Use of Faulty States in Cat-Code Error Correction

Abstract

Bosonic codes have seen a resurgence in interest for applications as varied as fault tolerant quantum architectures, quantum enhanced sensing, and entanglement distribution. Cat codes have been proposed as low-level elements in larger architectures, and the theory of rotationally symmetric codes more generally has been significantly expanded in the recent literature. The fault-tolerant preparation and maintenance of cat code states as a stand-alone quantum error correction scheme remains however limited by the need for robust state preparation and strong inter-mode interactions. In this work, we consider the teleportation-based correction circuit for cat code quantum error correction. We show that the class of acceptable ancillary states is broader than is typically acknowledged, and exploit this to propose the use of many-component ``bridge'' states which, though not themselves in the cat code space, are nonetheless capable of syndrome extraction in the regime where non-linear interactions are a limiting factor.

Figures (9)

• Figure 1: a) Tele-correction circuit for rotationally symmetric Bosonic codes grimsmoQuantumComputingRotationSymmetric2020, including cat codes, for arbitrary encoded input state ${\lvert\psi\rangle_{K}}$. Here the state subscripts ${K,M,N}$ are even and denote the numbers of coherent state components in the encoding. $\hat{R}_{MN}$ represents the controlled rotation gate: $e^{i\left(4\pi/MN\right)\hat{n}_{1}\hat{n}_{2}}$. Operators $\hat{\phi}$ denote general phase measurements, binned into indices ${k,k{'}}$ by the nearest coherent state components --- the pure output states shown here assume that components are perfectly distinguishable. b) Projective Hadamard circuit for rotation codes, which is repeated to form the full tele-correction circuit. c) Modular photon number measurement. Here ${\hat{P}_{k,K}}$ represents projection onto the subspace of photon numbers congruent to ${k}$ modulo ${K}$, and the phase angle in this case is binned into ${K}$ segments modulo ${4\pi/N}$.
• Figure 2: Sketch of the binning procedure for heterodyne phase measurements in the tele-correction circuit of Figure \ref{['fig:cat_tele_recovery_circuit']} (a), with the number of components ${K=6}$ and ${k\in\{0:K-1\}}$. For example, if the location indicated by heterodyne detection is given by the red asterisk, the nearest coherent state component would be represented by the red concentric circles, corresponding to index ${k=0}$.
• Figure 3: The full, modified tele-correction circuit. Preparation of ${\lvert + \rangle}$ ancillary states has been replaced by YS-state preparation from coherent states and modular photon number measurement, and the resulting necessary quadratic corrections to relative phases have been added. ${\hat{n}^{2}_{K}}$ denotes the self-Kerr evolution required to generate an ${K}$-component YS-state. The magnitude of the nonlinearity required for each such phase correction scales inversely as the cube of the number of coherent state components typical for an input state.
• Figure 4: Modular Photon Number Measurement shown in Figure \ref{['fig:cat_tele_recovery_circuit']} (c). Final state fidelities and their probabilities, with a probe cat state and a target coherent state of double amplitude. Fidelities are binned into histograms to indicate success rates when post-selecting based on measurement outcome. Histogram bars correspond to measurement outcome indices, as sketched in Figure \ref{['fig:cat_project_distance_sketch']}. Error bars represent Wilson score ${95\%}$ confidence intervals. Threshold probabilities ${p_{th}}$ are given for fidelities above ${0.9}$, ${0.95}$ and ${0.99}$ respectively, alongside the mean fidelity ${\mu}$ and its variance ${\sigma^{2}}$. (Left) Measuring the photon number ${\mathrm{mod}\:2}$, using a probe cat state with ${2}$ coherent state components and ${\alpha=4}$, and heterodyne local oscillators with amplitude ${\beta=6}$. ${10^{5}}$ samples taken. (Right) Measuring the photon number ${\mathrm{mod}\:4}$, using a probe cat state with ${4}$ coherent state components and ${\alpha=12}$, and heterodyne local oscillators of amplitude ${\beta=8}$. ${4\times 10^{4}}$ samples taken. (Inset) The ${4}$-component results display a statistically significant difference in high-fidelity probabilities for even and odd indices, located either side of ${p=0.99}$, a possible artifact of the small choice of ${\beta}$ and discretised grid of measurement outcomes.
• Figure 5: Final state fidelities and their probabilities for the tele-correction as shown in Figure \ref{['fig:cat_tele_recovery_circuit']} (a), with a Yurke-Stoler input state to the second rail. The first and third rails are initialised with ideal ${\lvert +\rangle_{6}}$ states, with ${\lvert\alpha\rvert=4}$. The second rail uses the Yurke--Stoler state initialisation to simulate preparation of a ${\lvert +\rangle_{12}}$ state with ${\lvert\alpha\rvert=8}$. ${80000}$ samples have been taken, and error bars represent Wilson score ${95\%}$ confidence intervals. Fidelities are binned into histograms to indicate success rates when post-selecting based on measurement outcome. Clustered histogram bars correspond to measurement outcome indices, as sketched in Figure \ref{['fig:cat_project_distance_sketch']}. The probabilities of achieving fidelities above ${\{0.9, 0.95, 0.99\}}$ are ${\{ 0.977, 0.963, 0.908 \}}$ respectively. The mean output fidelity ${\mu=0.989}$ and variance ${\sigma^{2}=0.0043}$ are also shown.