Linear optical quantum computing with a hybrid squeezed cat code

TL;DR

The paper introduces a hybrid squeezed cat (H-SC) code that combines squeezed-cat bosonic states with polarization qubits to enable fault-tolerant optical quantum information processing using only linear optics. It presents a generation scheme for entangled hybrid ancillae, a universal gate set via gate teleportation, and a loss-compensation protocol based on hybrid teleportation, showing through simulations that H-SC outperforms both the squeezed cat code and the hybrid cat code in generation and gate success at moderate photon numbers. The approach leverages linear optics to avoid strong nonlinearities and demonstrates a practical path toward scalable optical quantum computation, including potential concatenation with surface codes and applicability to other physical platforms. These results position the H-SC code as a versatile resource for robust optical quantum information processing with current or near-term technology.

Abstract

In recent years, squeezed cat codes with resilience to specific types of loss have been proposed as a step toward realizing fault-tolerant optical quantum computers. However, error correction for squeezed cat codes requires a strong nonlinearity, which makes its implementation challenging with current technology. We propose a novel hybrid code that combines the squeezed cat code and the polarization qubit. First, we propose a generation method and a universal gate set that can be implemented with a linear optical system. Then, we show the superiority of the hybrid squeezed cat code over the hybrid cat code and the squeezed cat code through numerical simulations. These results demonstrate that the hybrid squeezed cat code is a promising candidate as a new resource for optical quantum information processing.

Figures (6)

• Figure 1: The generation circuit of the hybrid entangled states. It consists of a displacement operation, a variable beam splitter, a half beam splitter (HBS), polarization beam splitters (PBSs), and photon number resolving detectors. Nonlinear transformations are unnecessary.
• Figure 2: Success probability of generating the H-SC code. The blue (solid) line shows the success probability for a hybrid cat state with squeezing parameter $\xi = 0$. The orange (broken line) and green (dots) represent the success probabilities for H-SC states with $\xi = 0.25$ and $\xi = 0.50$, respectively. For $\xi = 0.25$ and $0.50$, the corresponding amplitudes are approximately $|\alpha| \approx 1.39$ and $1.32$, respectively.
• Figure 3: Optical system for performing the H gate on the H-SC code. $B_D$ is the Bell measurement for the polarization qubit, and $B_C$ is the Bell measurement for the squeezed cat state. PNRDs represent photon number resolving detectors, and PDs represent photon detectors. The operation $\hat{U}$ can be implemented with wave plates and displacement operations.
• Figure 4: Optimal squeezing parameter for maximizing the success probability of hybrid Bell measurements, and the corresponding success probability. The blue (broken) line represents the optimal squeezing parameter, and the orange (dots) represents the success probability of the hybrid Bell measurement. The optimal squeezing parameter takes the maximum value $\xi \approx 0.17$ at $\bar{n} = 0.7$, where the corresponding amplitude is approximately $\alpha \approx 2$.
• Figure 5: Loss compensation circuit through quantum teleportation. $B_D$ represents the Bell measurement of the polarization qubit, and $B_C$ represents the Bell measurement of the squeezed cat state. The ancilla qubits required are $(\ket{0_L}\ket{0_L} + \ket{1_L}\ket{1_L})/\sqrt{2}$. Losses are assumed not to occur in the ancillary qubits.