Fail fast: techniques to probe rare events in quantum error correction

TL;DR

The paper tackles the infeasibility of directly estimating exceedingly rare logical failures in quantum error correction under circuit noise by introducing three complementary methods: (I) fitting a universal failure-spectrum ansatz to accessible data to extrapolate $f(w)$ and $P(q)$; (II) computing or bounding the min-weight onset and related logical-failure structures to bound the onset and quantify decoding limitations; (III) extending the splitting method with multi-seeded initialization to improve convergence for general quantum LDPC codes. These techniques are demonstrated on distance-6, -12, and -18 bivariate bicycle codes, unrotated/rotated toric codes, and rotated surface codes, revealing strong low-error-rate performance for Relay and room for decoder improvements. The three methods show consistent agreement across regimes, enabling reliable characterization of rare-event behavior and guiding decoder design and resource estimates for fault-tolerant quantum computing. The work offers practical strategies to push failure-rate predictions into the regime relevant for large-scale quantum architectures and is adaptable to more complex noise models and adaptive protocols."

Abstract

The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g. 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g. billions) of logical gates on many (e.g. thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.

Figures (21)

• Figure 1: Illustration of our techniques for characterizing the logical error rate $P(q)$ as a function of physical error rate $q$ for a given QEC system (a decoding problem and decoder). Sampling (black dots) can estimate $P(q)$ directly (left) or indirectly via the failure spectrum $f(w)$ (right), but both become infeasible when failures are too rare. Technique I fits the data to a failure spectrum ansatz (red line), which predicts $P(q)$ across all $q$, with the low $q$ limit (red dashed) determined by the onset point $(w_0, f(w_0))$. Technique II identifies the optimal onset $(w^*_0, f^*(w^*_0))$ achievable by an ideal min-weight decoder. This constrains the failure spectrum since $w_0 \leq w^*_0$ and $f(w^*_0) \geq f^*(w^*_0)$, and identifies the potential gain from an improved decoder. Technique III applies the splitting method to estimate ratios of logical error rates across a sequence of $q$ values using Metropolis sampling. Each technique has limitations, but agreement among them strengthens confidence in the predicted low-error regime.
• Figure 2: Logical error rate and failure spectrum for a distance-12 rotated toric code (see RT(12)-bitflip in \ref{['sec:system-examples']}). (a) Direct sampling estimates $\hat{P}(q)$ of the logical error rate (black), with the overlaid sum $\mathcal{T}\{\hat{f}\}(q)$ (shaded red to include error bars) derived from the failure spectrum estimated in (b). (b) Direct sampling estimates $\hat{f}(w)$ of the failure spectrum, showing the normalized contribution of each weight $w$ to $\mathcal{T}\{\hat{f}\}(q)$ for three values of $q$ (colors). At $q = 0.005$ (well below pseudo-threshold), the sum is dominated by the minimum-weight term $w_0$, while higher $q$ values show significant contributions from a broader weight range. We sample sufficiently to estimate each $f(w)$ to within 1% relative error for all $w=6,7,\dots,N$, observing that $f(w)$ rises monotonically to $\sim\!0.75$ near $w\approx 30$ and remains near this value. We note that $f(w)$ eventually declines near $w\approx 115$ (beyond the plotted range), but for $q<\tfrac{1}{2}$ this non-monotonic tail modifies $\mathcal{T}\{\hat{f}\}(q)$ only negligibly.
• Figure 3: Failure spectrum and logical error rate curves for example QEC systems under bit-flip noise from \ref{['sec:system-examples']}. Solid lines in the left subfigures show fits of the ansatz $f^{(5)}_\text{ansatz}(w)$ from \ref{['sec:model-ansatz']}, and we include dashed lines showing $f_0\cdot (w/w_0)^{w_0}$ for reference (with $f_0$ set by the onset of $f^{(5)}_\text{ansatz}(w_0)$). Solid lines in the right subfigures show the corresponding logical error curve $P^{(5)}_\text{ansatz}(p):=\sum_w f^{(5)}_\text{ansatz}(w) \binom{N}{w} p^{w}(1-p)^{N-w}$. Subfigures in the same column share a common $x$-axis.
• Figure 4: Failure spectrum and logical error rate curves for example QEC systems under circuit noise. Labeling conventions are as in \ref{['fig:QEC-system-examples-bitflip']}.
• Figure 5: (a) Comparison of four ansatz models $f^{(l)}_\text{ansatz}(w)$ fitted to the failure spectrum of the $\mathrm{RT}(12)$ surface code under bitflip noise, with the corresponding inferred logical error rates $P^{(l)}_\text{ansatz}(p)$ shown in (b). The fits for $l=5$ and $l=6$ both match the data very closely. (c) Ratio of empirical logical error rates to those predicted by each ansatz across all QEC systems shown in \ref{['fig:QEC-system-examples-bitflip']} and \ref{['fig:QEC-system-examples-circuit']}. For each system, we fix $w_0$ and fit the remaining $l-1$ parameters of $f^{(l)}_\text{ansatz}(w)$ to failure spectrum data. Agreement improves systematically with increasing ansatz complexity.

Theorems & Definitions (4)

• Proposition 1
• proof
• Proposition 2
• proof