Cluster States Generation with a Quantum Metasurface

TL;DR

This paper addresses scalable generation of photonic cluster states for one-way quantum computation and quantum communication. It proposes the use of Quantum Metasurfaces formed by sub-wavelength atomic arrays to achieve quantum-controlled reflectivity via EIT and Rydberg blockade, enabling high-fidelity gates between an ancilla and photons. Two protocols are developed: a 2D cluster-state generation sequence given by $|\psi_{2D}\rangle=(\prod_{i=1}^{N} \mathrm{CNOT}_{a,i+N} \mathrm{CZ}_{a,i} \mathrm{H}_{a})|g\rangle_{a}\bigotimes_{i}|0\rangle_i$ and a tree-cluster construction that employs CZ, CNOT, Hadamard, and an E gate to transfer the ancilla state to photons, with the E gate implemented via a CNOT followed by a $\pi$-pulse and decay to $|g\rangle_a$. The authors demonstrate high-fidelity gates ($>0.9$) in a cavity-free, free-space setting and analyze fidelity under realistic disorder and finite Rydberg blockade, providing quantitative estimates and stability assessments. The work offers a scalable, low-loss route to large photonic cluster states suitable for quantum computation and secure quantum communication, leveraging spatial parallelism on the quantum metasurface for throughput gains.

Abstract

We investigate the implementation of photonic cluster state generation protocols using quantum metasurfaces comprising sub-wavelength atomic arrays which enables quantum-controlled reflectivity. These cluster states are generated using fundamental quantum logic gates and enable wide-ranging applications in quantum computation and communication. In the past few years, certain protocols have been developed, but their physical realizations induces natural losses on the system mainly originated from coupling the photonic structures, setting a limit on the efficiency and maximal qubit number. In this paper, we examine a physical implementation of two specific protocols for generating distinct cluster states: a two-dimensional cluster state and a tree cluster state. Our approach leverages the unique properties of a quantum metasurface and its free space settings to implement two-qubit quantum-logic gates, namely CNOT, CZ, and E gates, with practical fidelities exceeding 0.9, and potential speed-up due to parallelism. In addition, we analyze these protocols fidelities for practical conditions of potential implementation experiments, such as thermal fluctuation of trapped atoms.

Figures (6)

• Figure 1: A Quantum Metasurface featuring an ancillary atom at its center. $a$ is the lattice constant. Upon superposition of the ancillary atom $\frac{1}{\sqrt{2}} \left(\left|g\right\rangle + \left|r\right\rangle\right)$, an incoming photon with left-handed circular polarization $\left|0\right\rangle_p$ transitions into a superposition of both reflection and transmission $\frac{1}{\sqrt{2}} \left( \left|g\right\rangle \left|0\right\rangle_p + \left|r\right\rangle \left|1\right\rangle_p \right)$, where the reflected photon was flipped to right-handed circular polarization $\left|1\right\rangle_p$, for ideal transmission and reflection coefficients.
• Figure 2: a. Scheme for generating a scalable 2D cluster state. Sequential application of CNOT, CZ, and Hadamard gates with the Quantum Metasurface (QM) as control qubit. The incident photon (red) interacts with the QM, which is initialized in a superposition state, resulting in a superposition of reflection (green) and transmission (red). After passing through a wave-plate and a polarizing beam splitter (PBS), the photon returns to the QM, which remains in a superposition state. The reflected portion of the photon is redirected by the PBS (orange) due to its orthogonal polarization. The orange path has a different length, resulting a desired phase difference. b. Scheme of a part of a 2D cluster state of width $N$. Qubit k is an arbitrary qubit inside the grid entangled to its four nearest neighbors. c. Quantum circuit representation equivalent to the generation of four nodes out of the 2D cluster state. Each node $k$ is being connected to its $k+1$ neighbor and to its $k+N$ neighbor using CNOT and CZ gate with the ancilla.
• Figure 3: a. Diagram illustrating the Tree cluster creation protocol. Two optical axes are depicted: Optical path 1 implements a CNOT gate as part of the E gate, where the incident photon (purple) interacts with the QM, which is initialized in a superposition state, resulting in a superposition of reflection (dark red) and transmission (purple). Optical path 2 is identical to the diagram in Figure \ref{['2D system']}a, and accommodates photonic qubits 5 and 6, enabling the creation of a CZ gate for the necessary entanglement, following a similar procedure as in Figure \ref{['2D system']}a. b. Scheme of a binary tree with a height of three, which can be employed for information recovery in quantum communication. c. Quantum circuit representation equivalent to the generation of a three-level binary tree. Both sub-trees are constructed independently, followed by the entanglement of their roots (nodes 5,6, see d) with the ancilla. Subsequently, the ancilla's state is transferred to another photon via the E gate, culminating in the tree's formation. d. Implementation of the E gate, using atomic levels of the ancilla. After a CNOT gate, a $\pi$ pulse is applied, causing the state $\left|r\right\rangle$ to transition to $\left|e\right\rangle$. This pulse selectively drives the $\left|e\right\rangle \leftrightarrow\left|r\right\rangle$ transition due to resonance, while leaving $\left|g\right\rangle$ unaffected because the pulse frequency is far detuned from any transition involving $\left|g\right\rangle$. Subsequently, by observing the population in the state $\left|g\right\rangle$ after a decay process, the success of the gate operation is verified, regardless of the initial state.
• Figure 4: a A quantum metasurface under fluctuations in atomic location. The s.d. of the movement is $0.4a$ in this illustration. As the temperature increases, thermal fluctuations induce movements in the atoms, leading to imperfections in the array. To investigate this, we conducted simulations to assess the loss of reflectivity under different random atomic movements. b Fidelity of optical path 2, defined as a CNOT gate followed by a CZ gate, as specified in the protocol. c Fidelity decay of the tree state as a function of the separation between the two optical paths, due to the finite Rydberg Blockade effect. Two configurations are examined: one with both optical axes equidistant from the center of the Rydberg Blockade, and another with optical path 1 is at the center and optical path 2 offset by a specified distance. d Fidelity of the 2D cluster state versus state size for two scenarios: one with minimal positional disorder corresponding to near-ideal reflection coefficient (0.99), and the other with a positional standard deviation of $0.05a$($0.01\lambda$), which corresponds to a simulated reflection coefficient of 0.88.
• Figure A1: a The simulated electromagnetic field for zero movement. There is a destructive interference behind the QM and instructive interference of the imaginary part when reflected. b Our results demonstrate a decrease in the reflectivity of the Quantum Metasurface as the average distance between the atoms deviates from their ideal positions due to thermal effects.