Optimal fault-tolerant threshold of the surface code under biased dephasing noise: exact determination

Tuckett, Bartlett, Flammia and Brown (Phys. Rev. Lett. 124, 130501 (2020)) introduced a symmetry-tailored minimum-weight perfect-matching decoder achieving surface-code fault-tolerant thresholds above 5% under dephasing-biased noise, explicitly flagged as open the absence of optimal fault-tolerant thresholds for the phenomenological biased-noise model, and — benchmarking against maximum-likelihood decoding at ideal measurements — proposed belief propagation as a remedy for the sub-ML performance of their efficient decoder observed at finite bias. It remains open to determine the optimal (maximum-likelihood) fault-tolerant threshold $p_{\mathrm{th}}(\eta, q)$ of the surface code family under the phenomenological model with noise bias $\eta$ and measurement-error rate $q$ — as an exact value or provably tight bounds, e.g. by statistical-mechanical methods. Code-capacity thresholds are well characterized (the XZZX code's thresholds match the zero-rate hashing bound for every single-qubit Pauli channel, with numerical evidence that they exceed it in some noise regimes; a 50% code-capacity threshold under pure dephasing is provable for Clifford-deformed zero-rate LDPC codes), but no optimal fault-tolerant threshold value at finite bias, rigorous or numerical, is known.

Background

Fault-tolerant quantum computation protects quantum information by encoding it non-locally so that localized physical errors can be detected and reversed without measuring the encoded data. The surface code places qubits on a two-dimensional lattice and repeatedly measures local commuting check operators (stabilizers); the record of flipped check outcomes over time (the syndrome) is processed by a classical decoder that infers a correction. The standard surface code is a CSS code — named for Calderbank, Shor and Steane — whose stabilizers separate into X-type and Z-type checks, so bit-flip and phase-flip errors are conventionally decoded as two independent matching problems. Physical qubit errors are Pauli operators: bit-flip $X$, phase-flip $Z$, or both $Y$. Many platforms — notably superconducting circuits and cat qubits (qubits encoded in superconducting microwave oscillators, engineered so that bit flips are exponentially suppressed relative to dephasing) — suffer noise biased toward dephasing: $Z$ errors occur much more often than $X$ or $Y$ errors. The bias parameter $\eta \ge 1/2$ used here compares the per-round probability of the high-rate $Z$ error, $p_{\mathrm{h.r.}} = p\,\eta/(\eta+1)$, with that of each low-rate error, $p_{\mathrm{l.r.}} = p/(2(\eta+1))$, so $\eta = 1/2$ is depolarizing (all Paulis equally likely) and $\eta \to \infty$ is pure dephasing (conventions in the literature differ by factors of 2). In the phenomenological noise model, each data qubit suffers such an error each round with total probability $p$, and each stabilizer measurement returns the wrong outcome with probability $q$; the code of distance $d$ is run for $d$ rounds. The decoder with the lowest possible logical failure rate is the maximum-likelihood (ML) decoder, which selects the most probable logical equivalence class of the error given the full syndrome history; ML decoding is computationally intractable (#P-hard) in general, so practical schemes use efficient heuristics, chiefly minimum-weight perfect matching (MWPM), which pairs syndrome defects along minimum-cost paths in a weighted graph. The threshold $p_{\mathrm{th}}$ is the critical physical error rate separating a phase in which the logical failure probability vanishes as $d \to \infty$ from one in which it does not. The code-capacity case ($q=0$, ideal measurements) can be mapped exactly onto disordered statistical mechanics — a random-bond Ising model on the Nishimori line, the codeword-averaged disorder distribution of an error-correcting code — as established by Dennis, Kitaev, Landahl and Preskill; the zero-rate hashing bound, the threshold achievable by random codes, serves as the natural benchmark. The fault-tolerant case ($q>0$) corresponds to a genuinely three-dimensional disordered gauge model (Wang, Harrington and Preskill) whose thresholds are known only numerically even for the single-Pauli-sector model with equally likely data and measurement errors: approximately 2.9% for MWPM versus about 3.3% for optimal decoding (Ohno et al.). Tailoring the code to the noise by local Clifford deformations (single-qubit basis changes that rotate the Pauli axes of each stabilizer) produced the XY ('tailored') code of Tuckett et al. and then the XZZX code of Bonilla Ataides et al., which with approximate maximum-likelihood decoding by tensor networks — networks of small tensors encoding the syndrome likelihoods, contracted approximately at bounded computational cost — matches the zero-rate hashing bound at code capacity for every single-qubit Pauli channel (with numerical evidence of exceeding it in some noise regimes) and whose efficient matching decoder attains fault-tolerant thresholds tending to about 10% at large bias. What remains unknown in this program is the following: at code capacity the thresholds are now well characterized (numerically matching the hashing bound for all $\eta$, with a provable 50% threshold at $\eta \to \infty$ for Clifford-deformed zero-rate LDPC codes), but with unreliable measurements no optimal fault-tolerant threshold of the code family is established at any bias — every published value is the numerically estimated performance of one particular decoder — and it is moreover unknown whether an efficient (polynomial-time) decoder can attain the ML threshold at finite bias in this regime.

Problem Statement

For the surface code family — the standard CSS surface code and its bias-tailored local Clifford deformations, specifically the XY (tailored) and XZZX codes studied in the source works — under the phenomenological biased-noise model in which each data qubit suffers, each of $d$ measurement rounds, a Pauli error with total probability $p$ distributed with bias $\eta = p_{\mathrm{h.r.}}/(2 p_{\mathrm{l.r.}}) \ge 1/2$ toward dephasing ($Z$ the high-rate error, $X$ and $Y$ the low-rate errors), and each stabilizer measurement returns an incorrect outcome with probability $q$ (e.g. $q = p$ or $q = p_{\mathrm{h.r.}} + p_{\mathrm{l.r.}}$):

1. Determine the optimal fault-tolerant threshold $p_{\mathrm{th}}(\eta, q)$ for each of these codes: the supremal physical error rate below which the logical failure probability under maximum-likelihood decoding of the full syndrome history tends to zero as the code distance grows, as an exact value or provably tight upper and lower bounds (provable bounds that coincide in the limit of large code distance), determined across the parameter domain $\eta \ge 1/2$, $q \ge 0$. Following the convention of the source works, the distance-$d$ code is taken with periodic boundary conditions and run for $d$ rounds, and a logical failure is counted whenever the combination of the error history and the correction contains a non-trivial cycle, spatial or temporal.

Any answer must be consistent with the established anchors: at $q = 0$ the XZZX code's code-capacity threshold matches the zero-rate hashing bound for every single-qubit Pauli channel (with numerical evidence that it exceeds the bound in some noise regimes), and a 50% code-capacity threshold under pure dephasing is provable for Clifford-deformed zero-rate LDPC codes; at $\eta = 1/2$ the bias-tailored matching decoders reduce to the conventional matching decoder, whose threshold follows the line $p_{\mathrm{h.r.}} + p_{\mathrm{l.r.}} \approx 0.029$ (the Wang–Harrington–Preskill phenomenological value) in the convention $q = p_{\mathrm{h.r.}} + p_{\mathrm{l.r.}}$, and even the single-Pauli-sector optimum with equally likely data and measurement errors is known only numerically (about 2.9% for matching, 3.3% optimal); and as $\eta \to \infty$ with $q > 0$ the problem reduces to decoding $d$ decoupled copies of the two-dimensional surface code, for which the matching threshold is about 10.3% while no exact maximum-likelihood value is known.

References

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