. have proposed an approach for realizing a universal set of elementary gates based on nonadiabatic holonomies [24], whose robustness against noises has been analyzed [25,26]. Following this approach, a universal gate set involving two non-commutable single-qubit gates and a two-qubit controlled-NOT (CNOT) gate have been experimentally realized with nuclear magnetic resonance [27] and solid-state spins [28,29]. Several groups have demonstrated holonomic single-qubit gates in superconducting circuits [3033], which represent a promising platform for quantum computation [34]. Recently, Egger et al. reported a holonomic operation for producing entangled states in a superconducting circuit [35]. However, a non-Abelian geometric entangling gate necessary for constructing a universal holonomic gate set has not been implemented in such scalable systems. More recently, Han et al. reported a universal set of time-optimal geometric gates with superconducting qubits [36], where single-qubit gates were realized using non-Abelian geometric phase, but the two-qubit gate was based on Abelian geometric phase.

In this paper, we propose and experimentally demonstrate a scheme for realizing a non-adiabatic, non-Abelian geometric CNOT gate for two qubits, one acting as the control qubit and the other as the target qubit. The two qubits are strongly coupled to a resonator, so that the energy levels of the target qubit depend on the state of the control qubit. This conditional energy-level shift enables the target qubit to be resonantly driven by classical fields, conditional on the state of the control qubit. With a suitable setting of the parameters, these classical fields can drive the degenerate subspace spanned by the two basis states of the target qubit to undergo a conditional cyclic evolution, realizing a CNOT gate between these two qubits. We realize this holonomic gate in a superconducting multi-qubit processor, where any two qubits can be selectively coupled to a common resonator but effectively decoupled from other qubits through frequency tuning. This flexibility enables direct implementation of holonomic gates between any pair of qubits on the chip, without the restriction of nearest-neighbor couplings. The measured process fidelity for the CNOT gate is above 0.9. With further improvements in the device design and fabrication, as confirmed by our numerical simulations, the gate fidelity can be significantly increased. Our scheme is applicable to other spin-boson systems, such as cavity QED and ion traps [37].

2. THEORETICAL MODEL

The system under consideration is composed of two qutrits coupled to a resonator. Each qutrit has three basis states, as shown in Fig. 1(a), with $| g \rangle$ and $| f \rangle$ serving as two logic states of a qubit, and $| e \rangle$, lying between $| g \rangle$ and $| f \rangle$, used as an auxiliary state for realizing the controlled logic operation. For simplicity, we will refer to the qutrits as qubits. As will be shown, the control qubit (${ Q}_1$) remains in its computational space, while the target qubit (${ Q}_2$) has a probability of being populated in the auxiliary level $| e \rangle$ during the gate operation. The transition $| g \rangle \leftrightarrow | e \rangle$ of each qubit resonantly interacts with the resonator, while $| f \rangle$ state is effectively decoupled from the resonator. In the interaction picture, the Hamiltonian describing the qubit–resonator interaction is given by

$${H_{{\rm int}}} = \hbar \sum\limits_{j = 1}^2 {\lambda _j}\left({a | {{e_j}} \rangle \langle {{g_j}} | + {a^\dagger}| {{g_j}} \rangle \langle {{e_j}} |} \right),$$
where $a$ and ${a^\dagger}$ are the photonic annihilation and creation operators for the resonator, respectively, and ${\lambda _j}$ is the coupling strength between the $j$th qubit and the resonator with angular frequency ${\omega _r}$. We here have set the energy of the ground state $| g \rangle$ for each qubit to be zero. To realize the CNOT gate, the transition $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ of ${Q}_2$ is driven by a classical field with angular frequency $({{\omega _r} - {\lambda _2}})$, and $| {{e_2}} \rangle \leftrightarrow | {{f_2}} \rangle$ is driven by a classical field with angular frequency $({{\omega _{f,2}} - {\omega _r} + {\lambda _2}})$, where $\hbar {\omega _{f,2}}$ is the energy of ${Q}_2$’s state $| {{f_2}} \rangle$ [Fig. 1(a)]. The interaction between the second qubit and the driving fields is described by
$${H_{{\rm dr}}} = \hbar \left[{{\Omega _{\textit{ge}}}{e^{i{\lambda _2}t}}| {{e_2}} \rangle \langle {{g_2}} | - {\Omega _{{ef}}}{e^{- i{\lambda _2}t}} | {{f_2}} \rangle \langle {{e_2}} |} \right] + {\rm h.c.},$$
where ${\Omega _{\textit{ge}}}$ and ${\Omega _{\textit{ef}}}$ denote the Rabi frequencies of the two fields driving $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ and $| {{e_2}} \rangle \leftrightarrow | {{f_2}} \rangle$, respectively. We here have assumed that the phases of the fields driving the transitions $| g \rangle \leftrightarrow | e \rangle$ and $| e \rangle \leftrightarrow | f \rangle$ are zero and $\pi$, respectively.
 figure: Fig. 1.

Fig. 1. Energy level configuration and excitation scheme for the two-qubit CNOT gate. The control and target qubits are denoted as ${Q}_1$ and ${Q}_2$, respectively. (a) Bare energy levels of ${Q}_2$ and frequencies of the drives. The quantum information of each qubit is encoded in states $| g \rangle$ and $| f \rangle$, with the auxiliary state $| e \rangle$ used for realizing the controlled-NOT gate. The transitions $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ and $| {{e_2}} \rangle \leftrightarrow | {{f_2}} \rangle$ of ${Q}_2$ are driven by classical fields of angular frequencies $({{\omega _r} - {\lambda _2}})$ and $({{\omega _{f,2}} - {\omega _r} + {\lambda _2}})$, respectively. Here ${\omega _r}$ is the angular frequency of the resonator that is strongly coupled to $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ with the coupling strength ${\lambda _2}$, and $\hbar {\omega _{f,2}}$ is the energy spacing between $| {{f_2}} \rangle$ and $| {{g_2}} \rangle$. (b) Dressed states and energy levels with ${Q}_1$ initially in $| {{f_1}} \rangle$. When being initially in $| {{f_1}} \rangle$, ${Q}_1$ is effectively decoupled from the resonator due to the large detuning. The strong coupling between ${Q}_2$ and the resonator results in dressed states $| {\psi _n^ \pm} \rangle$, whose energy levels are nonlinearly dependent on the coupling strength. The two driving fields are on-resonance with the transitions $| {{g_2}0} \rangle \leftrightarrow | {\psi _1^ -} \rangle$ and $| {\psi _1^ -} \rangle \leftrightarrow | {{f_2}0} \rangle$, respectively, but highly detuned from other transitions. (c) Dressed states and energy levels with ${Q}_1$ initially in $| {{g_1}} \rangle$. If initially in $| {{g_1}} \rangle$, ${Q}_1$, together with ${Q}_2$, is strongly coupled to the resonator, resulting in three dressed states $| {\Phi _1^ \pm} \rangle$ and $| {\Phi _1^0} \rangle$ in the single-excitation subspace. The driving fields are highly detuned from transitions of $| {{g_1}{g_2}0} \rangle$ and $| {{g_1}{f_2}0} \rangle$ to these dressed states.

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The strong couplings between the qubits and the resonator produce dressed states, whose energy levels depend on the total excitation number as well as on the number of qubits being initially populated in $| g \rangle$. When the control qubit is in the state $| {{f_1}} \rangle$, it does not interact with the resonator, and the coupling between the target qubit and the resonator is described by the Jaynes–Cummings model, whose eigenstates are given by

$$| {{\psi _0}} \rangle = | {{g_2}0} \rangle ,$$
$$| {\psi _n^ \pm} \rangle = \frac{1}{{\sqrt 2}}\left({| {{e_2}(n - 1)} \rangle \pm | {{g_2}n} \rangle} \right),n \ge 1.$$
Here the second symbol in each ket denotes the photon number in the resonator. The eigenenergies of the dressed states $| {\psi _n^ \pm} \rangle$ are $\hbar ({n{\omega _r} \pm \sqrt n {\lambda _2}})$. We here consider the case that the resonator is initially in the vacuum state $| 0 \rangle$. Consequently, the classical fields resonantly couple states $| {{g_2}0} \rangle$ and $| {{f_2}0}\rangle$ to the single-excitation dressed state $| {\psi _1^ -} \rangle$, as sketched in Fig. 1(b). We suppose that ${\Omega _{\textit{ge}}}$ and ${\Omega _{\textit{ef}}}$ are much smaller than ${\lambda _2}$, so that the classical fields cannot drive the transitions from $| {\psi _1^ -}\rangle$ to $| {\psi _2^ \pm} \rangle$ due to the large detunings. However, these off-resonant couplings shift the energy levels of $| {\psi _1^ -}\rangle$ by ${-}2\hbar {\delta _1}$, with ${\delta _1} = 2\Omega _{\textit{ge}}^2/{\lambda _2}$ (see Supplement 1). Furthermore, off-resonant coupling to $| {{h_1}} \rangle | {{g_2}0} \rangle$ and $| {{e_1}} \rangle | {\psi _2^ \pm} \rangle$ shifts the energy level of $| {{f_1}} \rangle | {\psi _1^ \pm} \rangle$ by an amount of ${-}\hbar {\delta _2}$, where ${\delta _2} = 9\lambda _1^2/4{\alpha _1}$ (see Supplement 1), and $| {{h_1}} \rangle$ is the fourth level of ${Q}_1$, and ${\alpha _1}$ is its anharmonicity (${\alpha _j} = 2{\omega _{e,j}} - {\omega _{f,j}}$, $j = 1,2$). To compensate for these shifts, the angular frequency of the field driving $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ should be set to ${\omega _{d,1}} = {\omega _r} - {\lambda _2} - {\delta _1} - {\delta _2}$, while that of the field driving $| {{e_2}} \rangle \leftrightarrow | {{f_2}} \rangle$ should be set to ${\omega _{d,2}} = {\omega _{f,2}} - {\omega _r} + {\lambda _2} + {\delta _1} + {\delta _2}$. With this setting and performing the transformation $\exp ({i{H_{{\rm int}}}t/\hbar})$, the system dynamics associated with ${\rm Q}_1$’s state $| {{f_1}} \rangle$ can be described by the effective Hamiltonian
$${H_{{\rm eff}}} = \hbar \Omega \left[{\cos \frac{\phi}{2}| {{g_2}0} \rangle \langle {\psi _1^ -} | + \sin \frac{\phi}{2}| {{f_2}0} \rangle \langle {\psi _1^ -}|} \right] | {{f_1}} \rangle \langle {{f_1}} | + {\rm h.c.},$$
where
$$\Omega = \sqrt {\Omega _{\textit{ge}}^2 + \Omega _{\textit{ef}}^2} /\sqrt 2 ,$$
$$\tan \frac{\phi}{2} = {\Omega _{\textit{ef}}}/{\Omega _{\textit{ge}}}.$$

When ${Q}_1$ is initially in the state $| {{g_1}} \rangle$, it is also strongly coupled to the resonator, and there are three dressed states in the single-excitation subspace:

$$| {\Phi _1^0} \rangle = \left({- \sin \theta | {{e_1}{g_2}0} \rangle + \cos \theta | {{g_1}{e_2}0} \rangle} \right),$$
$$| {\Phi _1^ \pm} \rangle = \frac{1}{{\sqrt 2}}\left[{\left({\cos \theta | {{e_1}{g_2}0} \rangle + \sin \theta | {{g_1}{e_2}0} \rangle} \right) \pm | {{g_1}{g_2}1} \rangle} \right],$$
where $\tan \theta = {\lambda _2}/{\lambda _1}$. The corresponding eigenenergies are $\hbar {\omega _e}$ and $\hbar ({{\omega _e} \pm \sqrt {\lambda _1^2 + \lambda _2^2}})$, as shown in Fig. 1(c). When $({\sqrt {\lambda _1^2 + \lambda _2^2} - {\lambda _2}})$ is much larger than ${\Omega _{\textit{ge}}}$ and ${\Omega _{\textit{ef}}}$, the qubits cannot make any transition between either of these single-excitation dressed states and the state $| {{g_1}{g_2}0} \rangle$ or $| {{g_1}{f_2}0} \rangle$, as each of these transitions is highly detuned from the driving fields. As a consequence, ${Q}_2$ is not affected by the driving fields when ${ Q}_1$ is initially in the state $| {{g_1}} \rangle$. Therefore, the system dynamics is described by the effective Hamiltonian of Eq. (5). The evolution of the initial basis states $| {{c_1}{d_2}0} \rangle$ ($c,d = g,f$) are given by
$$| {{\psi _{\textit{cd}}}(t)} \rangle = \exp \left({- i\int_0^t {H_{{\rm eff}}}dt/\hbar} \right)| {{c_1}{d_2}0} \rangle .$$
When ${\Omega _{\textit{ef}}}/{\Omega _{\textit{ge}}}$ remains unchanged during the interaction, the evolution satisfies the parallel-transport condition $\langle {{\psi _{\textit{cd}}}(t)} |{H_{{\rm eff}}}| {{\psi _{{c^{{\prime}}}{d^{{\prime}}}}}(t)} \rangle = 0$, and hence is purely geometric. If the Rabi frequencies of the driving fields and the interaction time are appropriately chosen so that $\int_0^T \Omega {\rm d}t = \pi$, the degenerate qubit subspace undergoes a cyclic evolution. Consequently, the qubits return to the computational space $\{{| {{g_1}{g_2}} \rangle ,| {{g_1}{f_2}} \rangle ,| {{f_1}{g_2}} \rangle ,| {{f_1}{f_2}} \rangle} \}$ with the resonator left in the vacuum state $| 0 \rangle$ after the time $T$. With this setting, the evolution operator of the qubits in the computational basis is
$$U = \left({\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&{- \cos \phi}&{\sin \phi}\\0&0&{\sin \phi}&{\cos \phi}\end{array}} \right),$$
which is a non-Abelian holonomy. For $\phi = \pi /2$, i.e., ${\Omega _{\textit{ge}}} = {\Omega _{\textit{ef}}}$, this corresponds to a CNOT gate, which flips the state of the target qubit conditional on the control qubit being in the state $| {{f_1}} \rangle$.

3. EXPERIMENTAL IMPLEMENTATION

The experiment is performed in a superconducting circuit involving five frequency-tunable qubits, labeled from ${Q}_1$ to ${Q}_5$, coupled to a resonator with a fixed frequency ${\omega _r}/2\pi = 5.584\;{\rm GHz}$ [21,38,39]. In our experiment, ${Q}_1$ and ${Q}_2$, whose anharmonicities are $2\pi \times 242\;{\rm MHz}$ and $2\pi \times 249\;{\rm MHz}$, are used as the control and target qubits, respectively. The on-resonance coupling strengths of the $g\! -\! e$ transitions of ${Q}_1$ and ${Q}_2$ to the resonator are respectively ${\lambda _1}= 2\pi \times 20.8\;{\rm MHz}$ and ${\lambda _2} = 2\pi \times 19.9\; {\rm MHz}$. The energy relaxation time ${T}_1$ and pure Gaussian dephasing time ${T}_2^ *$ for the basis state $| f \rangle$ of ${ Q}_1$ (${Q}_2$) are $13.0\, (10.7)\;\unicode{x00B5} {\rm s}$ and $2.1\,(1.5)\;\unicode{x00B5} {\rm s}$, while those for the intermediate state $| e \rangle$ are $23.9\,(15.9)\;\unicode{x00B5} {\rm s}$ and $2.7\,(2.1)\;\unicode{x00B5} {\rm s}$, respectively. The other qubits are on far off-resonance with the resonator so that their interactions with the resonator are effectively switched off throughout the gate operation. We note that during the gate operation, the two qubits have a probability of being populated in $| {{f_1}{e_2}} \rangle$, which is significantly coupled to $| {{e_1}{f_2}} \rangle$ via virtual photon exchange, as the two qubits almost have the same anharmonicity $\alpha \simeq 2\pi \times 240\; {\rm MHz}$. To suppress this coupling, ${Q}_1$ should be detuned from ${Q}_2$ by an amount much larger than ${\lambda _1}{\lambda _2}/\alpha$. This detuning slightly changes the energy level configuration of the dressed states associated with ${Q}_1$’s initial state $| {{g_1}} \rangle$, but does not affect the gate dynamics.

As shown in Fig. 2, the experiment starts with the initialization of ${Q}_1$ and ${Q}_2$ to the ground state $| g \rangle$ at their idle frequencies 5.47 GHz and 5.43 GHz, respectively, which is followed by the preparation of each qubit in one of the six states $\{| g \rangle$, $({| g \rangle - i| f \rangle})/\sqrt 2$, $({| g \rangle + i| f \rangle})/\sqrt 2$, $({| g \rangle + | f \rangle})/\sqrt 2$, $({| g \rangle - | f \rangle})/\sqrt 2$, $| f \rangle \}$. Except $| g \rangle$, each of the other single-qubit states is produced by a $g\! -\! e\;\pi /2$- or $\pi$-pulse followed by a $e\! -\! f\;\pi$-pulse.

 figure: Fig. 2.

Fig. 2. Pulse sequence. Before the gate operation, both qubits are initialized to their ground states at the corresponding idle frequencies, where single-qubit rotations are performed to prepare them in a product state. Then a ${Z}$ pulse is applied to ${Q}_1$, tuning $| {{g_1}} \rangle \leftrightarrow | {{e_1}} \rangle$ close to the resonator’s frequency; ${Q}_2$ is subjected to a ${Z}$ pulse, which brings $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ to the resonator’s frequency, and a driving pulse involving two frequency components on-resonance with the transitions $| {{g_2}0} \rangle \leftrightarrow | {\psi _1^ -} \rangle$ and $| {\psi _1^ -} \rangle \leftrightarrow | {{f_2}0} \rangle$. After the CNOT gate, realized with these pulses, both qubits are tuned back to their idle frequencies for quantum state tomography.

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After these effective single-qubit rotations, these two qubits are prepared in a product state. We then apply square Z pulses to both qubits, tuning their $| g \rangle \leftrightarrow | e \rangle$ transition frequencies to 5.58 GHz and 5.584 GHz and thus switching on their interactions with the resonator. Accompanying these ${Z}$ pulses, a driving pulse composed of two components with frequencies of 5.565 GHz and 5.369 GHz is applied to ${Q}_2$, resonantly connecting the computational states $| {{g_2}0} \rangle \;$ and $| {{f_2}0} \rangle$ to the dressed state $| {\psi _1^ -} \rangle$. The Rabi frequencies of these driving fields are ${\Omega _{\textit{ge}}} = {\Omega _{\textit{ef}}} = 2\pi \times 2.2\;{\rm MHz}$. Since the resonator is initially in the vacuum state, the system dynamics is governed by the effective Hamiltonian (5) and the time evolution given by Eq. (10). After a duration of $\tau = 205\; {\rm ns}$, the CNOT gate is realized.

One of the most important features of the CNOT gate is that it can convert a two-qubit product state into an entangled state. In particular, when the control qubit is initially in the superposition state $({| {{g_1}} \rangle + | {{f_1}} \rangle})/\sqrt 2$ and the target state in $| {{g_2}} \rangle$, they will evolve to the maximally entangled state $({| {{g_1}{g_2}} \rangle + | {{f_1}{f_2}} \rangle})/\sqrt 2$ after this gate. We measure this output state by quantum state tomography. This is realized by subsequently biasing each of the two qubits back to its idle frequency right after the gate operation, applying an $e\! -\! f\;\pi$-pulse to each qubit, and measuring its state along one of the three orthogonal ($x$, $y$, and $z$) axes of the corresponding Bloch sphere with respect to the basis $\{{| g \rangle ,| e \rangle} \}$. The $z$ measurement is directly realized by state readout, while the $x\;(y)$ measurement is realized by the combination of a $g\! -\! e\;\pi /2$-pulse and state readout. The reconstructed output two-qubit density matrix is displayed in Fig. 3, which has a fidelity of $0.935 \pm 0.016$ to the ideal maximally entangled state, and a concurrence of $0.888 \pm 0.029$.

 figure: Fig. 3.

Fig. 3. Measured density matrix of the output state with the input state $({| {{g_1}{g_2}} \rangle + | {{f_1}{g_2}} \rangle})/\sqrt 2$. Each matrix element is characterized by two colorbars, one for the real part and the other for the imaginary part. The black wire frames denote the matrix elements of the ideal output states.

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To fully characterize the performance of the implemented CNOT gate, we prepare a full set of 36 distinct two-qubit input states before the two-qubit gates, and measure these states and the corresponding output states. With these measured results, the process matrix for the gate operation is reconstructed. The measured process matrix, ${\chi _{{\rm meas}}}$, is presented in Fig. 4. The gate fidelity, defined as $F = tr({{\chi _{{\rm id}}}{\chi _{{\rm meas}}}})$, is $0.905 \pm 0.008$, where ${\chi _{{\rm id}}}$ is the ideal process matrix. The measured fidelity is in good agreement with the numerical simulation based on the Lindblad master equation, which yields a fidelity of 0.908. One of the error sources is the transitions from $| {{g_1}{g_2}0} \rangle$ and $| {{g_1}{f_2}0} \rangle$ to $| {\Phi _1^0} \rangle$ and $| {\Phi _1^ \pm} \rangle$ and the transition from $| {\psi _1^ -} \rangle$ to $| {\psi _2^ -} \rangle$ induced by the drive, which causes quantum information leakage to the noncomputational space. Such a leakage error can be mitigated through improvement of the qubit’s nonlinearity or by balancing the drive amplitude and the gate operation time provided the qubits’ coherence is bettered, which allows the gate fidelity to be increased by about 6.5% (see Supplement 1). On the other hand, the qubits’ energy relaxation and their dephasings contribute about 1.8% and 1.6% of the error, respectively. Our further numerical simulations show that the CNOT gate with a fidelity above 99% can be obtained with sufficiently large qubit nonlinearity ${\alpha _j}$ and qubit-resonator coupling strength ${\lambda _j}$. For instance, with the parameters ${\lambda _j}/2\pi = 110\; {\rm MHz}$, ${\alpha _j}/2\pi \; = - 3.69\; {\rm GHz}$ [40,41], ${\Omega _{\textit{ge}}}/2\pi = {\Omega _{\textit{ef}}}/2\pi = 5.9\; {\rm MHz}$, ${T_1} = 60\;\unicode{x00B5} {\rm s}$, and $T_{2,j}^ * = 86\;\unicode{x00B5} {\rm s}$, we find a CNOT gate with the operation time of about 87 ns and fidelity of 0.991, which is at the surface code threshold for fault tolerance [4244]. We note this gate is robust against the frequency fluctuations of the driving fields. Suppose that the angular frequencies of these drives deviate from the desired values by an amount of $\delta \omega = 2\pi \times 100\; {\rm kHz}$. The infidelity incurred by this deviation is about ${[\pi {(\delta \omega)^2}/(8\Omega _{ge/ef}^2)]^2} \simeq 0.1\%$.

 figure: Fig. 4.

Fig. 4. Measured process matrix for the realized CNOT gate. The process matrix is measured by preparing a set of 36 distinct input product states in the computational basis $\{{| {{g_1}{g_2}} \rangle ,| {{g_1}{f_2}} \rangle ,| {{f_1}{g_2}} \rangle ,| {{f_1}{f_2}} \rangle} \}$ and reconstructing the density matrices for these states and for the output states produced by the CNOT gate. The $| e \rangle$-state populations of ${Q}_1$ and ${Q}_2$, averaged over the 36 output states, are 2.2% and 2.8%, respectively.

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4. CONCLUSION

In conclusion, we have proposed and demonstrated a scheme for implementing a non-Abelian geometric gate between two superconducting qubits, whose ground and second excited states act as the computational basis states. The conditional dynamics is realized by resonantly driving the transitions between the basis states of the target qubit to the single-excitation dressed states formed by this qubit and the resonator. This entangling gate, together with the previously demonstrated non-Abelian geometric single-qubit gates [3033], constitutes a universal set of holonomic gates for realizing quantum computation with superconducting qubits. The method can be directly applied to other systems composed of qubits coupled to a bosonic mode, including cavity QED and ion traps.

Funding

National Natural Science Foundation of China (11674060, 11874114, 11875108, 11904393, 11934018, 92065114); the Strategic Priority Research Program of Chinese Academy of Sciences (XDB28000000); Beijing Municipal Natural Science Foundation (Z200009).

Acknowledgment

We thank Haohua Wang at Zhejiang University for technical support. S.-B.Z. conceived the experiment. K.X., W. N., and Z.-B.Y. performed the experiment and analyzed the data with the assistance of X.-J.H. and P.-R.H. H.L. and D.Z. provided the devices used for the experiment. S.-B.Z., Z.-B.Y., K.X., and H.F. wrote the manuscript with feedbacks from all authors. The experiment was performed at Fuzhou University.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

REFERENCES

1. M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984). [CrossRef]  

2. Y. Aharonov and A. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987). [CrossRef]  

3. F. Wilczek and A. Zee, “Appearance of gauge structure in simple dynamical systems,” Phys. Rev. Lett. 52, 2111–2114 (1984). [CrossRef]  

4. G. D. Chiara and G. M. Palma, “Berry phase for a spin 1/2 particle in a classical fluctuating field,” Phys. Rev. Lett. 91, 090404 (2003). [CrossRef]  

5. S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009). [CrossRef]  

6. A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, “Geometric phase in open systems,” Phys. Rev. Lett. 90, 160402 (2003). [CrossRef]  

7. S.-B. Zheng, “Geometric phase for a driven quantum field subject to decoherence,” Phys. Rev. A 91, 052117 (2015). [CrossRef]  

8. P. Zanardi and M. Rasetti, “Holonomic quantum computation,” Phys. Lett. A 264, 94–99 (1999). [CrossRef]  

9. J. Pachos, P. Zanardi, and M. Rasetti, “Non-Abelian Berry connections for quantum computation,” Phys. Rev. A 61, 010305 (2000). [CrossRef]  

10. J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation with NMR,” Nature (London) 403, 869–871 (2000). [CrossRef]  

11. X.-B. Wang and M. Keiji, “Nonadiabatic conditional geometric phase shift with NMR,” Phys. Rev. Lett. 87, 097901 (2001). [CrossRef]  

12. S.-L. Zhu and Z.-D. Wang, “Implementation of universal quantum gates based on nonadiabatic geometric phases,” Phys. Rev. Lett. 89, 097902 (2002). [CrossRef]  

13. A. Nazir, T. Spiller, and W. J. Munro, “Decoherence of geometric phase gates,” Phys. Rev. A 65, 042303 (2002). [CrossRef]  

14. A. Blais and A.-M. S. Tremblay, “Effect of noise on geometric logic gates for quantum computation,” Phys. Rev. A 67, 012308 (2003). [CrossRef]  

15. J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, “Holonomic surface codes for fault-tolerant quantum computation,” Phys. Rev. A 97, 022335 (2018). [CrossRef]  

16. Y.-H. Chen, W. Qin, R. Stassi, X. Wang, and F. Nori, “Generation of Fock-state superpositions and binomial-code holonomic gates via dressed intermediate states in the ultrastrong light-matter coupling regime,” arXiv:2012.06090 (2021).

17. D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003). [CrossRef]  

18. J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008). [CrossRef]  

19. C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016). [CrossRef]  

20. J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016). [CrossRef]  

21. C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017). [CrossRef]  

22. Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020). [CrossRef]  

23. Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020). [CrossRef]  

24. E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012). [CrossRef]  

25. M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012). [CrossRef]  

26. S.-B. Zheng, C.-P. Yang, and F. Nori, “Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts,” Phys. Rev. A 93, 032313 (2016). [CrossRef]  

27. G. Feng, G. Xu, and G. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013). [CrossRef]  

28. C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014). [CrossRef]  

29. S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, “Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin,” Nat. Commun. 5, 4870 (2014). [CrossRef]  

30. A. A. Abdumalikov Jr., J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013). [CrossRef]  

31. Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018). [CrossRef]  

32. T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019). [CrossRef]  

33. Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019). [CrossRef]  

34. J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42 (2005). [CrossRef]  

35. D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019). [CrossRef]  

36. Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

37. I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011). [CrossRef]  

38. W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019). [CrossRef]  

39. Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021). [CrossRef]  

40. M. A. Yurtalan, J. Shi, G. J. K. Flatt, and A. Lupascu, “Characterization of multi-level dynamics and decoherence in a high-anharmonicity capacitively shunted flux circuit,” arxiv:2008.00593 (2020).

41. J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B 75, 140515 (2007). [CrossRef]  

42. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012). [CrossRef]  

43. R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014). [CrossRef]  

44. R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015). [CrossRef]  

References

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  • |
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  • |

  1. M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
    [Crossref]
  2. Y. Aharonov and A. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
    [Crossref]
  3. F. Wilczek and A. Zee, “Appearance of gauge structure in simple dynamical systems,” Phys. Rev. Lett. 52, 2111–2114 (1984).
    [Crossref]
  4. G. D. Chiara and G. M. Palma, “Berry phase for a spin 1/2 particle in a classical fluctuating field,” Phys. Rev. Lett. 91, 090404 (2003).
    [Crossref]
  5. S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
    [Crossref]
  6. A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, “Geometric phase in open systems,” Phys. Rev. Lett. 90, 160402 (2003).
    [Crossref]
  7. S.-B. Zheng, “Geometric phase for a driven quantum field subject to decoherence,” Phys. Rev. A 91, 052117 (2015).
    [Crossref]
  8. P. Zanardi and M. Rasetti, “Holonomic quantum computation,” Phys. Lett. A 264, 94–99 (1999).
    [Crossref]
  9. J. Pachos, P. Zanardi, and M. Rasetti, “Non-Abelian Berry connections for quantum computation,” Phys. Rev. A 61, 010305 (2000).
    [Crossref]
  10. J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation with NMR,” Nature (London) 403, 869–871 (2000).
    [Crossref]
  11. X.-B. Wang and M. Keiji, “Nonadiabatic conditional geometric phase shift with NMR,” Phys. Rev. Lett. 87, 097901 (2001).
    [Crossref]
  12. S.-L. Zhu and Z.-D. Wang, “Implementation of universal quantum gates based on nonadiabatic geometric phases,” Phys. Rev. Lett. 89, 097902 (2002).
    [Crossref]
  13. A. Nazir, T. Spiller, and W. J. Munro, “Decoherence of geometric phase gates,” Phys. Rev. A 65, 042303 (2002).
    [Crossref]
  14. A. Blais and A.-M. S. Tremblay, “Effect of noise on geometric logic gates for quantum computation,” Phys. Rev. A 67, 012308 (2003).
    [Crossref]
  15. J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, “Holonomic surface codes for fault-tolerant quantum computation,” Phys. Rev. A 97, 022335 (2018).
    [Crossref]
  16. Y.-H. Chen, W. Qin, R. Stassi, X. Wang, and F. Nori, “Generation of Fock-state superpositions and binomial-code holonomic gates via dressed intermediate states in the ultrastrong light-matter coupling regime,” arXiv:2012.06090 (2021).
  17. D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
    [Crossref]
  18. J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
    [Crossref]
  19. C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
    [Crossref]
  20. J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
    [Crossref]
  21. C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
    [Crossref]
  22. Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
    [Crossref]
  23. Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
    [Crossref]
  24. E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
    [Crossref]
  25. M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
    [Crossref]
  26. S.-B. Zheng, C.-P. Yang, and F. Nori, “Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts,” Phys. Rev. A 93, 032313 (2016).
    [Crossref]
  27. G. Feng, G. Xu, and G. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013).
    [Crossref]
  28. C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
    [Crossref]
  29. S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, “Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin,” Nat. Commun. 5, 4870 (2014).
    [Crossref]
  30. A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
    [Crossref]
  31. Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
    [Crossref]
  32. T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
    [Crossref]
  33. Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
    [Crossref]
  34. J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42 (2005).
    [Crossref]
  35. D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
    [Crossref]
  36. Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).
  37. I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
    [Crossref]
  38. W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
    [Crossref]
  39. Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
    [Crossref]
  40. M. A. Yurtalan, J. Shi, G. J. K. Flatt, and A. Lupascu, “Characterization of multi-level dynamics and decoherence in a high-anharmonicity capacitively shunted flux circuit,” arxiv:2008.00593 (2020).
  41. J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B 75, 140515 (2007).
    [Crossref]
  42. A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
    [Crossref]
  43. R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
    [Crossref]
  44. R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
    [Crossref]

2021 (1)

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

2020 (2)

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

2019 (4)

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

2018 (2)

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, “Holonomic surface codes for fault-tolerant quantum computation,” Phys. Rev. A 97, 022335 (2018).
[Crossref]

2017 (1)

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

2016 (3)

C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
[Crossref]

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

S.-B. Zheng, C.-P. Yang, and F. Nori, “Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts,” Phys. Rev. A 93, 032313 (2016).
[Crossref]

2015 (2)

S.-B. Zheng, “Geometric phase for a driven quantum field subject to decoherence,” Phys. Rev. A 91, 052117 (2015).
[Crossref]

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

2014 (3)

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, “Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin,” Nat. Commun. 5, 4870 (2014).
[Crossref]

2013 (2)

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

G. Feng, G. Xu, and G. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013).
[Crossref]

2012 (3)

E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
[Crossref]

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

2011 (1)

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

2009 (1)

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

2008 (1)

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[Crossref]

2007 (1)

J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B 75, 140515 (2007).
[Crossref]

2005 (1)

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42 (2005).
[Crossref]

2003 (4)

A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, “Geometric phase in open systems,” Phys. Rev. Lett. 90, 160402 (2003).
[Crossref]

G. D. Chiara and G. M. Palma, “Berry phase for a spin 1/2 particle in a classical fluctuating field,” Phys. Rev. Lett. 91, 090404 (2003).
[Crossref]

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

A. Blais and A.-M. S. Tremblay, “Effect of noise on geometric logic gates for quantum computation,” Phys. Rev. A 67, 012308 (2003).
[Crossref]

2002 (2)

S.-L. Zhu and Z.-D. Wang, “Implementation of universal quantum gates based on nonadiabatic geometric phases,” Phys. Rev. Lett. 89, 097902 (2002).
[Crossref]

A. Nazir, T. Spiller, and W. J. Munro, “Decoherence of geometric phase gates,” Phys. Rev. A 65, 042303 (2002).
[Crossref]

2001 (1)

X.-B. Wang and M. Keiji, “Nonadiabatic conditional geometric phase shift with NMR,” Phys. Rev. Lett. 87, 097901 (2001).
[Crossref]

2000 (2)

J. Pachos, P. Zanardi, and M. Rasetti, “Non-Abelian Berry connections for quantum computation,” Phys. Rev. A 61, 010305 (2000).
[Crossref]

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation with NMR,” Nature (London) 403, 869–871 (2000).
[Crossref]

1999 (1)

P. Zanardi and M. Rasetti, “Holonomic quantum computation,” Phys. Lett. A 264, 94–99 (1999).
[Crossref]

1987 (1)

Y. Aharonov and A. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[Crossref]

1984 (2)

F. Wilczek and A. Zee, “Appearance of gauge structure in simple dynamical systems,” Phys. Rev. Lett. 52, 2111–2114 (1984).
[Crossref]

M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
[Crossref]

Abdumalikov, A. A.

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

Aharonov, Y.

Y. Aharonov and A. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[Crossref]

Anandan, A.

Y. Aharonov and A. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[Crossref]

Andersson, L. M.

E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
[Crossref]

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

Arroyo-Camejo, S.

S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, “Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin,” Nat. Commun. 5, 4870 (2014).
[Crossref]

Ashhab, S.

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B 75, 140515 (2007).
[Crossref]

Balasubramanian, G.

S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, “Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin,” Nat. Commun. 5, 4870 (2014).
[Crossref]

Ballance, C. J.

C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
[Crossref]

Barends, R.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Barkoutsos, P. K.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Barrett, M.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Benhelm, J.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[Crossref]

Berger, S.

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

Berry, M. V.

M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
[Crossref]

Blais, A.

A. Blais and A.-M. S. Tremblay, “Effect of noise on geometric logic gates for quantum computation,” Phys. Rev. A 67, 012308 (2003).
[Crossref]

Blatt, R.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[Crossref]

Bowler, R.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Britton, J.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Buluta, I.

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

Cai, W.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Campbell, B.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Carollo, A.

A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, “Geometric phase in open systems,” Phys. Rev. Lett. 90, 160402 (2003).
[Crossref]

Castagnoli, G.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation with NMR,” Nature (London) 403, 869–871 (2000).
[Crossref]

Chen, T.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Chen, Y.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Chen, Y.-H.

Y.-H. Chen, W. Qin, R. Stassi, X. Wang, and F. Nori, “Generation of Fock-state superpositions and binomial-code holonomic gates via dressed intermediate states in the ultrastrong light-matter coupling regime,” arXiv:2012.06090 (2021).

Chen, Z.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Chiara, G. D.

G. D. Chiara and G. M. Palma, “Berry phase for a spin 1/2 particle in a classical fluctuating field,” Phys. Rev. Lett. 91, 090404 (2003).
[Crossref]

Chiaro, B.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Chu, J.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Cleland, A. N.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Coakley, K.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Dai, C.-Y.

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

Dai, W.

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

DeMarco, B.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Deng, H.

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Devitt, S. J.

J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, “Holonomic surface codes for fault-tolerant quantum computation,” Phys. Rev. A 97, 022335 (2018).
[Crossref]

Dong, Y.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Duan, L.-M.

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

Duan, P.

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Dunsworth, A.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Egger, D. J.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Ekert, A.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation with NMR,” Nature (London) 403, 869–871 (2000).
[Crossref]

Ericsson, M.

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

Fan, H.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

Feng, G.

G. Feng, G. Xu, and G. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013).
[Crossref]

Filipp, S.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

Fink, J. M.

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

Flatt, G. J. K.

M. A. Yurtalan, J. Shi, G. J. K. Flatt, and A. Lupascu, “Characterization of multi-level dynamics and decoherence in a high-anharmonicity capacitively shunted flux circuit,” arxiv:2008.00593 (2020).

Fowler, A. G.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Fuentes-Guridi, I.

A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, “Geometric phase in open systems,” Phys. Rev. Lett. 90, 160402 (2003).
[Crossref]

Fuhrer, A.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Gaebler, J. P.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Ganzhorn, M.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Geltenbort, P.

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

Glancy, S.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Guo, G.

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Guo, Q.

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Han, J.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Han, P.-R.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

Han, Z.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Harty, T. P.

C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
[Crossref]

Hasegawa, Y.

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

He, L.

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

Hell, S. W.

S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, “Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin,” Nat. Commun. 5, 4870 (2014).
[Crossref]

Hessmo, B.

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
[Crossref]

Hoi, I.-C.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

Hu, L.

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Hu, X.

J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B 75, 140515 (2007).
[Crossref]

Hua, Z.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Huang, K.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Huang, T.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Huang, X.-J.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

Itano, W. M.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Jeffrey, E.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Jelenkovic, B.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Jia, Z.

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Johansson, M.

E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
[Crossref]

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

Jones, J. A.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation with NMR,” Nature (London) 403, 869–871 (2000).
[Crossref]

Juliusson, K.

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

Keiji, M.

X.-B. Wang and M. Keiji, “Nonadiabatic conditional geometric phase shift with NMR,” Phys. Rev. Lett. 87, 097901 (2001).
[Crossref]

Keith, A. C.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Kelly, J.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Kirchmair, G.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[Crossref]

Klepp, J.

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

Knill, E.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Korotkov, A. N.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Lan, D.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Langer, C.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Lazariev, A.

S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, “Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin,” Nat. Commun. 5, 4870 (2014).
[Crossref]

Leibfried, D.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Li, D.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Li, H.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

Li, X.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Lin, Y.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Linke, N. M.

C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
[Crossref]

Liu, B.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Liu, B.-J.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Liu, S.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Liu, W.

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Long, G.

G. Feng, G. Xu, and G. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013).
[Crossref]

Lucas, D.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Lucas, D. M.

C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
[Crossref]

Lupascu, A.

M. A. Yurtalan, J. Shi, G. J. K. Flatt, and A. Lupascu, “Characterization of multi-level dynamics and decoherence in a high-anharmonicity capacitively shunted flux circuit,” arxiv:2008.00593 (2020).

Ma, Y.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Mariantoni, M.

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Martinis, J. M.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Megrant, A.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Meyer, V.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Moll, N.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Mu, X.

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Mueller, P.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Munro, W. J.

A. Nazir, T. Spiller, and W. J. Munro, “Decoherence of geometric phase gates,” Phys. Rev. A 65, 042303 (2002).
[Crossref]

Mutus, J.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Mutus, J. Y.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

Nazir, A.

A. Nazir, T. Spiller, and W. J. Munro, “Decoherence of geometric phase gates,” Phys. Rev. A 65, 042303 (2002).
[Crossref]

Neill, C.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Ning, W.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

Nori, F.

J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, “Holonomic surface codes for fault-tolerant quantum computation,” Phys. Rev. A 97, 022335 (2018).
[Crossref]

S.-B. Zheng, C.-P. Yang, and F. Nori, “Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts,” Phys. Rev. A 93, 032313 (2016).
[Crossref]

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B 75, 140515 (2007).
[Crossref]

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42 (2005).
[Crossref]

Y.-H. Chen, W. Qin, R. Stassi, X. Wang, and F. Nori, “Generation of Fock-state superpositions and binomial-code holonomic gates via dressed intermediate states in the ultrastrong light-matter coupling regime,” arXiv:2012.06090 (2021).

O’Malley, P.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

O’Malley, P. J. J.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

Pachos, J.

J. Pachos, P. Zanardi, and M. Rasetti, “Non-Abelian Berry connections for quantum computation,” Phys. Rev. A 61, 010305 (2000).
[Crossref]

Palma, G. M.

G. D. Chiara and G. M. Palma, “Berry phase for a spin 1/2 particle in a classical fluctuating field,” Phys. Rev. Lett. 91, 090404 (2003).
[Crossref]

Pan, X.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Pechal, M.

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

Plonka-Spehr, C.

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

Qin, W.

Y.-H. Chen, W. Qin, R. Stassi, X. Wang, and F. Nori, “Generation of Fock-state superpositions and binomial-code holonomic gates via dressed intermediate states in the ultrastrong light-matter coupling regime,” arXiv:2012.06090 (2021).

Qiu, L.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Quintana, C.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

Rasetti, M.

J. Pachos, P. Zanardi, and M. Rasetti, “Non-Abelian Berry connections for quantum computation,” Phys. Rev. A 61, 010305 (2000).
[Crossref]

P. Zanardi and M. Rasetti, “Holonomic quantum computation,” Phys. Lett. A 264, 94–99 (1999).
[Crossref]

Rauch, H.

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

Rong, H.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Roos, C. F.

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[Crossref]

Rosenband, T.

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Roushan, P.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Salis, G.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Sank, D.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Santos, M. F.

A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, “Geometric phase in open systems,” Phys. Rev. Lett. 90, 160402 (2003).
[Crossref]

Schmidt, U.

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

Sepiol, M. A.

C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
[Crossref]

Shi, J.

M. A. Yurtalan, J. Shi, G. J. K. Flatt, and A. Lupascu, “Characterization of multi-level dynamics and decoherence in a high-anharmonicity capacitively shunted flux circuit,” arxiv:2008.00593 (2020).

Singh, K.

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
[Crossref]

Sjöqvist, E.

E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
[Crossref]

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

Song, C.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Song, S.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Song, Y. P.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Song, Y.-P.

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Spiller, T.

A. Nazir, T. Spiller, and W. J. Munro, “Decoherence of geometric phase gates,” Phys. Rev. A 65, 042303 (2002).
[Crossref]

Stassi, R.

Y.-H. Chen, W. Qin, R. Stassi, X. Wang, and F. Nori, “Generation of Fock-state superpositions and binomial-code holonomic gates via dressed intermediate states in the ultrastrong light-matter coupling regime,” arXiv:2012.06090 (2021).

Sun, L.

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Tan, T. R.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Tan, X.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Tavernelli, I.

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Tong, D. M.

E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
[Crossref]

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

Tong, D.-M.

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Tremblay, A.-M. S.

A. Blais and A.-M. S. Tremblay, “Effect of noise on geometric logic gates for quantum computation,” Phys. Rev. A 67, 012308 (2003).
[Crossref]

Vainsencher, A.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Vedral, V.

A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, “Geometric phase in open systems,” Phys. Rev. Lett. 90, 160402 (2003).
[Crossref]

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation with NMR,” Nature (London) 403, 869–871 (2000).
[Crossref]

Veitia, A.

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Wallraff, A.

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

Wan, Y.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Wang, F.

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

Wang, H.

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Wang, T.

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Wang, W.

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Wang, W.-B.

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

Wang, X.

Y.-H. Chen, W. Qin, R. Stassi, X. Wang, and F. Nori, “Generation of Fock-state superpositions and binomial-code holonomic gates via dressed intermediate states in the ultrastrong light-matter coupling regime,” arXiv:2012.06090 (2021).

Wang, X.-B.

X.-B. Wang and M. Keiji, “Nonadiabatic conditional geometric phase shift with NMR,” Phys. Rev. Lett. 87, 097901 (2001).
[Crossref]

Wang, Z.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Wang, Z.-D.

S.-L. Zhu and Z.-D. Wang, “Implementation of universal quantum gates based on nonadiabatic geometric phases,” Phys. Rev. Lett. 89, 097902 (2002).
[Crossref]

Wenner, J.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

White, T. C.

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

Wilczek, F.

F. Wilczek and A. Zee, “Appearance of gauge structure in simple dynamical systems,” Phys. Rev. Lett. 52, 2111–2114 (1984).
[Crossref]

Wineland, D. J.

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

Xia, Y.

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

Xiang, L.

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Xu, D.

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Xu, G.

G. Feng, G. Xu, and G. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013).
[Crossref]

Xu, J.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Xu, K.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Xu, Y.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Xue, Z.-Y.

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

Yan, T.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Yan, Z.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Yang, C.-P.

S.-B. Zheng, C.-P. Yang, and F. Nori, “Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts,” Phys. Rev. A 93, 032313 (2016).
[Crossref]

Yang, X.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Yang, Z.-B.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

Yin, Y.

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Yin, Z.-Q.

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

You, J. Q.

J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, “Holonomic surface codes for fault-tolerant quantum computation,” Phys. Rev. A 97, 022335 (2018).
[Crossref]

J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B 75, 140515 (2007).
[Crossref]

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42 (2005).
[Crossref]

Yu, D.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Yu, X.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Yu, Y.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Yung, M.-H.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Yurtalan, M. A.

M. A. Yurtalan, J. Shi, G. J. K. Flatt, and A. Lupascu, “Characterization of multi-level dynamics and decoherence in a high-anharmonicity capacitively shunted flux circuit,” arxiv:2008.00593 (2020).

Zanardi, P.

J. Pachos, P. Zanardi, and M. Rasetti, “Non-Abelian Berry connections for quantum computation,” Phys. Rev. A 61, 010305 (2000).
[Crossref]

P. Zanardi and M. Rasetti, “Holonomic quantum computation,” Phys. Lett. A 264, 94–99 (1999).
[Crossref]

Zee, A.

F. Wilczek and A. Zee, “Appearance of gauge structure in simple dynamical systems,” Phys. Rev. Lett. 52, 2111–2114 (1984).
[Crossref]

Zhang, J.

J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, “Holonomic surface codes for fault-tolerant quantum computation,” Phys. Rev. A 97, 022335 (2018).
[Crossref]

Zhang, L.

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Zhang, P.

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Zhang, W.-G.

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

Zhang, Z.

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Zhao, P.-Z.

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

Zheng, D.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Zheng, S.-B.

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

S.-B. Zheng, C.-P. Yang, and F. Nori, “Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts,” Phys. Rev. A 93, 032313 (2016).
[Crossref]

S.-B. Zheng, “Geometric phase for a driven quantum field subject to decoherence,” Phys. Rev. A 91, 052117 (2015).
[Crossref]

Zheng, W.

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Zhong, Z.-R.

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

Zhu, S.-L.

S.-L. Zhu and Z.-D. Wang, “Implementation of universal quantum gates based on nonadiabatic geometric phases,” Phys. Rev. Lett. 89, 097902 (2002).
[Crossref]

Zhu, X.

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

Zu, C.

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

Nat. Commun. (2)

C. Song, S.-B. Zheng, P. Zhang, K. Xu, L. Zhang, Q. Guo, W. Liu, D. Xu, H. Deng, K. Huang, D. Zheng, X. Zhu, and H. Wang, “Continuous-variable geometric phase and its manipulation for quantum computation in a superconducting circuit,” Nat. Commun. 8, 1061 (2017).
[Crossref]

S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, “Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin,” Nat. Commun. 5, 4870 (2014).
[Crossref]

Nat. Phys. (1)

J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, “Towards fault-tolerant quantum computing with trapped ions,” Nat. Phys. 4, 463–466 (2008).
[Crossref]

Nature (London) (6)

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation with NMR,” Nature (London) 403, 869–871 (2000).
[Crossref]

A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, “Experimental realization of non-Abelian nonadiabatic geometric gates,” Nature (London) 496, 482–485 (2013).
[Crossref]

D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenković, C. Langer, T. Rosenband, and D. J. Wineland, “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate,” Nature (London) 422, 412–415 (2003).
[Crossref]

C. Zu, W.-B. Wang, L. He, W.-G. Zhang, C.-Y. Dai, F. Wang, and L.-M. Duan, “Experimental realization of universal geometric quantum gates with solid-state spins,” Nature (London) 514, 72–75 (2014).
[Crossref]

R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, C. Neill, P. O’Malley, P. Roushan, A. Vainsencher, J. Wenner, A. N. Korotkov, A. N. Cleland, and J. M. Martinis, “Superconducting quantum circuits at the surface code threshold for fault tolerance,” Nature (London) 508, 500–503 (2014).
[Crossref]

R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, I.-C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A. Vainsencher, J. Wenner, A. N. Cleland, and J. M. Martinis, “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature (London) 519, 66–69 (2015).
[Crossref]

New J. Phys. (2)

E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, “Nonadiabatic holonomic quantum computation,” New J. Phys. 14, 103035 (2012).
[Crossref]

Z. Zhang, P.-Z. Zhao, T. Wang, L. Xiang, Z. Jia, P. Duan, D.-M. Tong, Y. Yin, and G. Guo, “Single-shot realization of nonadiabatic holonomic gates with a superconducting Xmon qutrit,” New J. Phys. 21, 073024 (2019).
[Crossref]

npj Quantum Inf. (1)

Z.-B. Yang, P.-R. Han, X.-J. Huang, W. Ning, H. Li, K. Xu, D. Zheng, H. Fan, and S.-B. Zheng, “Experimental demonstration of entanglement-enabled universal quantum cloning in a circuit,” npj Quantum Inf. 7, 44 (2021).
[Crossref]

Phys. Lett. A (1)

P. Zanardi and M. Rasetti, “Holonomic quantum computation,” Phys. Lett. A 264, 94–99 (1999).
[Crossref]

Phys. Rev. A (8)

J. Pachos, P. Zanardi, and M. Rasetti, “Non-Abelian Berry connections for quantum computation,” Phys. Rev. A 61, 010305 (2000).
[Crossref]

S.-B. Zheng, “Geometric phase for a driven quantum field subject to decoherence,” Phys. Rev. A 91, 052117 (2015).
[Crossref]

A. Nazir, T. Spiller, and W. J. Munro, “Decoherence of geometric phase gates,” Phys. Rev. A 65, 042303 (2002).
[Crossref]

A. Blais and A.-M. S. Tremblay, “Effect of noise on geometric logic gates for quantum computation,” Phys. Rev. A 67, 012308 (2003).
[Crossref]

J. Zhang, S. J. Devitt, J. Q. You, and F. Nori, “Holonomic surface codes for fault-tolerant quantum computation,” Phys. Rev. A 97, 022335 (2018).
[Crossref]

M. Johansson, E. Sjöqvist, L. M. Andersson, M. Ericsson, B. Hessmo, K. Singh, and D. M. Tong, “Robustness of non-adiabatic holonomic gates,” Phys. Rev. A 86, 062322 (2012).
[Crossref]

S.-B. Zheng, C.-P. Yang, and F. Nori, “Comparison of the sensitivity to systematic errors between nonadiabatic non-Abelian geometric gates and their dynamical counterparts,” Phys. Rev. A 93, 032313 (2016).
[Crossref]

A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, “Surface codes: towards practical large-scale quantum computation,” Phys. Rev. A 86, 032324 (2012).
[Crossref]

Phys. Rev. Appl. (1)

D. J. Egger, M. Ganzhorn, G. Salis, A. Fuhrer, P. Mueller, P. K. Barkoutsos, N. Moll, I. Tavernelli, and S. Filipp, “Entanglement generation in superconducting qubits using holonomic operations,” Phys. Rev. Appl. 11, 014017 (2019).
[Crossref]

Phys. Rev. B (1)

J. Q. You, X. Hu, S. Ashhab, and F. Nori, “Low-decoherence flux qubit,” Phys. Rev. B 75, 140515 (2007).
[Crossref]

Phys. Rev. Lett. (15)

W. Ning, X.-J. Huang, P.-R. Han, H. Li, H. Deng, Z.-B. Yang, Z.-R. Zhong, Y. Xia, K. Xu, D. Zheng, and S.-B. Zheng, “Deterministic entanglement swapping in a superconducting circuit,” Phys. Rev. Lett. 123, 060502 (2019).
[Crossref]

Y. Xu, W. Cai, Y. Ma, X. Mu, L. Hu, T. Chen, H. Wang, Y.-P. Song, Z.-Y. Xue, Z.-Q. Yin, and L. Sun, “Single-loop realization of arbitrary non-adiabatic holonomic single-qubit quantum gates in a superconducting circuit,” Phys. Rev. Lett. 121, 110501 (2018).
[Crossref]

T. Yan, B.-J. Liu, K. Xu, C. Song, S. Liu, Z. Zhang, H. Deng, Z. Yan, H. Rong, K. Huang, M.-H. Yung, Y. Chen, and D. Yu, “Experimental realization of nonadiabatic shortcut to non-Abelian geometric gates,” Phys. Rev. Lett. 122, 080501 (2019).
[Crossref]

G. Feng, G. Xu, and G. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013).
[Crossref]

Y. Xu, Z. Hua, T. Chen, X. Pan, X. Li, J. Han, W. Cai, Y. Ma, H. Wang, Y. P. Song, Z.-Y. Xue, and L. Sun, “Experimental implementation of universal nonadiabatic geometric quantum gates in a superconducting circuit,” Phys. Rev. Lett. 124, 230503 (2020).
[Crossref]

Y. Xu, Y. Ma, W. Cai, X. Mu, W. Dai, W. Wang, L. Hu, X. Li, J. Han, H. Wang, Y. P. Song, Z.-B. Yang, S.-B. Zheng, and L. Sun, “Demonstration of controlled-phase gates between two error-correctable photonic qubits,” Phys. Rev. Lett. 124, 120501 (2020).
[Crossref]

X.-B. Wang and M. Keiji, “Nonadiabatic conditional geometric phase shift with NMR,” Phys. Rev. Lett. 87, 097901 (2001).
[Crossref]

S.-L. Zhu and Z.-D. Wang, “Implementation of universal quantum gates based on nonadiabatic geometric phases,” Phys. Rev. Lett. 89, 097902 (2002).
[Crossref]

C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, “High-fidelity quantum logic gates using trapped-ion hyperfine qubits,” Phys. Rev. Lett. 117, 060504 (2016).
[Crossref]

J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, “High-fidelity universal gate set for 9Be+ ion qubits,” Phys. Rev. Lett. 117, 060505 (2016).
[Crossref]

Y. Aharonov and A. Anandan, “Phase change during a cyclic quantum evolution,” Phys. Rev. Lett. 58, 1593–1596 (1987).
[Crossref]

F. Wilczek and A. Zee, “Appearance of gauge structure in simple dynamical systems,” Phys. Rev. Lett. 52, 2111–2114 (1984).
[Crossref]

G. D. Chiara and G. M. Palma, “Berry phase for a spin 1/2 particle in a classical fluctuating field,” Phys. Rev. Lett. 91, 090404 (2003).
[Crossref]

S. Filipp, J. Klepp, Y. Hasegawa, C. Plonka-Spehr, U. Schmidt, P. Geltenbort, and H. Rauch, “Experimental demonstration of the stability of Berry’s phase for a spin-1/2 particle,” Phys. Rev. Lett. 102, 030404 (2009).
[Crossref]

A. Carollo, I. Fuentes-Guridi, M. F. Santos, and V. Vedral, “Geometric phase in open systems,” Phys. Rev. Lett. 90, 160402 (2003).
[Crossref]

Phys. Today (1)

J. Q. You and F. Nori, “Superconducting circuits and quantum information,” Phys. Today 58(11), 42 (2005).
[Crossref]

Proc. R. Soc. London A (1)

M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. London A 392, 45–57 (1984).
[Crossref]

Rep. Prog. Phys. (1)

I. Buluta, S. Ashhab, and F. Nori, “Natural and artificial atoms for quantum computation,” Rep. Prog. Phys. 74, 104401 (2011).
[Crossref]

Other (3)

M. A. Yurtalan, J. Shi, G. J. K. Flatt, and A. Lupascu, “Characterization of multi-level dynamics and decoherence in a high-anharmonicity capacitively shunted flux circuit,” arxiv:2008.00593 (2020).

Y.-H. Chen, W. Qin, R. Stassi, X. Wang, and F. Nori, “Generation of Fock-state superpositions and binomial-code holonomic gates via dressed intermediate states in the ultrastrong light-matter coupling regime,” arXiv:2012.06090 (2021).

Z. Han, Y. Dong, B. Liu, X. Yang, S. Song, L. Qiu, D. Li, J. Chu, W. Zheng, J. Xu, T. Huang, Z. Wang, X. Yu, X. Tan, D. Lan, M.-H. Yung, and Y. Yu, “Experimental realization of universal time-optimal non-Abelian geometric gates,” arXiv:2004.10364 (2020).

Supplementary Material (1)

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)

Fig. 1.
Fig. 1. Energy level configuration and excitation scheme for the two-qubit CNOT gate. The control and target qubits are denoted as ${Q}_1$ and ${Q}_2$, respectively. (a) Bare energy levels of ${Q}_2$ and frequencies of the drives. The quantum information of each qubit is encoded in states $| g \rangle$ and $| f \rangle$, with the auxiliary state $| e \rangle$ used for realizing the controlled-NOT gate. The transitions $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ and $| {{e_2}} \rangle \leftrightarrow | {{f_2}} \rangle$ of ${Q}_2$ are driven by classical fields of angular frequencies $({{\omega _r} - {\lambda _2}})$ and $({{\omega _{f,2}} - {\omega _r} + {\lambda _2}})$, respectively. Here ${\omega _r}$ is the angular frequency of the resonator that is strongly coupled to $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ with the coupling strength ${\lambda _2}$, and $\hbar {\omega _{f,2}}$ is the energy spacing between $| {{f_2}} \rangle$ and $| {{g_2}} \rangle$. (b) Dressed states and energy levels with ${Q}_1$ initially in $| {{f_1}} \rangle$. When being initially in $| {{f_1}} \rangle$, ${Q}_1$ is effectively decoupled from the resonator due to the large detuning. The strong coupling between ${Q}_2$ and the resonator results in dressed states $| {\psi _n^ \pm} \rangle$, whose energy levels are nonlinearly dependent on the coupling strength. The two driving fields are on-resonance with the transitions $| {{g_2}0} \rangle \leftrightarrow | {\psi _1^ -} \rangle$ and $| {\psi _1^ -} \rangle \leftrightarrow | {{f_2}0} \rangle$, respectively, but highly detuned from other transitions. (c) Dressed states and energy levels with ${Q}_1$ initially in $| {{g_1}} \rangle$. If initially in $| {{g_1}} \rangle$, ${Q}_1$, together with ${Q}_2$, is strongly coupled to the resonator, resulting in three dressed states $| {\Phi _1^ \pm} \rangle$ and $| {\Phi _1^0} \rangle$ in the single-excitation subspace. The driving fields are highly detuned from transitions of $| {{g_1}{g_2}0} \rangle$ and $| {{g_1}{f_2}0} \rangle$ to these dressed states.
Fig. 2.
Fig. 2. Pulse sequence. Before the gate operation, both qubits are initialized to their ground states at the corresponding idle frequencies, where single-qubit rotations are performed to prepare them in a product state. Then a ${Z}$ pulse is applied to ${Q}_1$, tuning $| {{g_1}} \rangle \leftrightarrow | {{e_1}} \rangle$ close to the resonator’s frequency; ${Q}_2$ is subjected to a ${Z}$ pulse, which brings $| {{g_2}} \rangle \leftrightarrow | {{e_2}} \rangle$ to the resonator’s frequency, and a driving pulse involving two frequency components on-resonance with the transitions $| {{g_2}0} \rangle \leftrightarrow | {\psi _1^ -} \rangle$ and $| {\psi _1^ -} \rangle \leftrightarrow | {{f_2}0} \rangle$. After the CNOT gate, realized with these pulses, both qubits are tuned back to their idle frequencies for quantum state tomography.
Fig. 3.
Fig. 3. Measured density matrix of the output state with the input state $({| {{g_1}{g_2}} \rangle + | {{f_1}{g_2}} \rangle})/\sqrt 2$. Each matrix element is characterized by two colorbars, one for the real part and the other for the imaginary part. The black wire frames denote the matrix elements of the ideal output states.
Fig. 4.
Fig. 4. Measured process matrix for the realized CNOT gate. The process matrix is measured by preparing a set of 36 distinct input product states in the computational basis $\{{| {{g_1}{g_2}} \rangle ,| {{g_1}{f_2}} \rangle ,| {{f_1}{g_2}} \rangle ,| {{f_1}{f_2}} \rangle} \}$ and reconstructing the density matrices for these states and for the output states produced by the CNOT gate. The $| e \rangle$-state populations of ${Q}_1$ and ${Q}_2$, averaged over the 36 output states, are 2.2% and 2.8%, respectively.

Equations (11)

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H i n t = j = 1 2 λ j ( a | e j g j | + a | g j e j | ) ,
H d r = [ Ω ge e i λ 2 t | e 2 g 2 | Ω e f e i λ 2 t | f 2 e 2 | ] + h . c . ,
| ψ 0 = | g 2 0 ,
| ψ n ± = 1 2 ( | e 2 ( n 1 ) ± | g 2 n ) , n 1.
H e f f = Ω [ cos ϕ 2 | g 2 0 ψ 1 | + sin ϕ 2 | f 2 0 ψ 1 | ] | f 1 f 1 | + h . c . ,
Ω = Ω ge 2 + Ω ef 2 / 2 ,
tan ϕ 2 = Ω ef / Ω ge .
| Φ 1 0 = ( sin θ | e 1 g 2 0 + cos θ | g 1 e 2 0 ) ,
| Φ 1 ± = 1 2 [ ( cos θ | e 1 g 2 0 + sin θ | g 1 e 2 0 ) ± | g 1 g 2 1 ] ,
| ψ cd ( t ) = exp ( i 0 t H e f f d t / ) | c 1 d 2 0 .
U = ( 1 0 0 0 0 1 0 0 0 0 cos ϕ sin ϕ 0 0 sin ϕ cos ϕ ) ,

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