Abstract
Single-shot error correction corrects data noise using only a single round of noisy measurements on the data qubits, removing the need for intensive measurement repetition. We introduce a general concept of confinement for quantum codes, which roughly stipulates qubit errors cannot grow without triggering more measurement syndromes. We prove confinement is sufficient for single-shot decoding of adversarial errors and linear confinement is sufficient for single-shot decoding of local stochastic errors. Further to this, we prove that all three-dimensional homological product codes exhibit confinement in their $X$-components and are therefore single-shot for adversarial phase-flip noise. For local stochastic phase-flip noise, we numerically explore these codes and again find evidence of single-shot protection. Our Monte Carlo simulations indicate sustainable thresholds of $3.08(4)\%$ and $2.90(2)\%$ for 3D surface and toric codes respectively, the highest observed single-shot thresholds to date. To demonstrate single-shot error correction beyond the class of topological codes, we also run simulations on a randomly constructed 3D homological product code.
Highlights
Quantum error correction encodes logical quantum information into a codespace [1]
This paper is in two parts: on the one hand, we propose the concept of confinement as an essential characteristic for a code family to display single-shot properties; on the other, we investigate the single-shot performances of the class of 3D homological product codes [19,28,29], which we call 3D product codes
We prove that good linear confinement is a sufficient condition for a family of codes to exhibit a sustainable single-shot threshold for local stochastic noise (Appendix A)
Summary
Quantum error correction encodes logical quantum information into a codespace [1]. Given perfect measurement of the codespace stabilizers we obtain the syndrome of any error present. Bombín coined the phrase single-shot error correction and remarked that it “is related to self-correction and confinement phenomena in the corresponding quantum Hamiltonian model.” [10]. He defined confinement for subsystem codes, and showed that it is sufficient for singleshot error correction with a limited class of subsystem codes. Quantum expander codes lack the soundness property so neither Bombín’s notion of confinement or Campbell’s notion of soundness is sufficient to encompass all known examples of single-shot error correction. VI, we discuss future research directions that flow from this work
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