Abstract
Qubit operation belonging to unitary transformation is the fundamental operation to realize quantum computing and information processing. Here, we show that the complex and flexible light-matter interaction between dielectric metasurfaces and incident light can be used to perform arbitrary U(2) operations. By incorporating both coherent spatial-mode operation together with two polarizations on a single metasurface, we further extend the discussion to single-photon two-qubit U(4) operations. We believe the efficient usage of metasurfaces as a potential compact platform can simplify optical qubit operation from bulky systems into conceptually subwavelength elements.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum computing and information processing have drawn a lot of interests in the past decade due to the potential speedup over their classical counterparts. Mathematically, an overall quantum operation can be regarded as a series of unitary transformations on the input quantum qubits in constructing a quantum network. There exist many physical systems to realize quantum computing, such as ion traps, Josephson junctions, and nitrogen-vacancy centers, etc. [1]. Among these physical systems, linear optics scheme is appealing because the quantum information carrier is the photon, which is potentially free from decoherence [2,3]. When applying quantum computing on the input photon, the fundamental qubit is usually prepared by single photon in two orthogonal modes or in two polarization channels. To generate the desired evolutions in quantum information processing, each corresponding qubit operation is implemented by some simple optical elements or their combinations, such as beam splitters, phase shifters, and wave plates [4,5]. One-qubit operation belongs to the class of U(2) transformations which have been theoretically discussed and experimentally realized by a combination of these elements [2–6] However, the physical implementation using traditional linear optical elements seem to be bulky and difficult to be integrated to miniaturize the physical system, a potential simplification of the current optics implementation is highly desired.
On the other hand, metasurfaces, a single or multiple layers of metamaterial structures, allow a flat and compact implementation for miniaturizing different optical elements in the classical optical regime [7,8]. Based on the rich degrees of freedom in fabricating any tailor-made and resonating metamaterial structures, they have been applied to different scenarios requiring complicated degrees of freedom, including holograms [9,10], optical flat lens [11,12], Stokes polarimeters [13–15], and analog computation [16–18]. Specifically, metamaterials have been used to perform information or image processing. By pixelating metamaterial into a discrete set of structures, these “digital metamaterials” can be further used to perform different mathematical operations, such as Fourier transform and differentiation [15–22]. Extending to the quantum optical regime, metasurfaces can be useful to substitute conventional linear optical elements in either deterministic schemes [23,24] or probabilistic schemes [25] with post-selection for quantum information processing. The complexity embedded with metasurfaces may allow future integration of quantum optical elements, again based on tailor-made resonances and the manipulation of constitutive parameters. Furthermore, it has been suggested in both theory and experiments that metasurfaces are able to achieve fundamental quantum entanglement and quantum interference [26–33]. It is therefore plausible to ask how metasurfaces can be generally used to perform quantum operations, allowing quantum optical schemes to be simplified. To answer this question, it is unavoidable to ask how metasurfaces can be designed to perform a tailor-made unitary operation, which is the common language in a quantum operation.
In this paper, we propose a metasurface platform capable of realizing arbitrary U(2) operations, which may make the general qubit operation migrate from bulky optical systems into conceptually subwavelength elements. Designated unitary transformations can be obtained by varying the geometric parameters of a single metasurface replacing the combination of a beam splitter and a phase shifter [4]. By considering two orthogonal polarizations impinging on a metasurface, we also explore single-photon two-qubit U(4) operations.
2. Theoretical and numerical demonstration
Any U(2) transformations can be performed optically by using one beam splitter and two phase shifters with 2 ports, as shown in Fig. 1(a). In general, for a U(N) transformation, one can decompose into a network of ${{N({N - 1} )} / 2}$ sets of U(2) elements with N input and output ports [4]. To simplify this bulky optical implementation, we consider a metasurface with oblique incidence. It can be described by a scattering matrix ${\textbf S}$ relating two input to two output beams/ports, with the definitions of the ports and polarizations shown in Fig. 1(a). To start with, we demonstrate the designing principle in U(2) for simplicity. In order to suppress polarization conversion, the metasurface has a (mirror) ${{\textbf M}_y}$-symmetry and we consider only the vertical polarization (E-field along $y$) at the moment. The logical states $|0 \rangle$ and $|1 \rangle$ correspond to the two ports, i.e. a dual-rail qubit. An unitary operation on such a qubit corresponds to an unitary matrix ${\textbf S}$, i.e. ${\textbf S}{{\textbf S}^\dagger } = {\textbf I}$, which can be fulfilled by a dielectric metasurface with time-reversal symmetry and negligible loss. Two-beam incidence on metasurfaces has already been used for coherent control of absorption and polarization at normal incidence [34,35]. Here, an oblique incidence on both sides is necessary to have different transmission/reflection phases for the two beams in constructing an arbitrary qubit unitary operation. In general for passive optical multiport, the scattering matrix relates the input and output mode operators in the same way to relate the input and output mode functions [36], as a $2 \times 2$ unitary matrix written as
As a starting point, we first construct ${\textbf S} \propto {{\boldsymbol {\sigma} }_z}$ as a comparatively simple example. Oblique incidence is set on both sides at 45 degree with vertical polarization and the unit cell (periodic in $x$ with infinite length in $y$) is displayed in the inset of Fig. 1(b) (Si parallelogram, with details in caption). The design breaks both ${{\textbf M}_x}$ and ${{\textbf M}_z}$ symmetries, allowing ${t_{11}} \ne {t_{22}}$($\varDelta {\varphi _t} \ne 0$) but retains inversion symmetry ${\textbf P}$ to have ${r_{21}} = {r_{12}}$ ($\varDelta {\varphi _r} = 0$). The unitary scattering matrix is then obtained by full-wave simulation using COMSOL Multiphysics. As shown in the Fig. 1(b) and (c), a resonance is used to achieve zero reflection amplitudes ($|r |= 0$) while the transmission amplitudes for the two input ports stay at one with $\pi $-phase difference ($\varDelta {\varphi _t} ={-} \pi$), at the operational wavelength 678 nm in our current case. To further understand the resonance, a coupled mode theory (CMT) [37,38] is used to capture its essential feature:
As the global phase $\varphi$ carries no physical importance, the general SU(2) scattering matrix is written as
where ${{\boldsymbol {\sigma} }_x}$, ${{\boldsymbol {\sigma} }_y}$, ${{\boldsymbol {\sigma} }_z}$ are the Pauli matrices and ${\textbf u} = ({{u_1},{u_2},{u_3}} )$ is a coordinate in the phase space, which lies either on or within the unit sphere, while ${u_0} = \sqrt {1 - {{|{\textbf u} |}^2}}$. Our previous example ${\textbf S} \propto {{\boldsymbol {\sigma} }_z}$ corresponds to ${\textbf u} = ({0,0,1} )$, i.e. the north pole of the unit sphere. By varying the geometric parameters of the structure continuously, one obtains a continuous trajectory in the phase space. Here, we keep the periodicity, thickness and permittivity constant, the other geometrical parameters are optimized variables. In principle, any desired trajectory could be obtained by morphing the designed structure with optimization. As an example, we morph the parallelogram in the previous example. The symmetry ensures $\varDelta {\varphi _r} = 0$ corresponding to trajectories on the ${u_1} - {u_3}$ plane, i.e. requiring two geometric parameters, chosen as $\alpha$ (intersection angle) and a (width) to obtain a desired trajectory. Figure 2(a) and (c) plot the resultant $|r |$ and $\varDelta {\varphi _t}$ as color maps against these two parameters (inset show definitions). Now, we walk along the $|r |= 0$ contour (dashed line in Fig. 2(a)) from $\alpha = {90^ \circ }$ to ${45^ \circ }$. As a result, $\varDelta {\varphi _t}$ decreases from $0$ to $- \pi$, with details shown in Fig. 2(b). It corresponds to a trajectory in the SU(2) phase space from the origin (${\textbf I}$ operation) to the north pole (${{\boldsymbol {\sigma} }_z}$ operation), shown as blue dots in Fig. 3(a), and is given byWe note that a ${{\textbf M}_z}$ operation on the structure flips the sign of $\varDelta {\varphi _t}$, reversing the phase space trajectory from the origin to the south-pole. Similarly, for the contour $\varDelta {\varphi _t} ={-} \pi$ (dashed line in Fig. 2(c)), the resultant $|r |$ varied from $1$ to $0$ when $\alpha$ varies from ${90^ \circ }$ to ${45^ \circ }$. The corresponding phase space trajectory (red dots in Fig. 3(a)) is on a unit circle on the ${u_1} - {u_3}$ plane from ${{\boldsymbol {\sigma} }_x}$ to ${{\boldsymbol {\sigma} }_z}$. This family of operations is given by
We have demonstrated the principle to obtain a desired unitary transformation by morphing the geometric configuration of our structure. Up to now, we have investigated the ${u_1}$ and ${u_3}$ components in the phase space. In fact, it is also possible to add a ${u_2}$ component (need $\varDelta {\varphi _r} \ne 0$) by further breaking ${{\textbf M}_z}$ and ${\textbf P}$ symmetry. As an example, we consider a family of “bulge-pit” structures shown in the inset in Fig. 3(b) and located a point corresponds to the ${{\boldsymbol {\sigma} }_y}$ operation with geometric parameters found in the caption in Fig. 3. The proposed structure is compatible with the previous parallelogram/rectangle design and can be continuously morphed among each other. The phase space trajectory from ${\textbf I}$ to ${{\boldsymbol {\sigma} }_y}$ along the ${u_2}$ axis is generated by preserving ${{\textbf M}_x}$ symmetry to keep $\varDelta {\varphi _t} = 0$(${u_3} = 0$) and allow $\varDelta {\varphi _r} \ne 0$(${u_2} \ne 0$). Here, we select two geometric parameters $({b,a} )$ to construct this unitary transformation, while the width of the bulge/pit, w, is constant. Along the dashed line in Fig. 3(b)/(c), we see that $\varDelta {\varphi _r} = \pi$ and $|r |$ varied from $0$ to $1$, as our desired family of operations
Again, a ${{\textbf M}_z}$ operation on the structure allows us to walk from $({0,0,0} )$ to $({0, - 1,0} )$ along the ${u_2}$ axis.Next, we turn to the discussion of U(4). Figure 4(a) shows the conventional optical implementation by cascading into six U(2) elements, using a network of beam splitters and phase shifters with 4 ports [4]. The working principle of the discussed metasurface could be extended to control 2 or more qubits based on the additional degree of freedom of photon. By adding the polarization degree of freedom, the scattering matrix extends to the class of U(4) transformation. The photonic state is now written as
As usual ${\textbf S}$ is defined as reference to the middle-plane ($z = 0$) of the metasurface and any global phase factor can be ignored. In the single-photon language, we can replace the mode function ${\textbf E}$ with the mode (annihilation) operator $\hat{{\textbf a}}$ in Eq. (11) [34]. Here, using the polarization qubit as control and path qubit as target, we can simulate a family of single-photon two-qubit controlled-U-like operations. Figure 4(a) shows our design as a square array of silicon rods with on-plane rotational angle $\beta$. In this setting there is no polarization conversion due to ${{\textbf M}_y}$ symmetry and the scattering matrix is in a simple block form
Further by considering the out-of-plane rotation ($\alpha \ne 0$) of the dielectric rods (see Fig. 5(a)), vertical and horizontal polarization channels are coupled and an U(4) operation with non-zero off-diagonal block in Eq. (12) can be generalized. We write the SU(4) scattering matrix as
Although these single-photon two-qubit operations are not scalable for universal quantum computation, they can be applied for single-photon few-qubit quantum information processing [43]. The above metasurface can be potentially useful to simplify or replace some of the U(4) operations within a specific quantum gate originally constructed from conventional optical elements. One example is a non-linear sign gate, using only linear optical elements with ancilla photons and post-selection, which can be applied in the KLM quantum computation protocol [44]. To generalize to larger number of qubit operation, the usual decomposition into multiport network of U(2) elements [4] is not favorable due to the propagation loss originated from the long optical depth (see Fig. 4(a)). Improved multiport network [45] has been studied for this purpose to reduce the optical depth. However, if factorization scheme is available, decomposition into multiport network of our U(4) metasurface elements can help significantly reduce optical depth and propagation loss. Finally, by introducing more than one incident photon to the metasurface, two- or multi-photon interference effect can be studied and provide potential application, such as quantum state preparation.
3. Discussion and conclusion
Benefit from the metasurfaces with structure design freedom, we theoretically construct a suite of element unitary transformations, which composes the general U(2) transformation on a photonic qubit. By adding the polarization degree of freedom, U(4) transformation can also be implemented, a CNOT-like gate, a family of controlled-U-like and other U(4) operations are numerically demonstrated as examples of deterministic single-photon two-qubit operations. These theoretical verifications clearly show that the metasurface approach can be potential candidate substituting the conventional linear optical elements to generate designer unitary transformations in the future few-qubit quantum information processing protocols with more integrated feasibility.
Funding
Research Grants Council, University Grants Committee of Hong Kong (16303019, 16303417, 16304520); National Natural Science Foundation of China (11974259).
Disclosures
The authors declare no conflicts of interest.
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