Cavity-QED tools for MBQC with optical binomial-codes

TL;DR

This work addresses MBQC with photonic encodings by proposing a practical toolkit for optical binomial codes, a simpler alternative to GKP states. It introduces a cavity-QED protocol to conditionally generate binomial states, a deterministic atom-cavity CZ-gate, deterministic cluster-state construction, and ancilla-assisted, XY-plane Pauli measurements, all modeled under realistic optical losses. The results show high-fidelity preparation of binomial superpositions and magic states (e.g., $T$-type and $H$-type) with fidelities exceeding 0.98, small CZ-map deviations $|\Delta R_{CZ}|$ (≈0.018–0.099), and robust cluster-state stabilizers and teleportation fidelities, indicating viability for fault-tolerant MBQC. This work provides a feasible path to implement MBQC with optical binomial codes and paves the way for hybrid atom-photon architectures using existing cavity QED technology.

Abstract

Measurement-based quantum computation (MBQC) offers a promising paradigm for photonic quantum computing, but its implementation requires the generation of specific non-Gaussian resource states. While continuous-variable encodings such as the highly complex (GKP) states have been widely studied, the much simpler binomial codes offer an experimentally accessible alternative, though they demand a distinct set of operational tools. Here, we present a toolkit for MBQC using optical binomial codes, detailing a cavity-QED protocol for conditional generation of cluster states and the implementation of Pauli measurements. Our work proposes the first steps for existing optical atom-cavity architectures to lay the groundwork for their use in quantum computation.

Figures (4)

• Figure 1: \ref{['cz0']} Preparing superposition states of binomial codes using Gaussian inputs, $\hat{O}(\varphi,R,m)$ denotes a single atom-cavity iteration Eq. \ref{['eq:ite']}. \ref{['den']}$\ket{+}$ state is generated using the circuit in \ref{['cz0']}, other density matrix elements are negligible. $\beta$ is the cavity efficiency. A single iteration $\hat{O}(\pi, H, m)$ using cavity QED is demonstrated in Hacker_2019 to generate cat states.
• Figure 2: CZ-gate between two optical modes encoded in binomial code states. \ref{['cz']} Implementation of the CZ-gate between optical modes; the feed-forward phase shift $e^{i\frac{\pi}{2}\hat{n}}$ is applied only for $\ket{s}$ measurement outcomes. Here, the atom-photon CZ-gate is implemented as a single iteration $\hat{O}(\frac{\pi}{2}, I, I)$, without the atomic rotation and projective measurement. \ref{['cz1']} Difference between the process map and the ideal CZ-gate ($\Delta R_\text{CZ}$). For cavity loss $\beta= \qty{0.999, 0.99}$, $\max \abs{\Delta R_\text{CZ}} = \qty{ 0.018, 0.099}$ is obtained using Eq. \ref{['eq:dcz']}.
• Figure 3: Deterministic generation of cluster states with binomial codes. \ref{['cl5']} 5-Star shaped cluster state generation, which can be foliated to construct a unit cell of the RHG lattice as shown in \ref{['cube']}. Vertices represent $\ket{+}$ states in Fig. \ref{['fg:pst']} and solid lines denote CZ-gate in Fig. \ref{['fg:czgat']}. \ref{['stb']} Stabilizer measurements for the star-shaped cluster state generated using the circuit in \ref{['cl5']}.\ref{['mc']} Conditional projective measurement $\ket{\chi}_t$ in the XY-plane on a cluster graph using PNRD's and ancillary state. Table \ref{['tab:sol1']} shows the fidelities of post-measurement states with respect to ideal projected states for both 3-chain \ref{['cl5']} and 5-star cluster states \ref{['mc']}. In both configurations, we take an ideal cluster state and project a qubit to $\ket{\chi}_{\pi/3}$, and similar results are observed varying the qubit location and the projection angle $t$.
• Figure 4: \ref{['p4']} and \ref{['p33']}: Optimization of amplitude ratios versus higher-order Fock state contributions for squeezed-displaced input states. It is clear that the state $\ket{+} = \ket{B}_{(\frac{\pi}{4},0)}$ is not viable through input state optimization and only the state $\ket{\mathrm{B}}_{(\frac{\pi}{3.3},0)}$ is achievable. \ref{['af:sol']} and \ref{['af:sol1']}: contour plots of Eq. \ref{['eq:arn']} for $\ket{T}_1$ and $\ket{H}$ states. Table.\ref{['tab:sol']} shows the $\qty{\beta, \zeta}$ corresponding to $\qty{\theta, \Phi}$ obtained from similar plots. The $\theta$ for $\ket{T}_2$ is defined as $\cos(2 \text{T}) = 1/\sqrt{3}$. \ref{['fg:feed']}$\mathcal{F}_{\text{net}} \geq 0$ along with $\sum_{n \notin \{0,2,4\}} |C_n|^2$ demonstrate that the fidelity, after the atom-cavity iteration exceeds the fidelity with Gaussian operations. \ref{['cz_st']} Comparison of density matrices from $R_\text{CZ}$ with ideal CZ gate for $\beta=0.999$, here we use $\psi_1= \cos(\pi/4) \ket{\tilde{1}} + e^{\mathrm{i} \pi/4} \sin(\pi/4) \ket{\tilde{0}}$ and $\psi_2 = \cos(\pi/3) \ket{\tilde{1}} + e^{\mathrm{i} \pi/5} \sin(\pi/3) \ket{\tilde{0}}$ and the fidelity with ideal gate operation is 0.981.