Charge-Informed Quantum Error Correction

TL;DR

This work develops a charge-informed quantum error-correction framework for U(1) symmetry-enriched topological memories under charge-conserving noise. By mapping decoding to a disordered 2D integer loop model on the Nishimori line, it reveals a Berezinskii-Kosterlitz-Thouless–like transition with a modified stiffness jump, and identifies a disorder-driven loop-glass regime and a zero-temperature minimum-cost-flow decoder as a U(1) analogue of MWPM. Suboptimal decoders obtained by varying a replica temperature discover a rich phase structure, including a confidently incorrect loop-glass phase, while the optimal and MCF decoders significantly outperform charge-agnostic decoders. The results illuminate how measuring U(1) charges can dramatically enhance decoding thresholds and provide a broader view of decoding behavior in symmetry-enriched topological codes with potential implications for fault-tolerant quantum computation.

Abstract

We investigate the statistical physics of quantum error correction in ${\rm U}(1)$ symmetry-enriched topological quantum memories. Starting from a phenomenological error model of charge-conserving noise, we study the optimal decoder assuming the local charges of each anyon can be measured. The error threshold of the optimal decoder corresponds to a continuous phase transition in a disordered two-dimensional integer loop model on the Nishimori line. Using an effective replica field theory analysis and Monte Carlo numerics, we show that the optimal decoding transition exhibits Berezinskii-Kosterlitz-Thouless universality with a modified universal jump in winding number variance. We further generalize the model beyond the Nishimori line, which defines a large class of suboptimal decoders. At low nonzero temperatures and strong disorder, we find numerical evidence of a disorder-dominated loop-glass phase which corresponds to a "confidently incorrect" decoder. The zero-temperature limit defines the minimum-cost flow decoder, which serves as the ${\rm U}(1)$ analog of minimum-weight perfect matching in $\mathbb{Z}_2$ topological codes. Both the optimal and minimum-cost flow decoders are shown to dramatically outperform the charge-agnostic optimal decoder in symmetry-enriched topological codes.

Figures (12)

• Figure 1: (a) Visualization of the quantum error correction problem investigated in this work. U(1) charges are created at the endpoints of an error configuration $k$ (red); an observer measures these charges, and annihilates them with a correction $r$ (blue). The combined error and correction form a collection of closed loops $J \sim (k - r)$. The bottom loop is topologically trivial and results in a successful recovery, while the top loop exhibits nontrivial winding and causes a logical error. (b) Numerical phase diagram of the model \ref{['eq:partition_fn']} with disorder strength $\alpha$ and temperature $\beta$; the optimal decoder is described by the Nishimori line $\alpha = \beta$ (gray dashed line). Black dots denote numerically estimated phase boundaries, with error bars omitted. The phase boundary between the loop-glass and the long-loop phases is conjectured.
• Figure 2: (a) Disorder-averaged helicity modulus $\overline{\Upsilon}$ on the Nishimori line as a function of the error strength $\alpha$, for various system sizes. Vanishing $\overline{\Upsilon}$ at small $\alpha$ indicates successful decoding, while nonvanishing $\overline{\Upsilon}$ at larger $\alpha$ implies a finite logical error probability. Inset: BKT finite-size scaling collapse with $f\left ( \alpha, L\right ) \equiv -\ln L + a \left ( \alpha^{-1} - \alpha_c^{-1}\right )^{-1/2}$ and $g\left ( \alpha, L\right ) \equiv \overline{\Upsilon} \left[ 1 + \left ( \ln L + c\right )^{-1}\right]^{-1}$ with $a = 7.6$, $c = 7.11$. This scaling collapse is consistent with a universal jump of $1/\pi$. (b) Edwards-Anderson helicity modulus $\chi \equiv \frac{1}{2} \sum_{\mu} \overline{\langle W_{\mu} \rangle^2}$ on the line $\beta = 0.15$ as a function of the error strength $\alpha$, for various system sizes. Inset: finite-size scaling collapse with $\delta\alpha = \alpha - 0.305$ and $\nu_g = 2.5$. (c) Disorder-averaged helicity modulus $\overline{\Upsilon}$ on the $\beta = 0.15$ line. The clear crossing and scaling collapse in $\chi$ at $\alpha_g \approx 0.305$, combined with the systematic decrease in $\overline{\Upsilon}$ with increasing system size, is suggestive of a loop-glass phase for $\alpha > \alpha_g$. (d) Edwards-Anderson helicity modulus $\chi$ from the (linear) MCF decoder, corresponding to the $\beta = 0$ line. Inset: finite-size scaling collapse with $\alpha_{\text{MCF}} = 0.265$ and $\nu_{\text{MCF}} = 2.19$.
• Figure S1: (a) The winding number $W_{x}$ of a current loop configuration $J$ (solid black lines), obtained by summing over all $J_{i\mu}$ cut by the curve $C_{y}$ (dotted red line). In this case, $W_{x} = 2$. (b) The same winding number computed along a different equivalent curve. $W_{x}$ is invariant under sequential deformations of the curve $C_{x}$.
• Figure S2: Examples of the MCF solutions for (a) $\alpha = 0.15$, (b) $\alpha = 0.25$, (c) $\alpha = 0.35$ in an $L = 8$ system. Magenta arrows represent the sampled errors, the cyan arrows are the linear MCF solutions. The positive charges are blue, the negative are red. The winding number can be computed along the red dashed cuts. The correction succeeds in (a) and (b) but fails in (c) due to non-trivial winding.
• Figure S3: Failure rate of the minimum-cost flow decoder, for system sizes up to $L=512$. Inset: finite-size scaling collapse with $\nu_{\rm MCF} = 2.19$ and $\delta \alpha = \alpha - \alpha_{\text{MCF}}$ with $\alpha_{\text{MCF}} = 0.265$.