## Abstract

We propose a protocol to protect the quantum states and entanglements from finite-temperature thermal noise via quantum gates. Compared to the common protocols protecting the quantum states and entanglements by using weak measurements and their reversals, no time-consuming weak measurements are needed in the present protocol and consequently, it is much faster. We also discuss the possible implementation of the protocol in cavity QED system.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Due to the inevitable interaction with the noisy environment, quantum state decoheres and quantum entanglement vanishes, which have been the main barriers that hinder the wide usage of quantum technology. As a consequence, the protection of the quantum states and quantum entanglements for a long time is important. Consequently several schemes, including those based on dynamic decoupling and quantum Zeno effect, have been proposed to overcome the effect of environment on quantum systems [1–5].

Another interesting method to protect the quantum states or entanglements is by using weak measurements and their reversals [6–19]. In these schemes, the quantum state is firstly transferred to more robust states by a weak measurement to resist the decoherence. After the decoherence process, the state is reversed back to the original state by another weak measurement. The behind physics is that the initial state is firstly projected to a more robust state. For example for an initial state $\alpha _1|0\rangle +\alpha _2|1\rangle$, we can project the state to $(\alpha _1|0\rangle +\alpha _2x|1\rangle )/\sqrt {|\alpha _1|^2+|\alpha _2|^2x^2}$ with $x<1$. This state has a larger probability to stay and keep on state $|0\rangle$, which decreases the influence of the decay. After the interaction with the environment, another weak measurement with $x>1$ is applied to reverse the quantum state. Due to the failure rates of the weak measurements, these schemes have limited success probabilities. In our previous works, we have shown that quantum gates can replace weak measurements in reversing the quantum states and protecting the entanglements [20–22]. In those works, ancilla qubits prepared in certain states are used to interact with the target qubits to construct the quantum gates. After the quantum gates, the ancilla qubits are detected. If the ancilla qubits are detected in certain state, such as a ground state, the reversal processes succeed and the states of the target qubits are reversed completely or approximately.

In all the above schemes, the quantum states decohere in a zero temperature environment. In this situation, only the quantum fluctuation is considered and the qubit decays from the excited state exponentially. However, in many quantum systems, such as superconducting circuits and Rydberg atom systems, this approximation is usually not valid and thermal environment has to be taken into account. The qubit not only decays to the ground state but also gets excited to the excited state incoherently by thermal photons. Protecting the quantum states and entanglements in such situation by using weak measurements and their reversals has been investigated [23,24]. In these procedures, the time-consuming weak measurements are required, which makes the whole process slow.

In this article, we present a scheme using quantum gates to protect the quantum states and entanglements of qubits under finite-temperature thermal noise (FTTN), which has in principle almost the same success probability and fidelity as that of weak measurement. We also present a physical implementation in a cavity QED system, where the quantum states that we consider are stored in superconducting cavities. They can be encoded to flying atoms adiabatically and then undergo spontaneous decay and thermal photon excitation. Finally, the atomic states are encoded back to other identical cavities. Ancilla qubits are used to interact with the cavities to construct quantum phase gates and subsequently control not (CNOT) gates. The CNOT gates are applied to the system before and after the cavity states undergo the thermal noise. The quantum state reversal is achieved with increased fidelity by projecting the ancilla qubits to certain states. This proposal does’t require time-consuming weak measurement (such as monitoring the cavities by photon counting devices) and consequently save the reversal time. Additionally, compared to the implementation of previous simulation schemes based on linear optics [11,24], the present implementation allows the qubits to undergo real decay and excitation processes, which can be useful in quantum state protection and quantum information.

The article is organized as follows: In Sec. 2, we present the general theory to protect the quantum states and entanglement via quantum gates. In Sec. 3, we give the physical implementation in a cavity QED system. Finally, in Sec. 4, we present the concluding remarks.

## 2. Protection theory and mathematical simulations

#### 2.1. Quantum state under finite-temperature and its reversal

In a thermal bath of finite-temperature $T$, a quantum state $\rho _s^k$ of a qubit evolves to [25]

whereThe scheme for the quantum state reversal is shown in Fig. 1(a). In order to reverse the state, we need an ancilla qubit prepared in the state [20]

where to interact with the system. Here $|0\rangle$ and $|1\rangle$ are the two eigen states of the ancilla qubit.After a CNOT gate

If the ancilla qubit is detected in the state $|1\rangle$, the target qubit is projected to

In order to improve the fidelity further, we modify our protocol as shown in Fig. 1(b). Here we use another ancilla qubit to interact with the target system and apply a CNOT gate on them before $\rho _s^k$ undergoes the thermal noise. This operator maps $\rho _s^k$ to be more robust against the thermal noise, which lowers the decoherence of the qubit. The ancilla qubit has a similar initial state as shown in Eq. (3) but different $\theta _l$ with $\tan \theta _l=x_l$. In Fig. 3, we set $x_l=0.65, x_r=2$ and plot the fidelity varying with $|\alpha _1|$. The result shows that the fidelity increases when two CNOT gates are applied. These results demonstrate that, compared to using weak measurements and their reversals used in the previous schemes [23,24], the quantum gates can also reverse the quantum states efficiently.

In FIg. 4, we plot the average fidelity increase as a function of $x_1$ and $x_2$ for $1000$ random generated initial mixed state. The figure shows that by choosing $x_1$ and $x_2$ appropriately, the present protocol can protect the mixed states efficiently with a maximum average fidelity increase about $0.06$.

#### 2.2. Protecting the entanglement from finite-temperature thermal noise

In this section, we show that the quantum gates can also protect the entanglement from thermal noise. We assume that the initial state of the quantum system to be a two-qubit entangled state. As described by Eq. (1), after time $t$, the state evolves to $\rho _s^n$. The corresponding elements $\rho _{ij}$ can be expressed as

As an example, we assume $\rho _s^k$ to be $\alpha _1|gg\rangle +\alpha _2|ee\rangle$. After the first pair of CNOT gates, the state is projected to

## 3. Protecting the quantum states and entanglements of cavity states

In this section we present a physical implementation of the above proposal in the cavity QED system. Our setup is schematically shown in Fig. 8(a). Cavities 1 and 2 are two identical superconducting cavities. we consider an initial state of the cavities, which have 0 or 1 photon in them and can be considered as qubits. As already discussed, in order to reverse the quantum states and entanglements between the two cavity fields, we need to construct CNOT gates between the two cavities and two ancilla qubits separately.

#### 3.1. CNOT gates

First, we discuss how to realize the CNOT gates used in the reversal processes. Here we utilize ancilla qubits to interact with the cavities. The ancilla qubit, as a Rydberg atom shown in Fig. 8(c), has a down state $|D\rangle$, an upper state $|U\rangle$, and an additional state $|A\rangle$. The transition dipole from $|U\rangle$ to $|D\rangle$ can interact with the Ramsey fields $R_{1,2,3}$ resonantly. By manipulating the field intensity of $R_1$, the ancilla qubit initially in state $\rho _{a}=|D\rangle \langle D|$ evolves to [20,22,30]

where $H_{\theta }$ is the Hadamard gate having the form#### 3.2. Detailed processes to protect the quantum states and entanglements

In the following steps, we give the detailed processes to protect the quantum entanglement of the two qubits from FTTN.

** Step** 1: As shown in Fig. 8(a), before the system undergoes the finite temperature thermal noise, we use two ancilla qubits to interact with the two cavities separately to realize the CNOT gates. After the two CNOT gates, we detect the ancilla qubits. If both qubits are in the ground states $|D\rangle$, we move on to the next step. Otherwise we modify the intensity of laser $R_1$ and do the above process again. The cavity state can be expressed as $\rho _s^{C_l}$ in Eq. (14).

** Step** 2: We pass two identical atoms shown in Fig. 8(b) through cavities 1 and 2 separately. The atoms are both initially prepared in ground state $|g\rangle$. Before the atom enters the cavity, it interacts with a Ramsey field resonantly with $|a\rangle$ to $|c\rangle$ transition and Rabi strength $\Omega _1$. When the atom enters the cavity, the transition dipole from $|a\rangle$ to $|g\rangle$ interacts with the cavity field resonantly with a coupling strength $g_c$. The dark state of the atom interacting with the two kinds of fields has the following form [12,31–35]:

** Step** 3: After the atom flies out of the cavity, it meets a $\pi$ beam. The beam interacts with transition dipole from $|c\rangle$ to $|e\rangle$ with Rabi frequency $\Omega _2$, such as by Raman process. The atom evolves from

** Step** 4: The atom passes through the area with finite-temperature, which can be realized by illuminating this area by thermal light [36,37]. The atom on the superposition state of $|e\rangle$ and $|g\rangle$ interacts with the vacuum modes and the thermal photons incoherently. It can both lose and gain excitations; the dynamic evolution function can be expressed as [38]

** Step** 5: The atom passes through another $\pi$ beam and cavity. As the inverse processes of $steps$ 3 and 2, the atom state transfers from $|e\rangle$ to $|c\rangle$ firstly by interacting with the $\pi$ beam. A photon is emitted back to cavity 2 and the atom goes back to level $|g\rangle$ from $|c\rangle$ adiabatically. The state of the cavity 1, influenced by the FTTN, is transferred to cavity 2. Meanwhile, the state of cavity 2, influenced by the FTTN, is transferred to cavity 1.

** Step** 6: As the same as $step$ 1, we perform two other CNOT gates on the cavities 1 and 2 separately. By detecting the states of the ancilla qubits to the ground states, we can project the cavity state to the final state with increasing fidelity and entanglement compared to that without the protection processes.

By the above processes, we can protect the quantum entanglement of two cavities. Note that the CNOT gates on the two cavity states are separated. They can be different, which is similar to the different weak measurement parameters in [23,24].

To reverse the quantum state, only one atom shown in Fig. 8(b) is required to pass the two cavities. The CNOT gates can be constructed as the same method discussed in the above subsection. After the same steps as the case to protect the entanglement, the cavity state can be reversed and stored in another cavity.

In a real experiment, the time for an atom passing through the cavity to construct a quantum phase gate or absorb and emit a photon adiabatically is about $20\mu s$ [30]. The total required time for reversal process is about hundreds of microseconds. The lifetime of the atom can be millisecond. Before and after the atoms fly in the thermal noise region, they stay in state $|g\rangle$ or $|c\rangle$, which always have long life times and can restrain the influence of the spontaneous decay on the system. The main experimental requirement is the slow decay rate of the cavities, which can hold the cavity states when the CNOT gates are applied. Since the technology of micromachining has improved, the damping time of the superconducting cavity is about 130ms [39,40]. As a consequence, we believe our implementation can be realized experimentally. Meanwhile, the interaction times between the cavities and the atoms are much shorter than the lifetime of the Rydberg atoms as well as the flying time of the atom through the FTTN area. If the entire system is placed in an environment with finite temperature, the effect of the thermal photons on the cavity state can still be expressed as Eq. (1). This means that our protocol is still valid even the entire system is placed in FTTN, which makes the implementation be easily to be realized. These effects can be approximately involved to the $p$ and $r$ in Eq. (1). However, similar to weak measurement requiring to swap the quantum state, such as swapping the cavity state from $\alpha _1|0\rangle +\alpha _2|1\rangle$ to $\alpha _2|0\rangle +\alpha _1|1\rangle$ by interacting with atoms, and having limited succeed probability [19], the quantum gate also has limited fidelity, which will decrease the fidelity of the final state. As the technology advaces, the fidelity of quantum gate, such as CNOT gate, based on Rydberg states are very high, even more than 0.9999 in [41]. If we can realize a CNOT gate with fidelity 0.99, the state after the CNOT gate can be expressed $0.99\rho _s^C+0.01\rho _s^F$ with $\rho _s^C$ being the state we want and $\rho _s^F$ being the state we donot want. Even $\rho _s^F$ is orthogonal to the initial state, it just has about 0.02 fidelity decrease, which is much smaller than the fidelity increase due to the protocol, i.e., on average 0.06 in Fig. 4 for mixed state. As a consequence, the physical implementation is still valid.

## 4. Conclusions

In this article, we presented a protocol to protect the quantum states and entanglements from FTTN via quantum gates. The fidelity and entanglement can be improved efficiently with a large probability. In the real physical implementation, by using the adiabatic passage of multi-level Rydberg atoms passing through the superconducting cavities, the quantum states stored in the cavities can be transferred to the atoms and subsequently undergo the FTTN effectively. Meanwhile, it is convenient to realize a quantum phase gate and subsequently a CNOT gate by passing an ancilla qubit through the cavity. By detecting the ancilla qubits, we can reverse the quantum states and entanglements with certain success probability. Compared to the protocols based on weak measurements and their reversals, the present protocol can be much faster. Additionally, compared to the previous simulation schemes based on linear optics, the qubits in the implementation of our proposal based on cavity QED system undergoes the real decay and thermal photon excitation processes as described in Eq. (1) and the physical setup is relatively easy to be realized with current experimental technologies. Our investigation can be useful in quantum state protection and quantum information.

## Funding

National Natural Science Foundation of China (11804219); King Abdulaziz City for Science and Technology.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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