MaxSAT decoders for arbitrary CSS codes

TL;DR

The paper addresses the problem of decoding CSS quantum codes under realistic noise by mapping the quantum maximum likelihood decoding problem to MaxSAT. It introduces a general CNF/3-SAT construction with hard syndrome constraints and weighted soft clauses that encode log-likelihoods, enabling decoding for arbitrary CSS codes and noisy measurements. The approach yields higher thresholds and better noise-suppression scaling than BP-OSD for color codes and improves accuracy for QLDPC codes, while remaining in the computationally easy phase due to favorable clause densities. The work also highlights hardware-friendly prospects, suggesting substantial speedups on ASICs/FPGAs and the potential for real-time quantum error correction in scalable quantum computers.

Abstract

Quantum error correction (QEC) is essential for operating quantum computers in the presence of noise. Here, we accurately decode arbitrary Calderbank-Shor-Steane (CSS) codes via the maximum satisfiability (MaxSAT) problem. We show how to map quantum maximum likelihood problem of CSS codes of arbitrary geometry and parity check weight into MaxSAT problems. We incorporate the syndrome measurements as hard clauses, while qubit and measurement error probabilities, including biased and non-uniform, are encoded as soft MaxSAT clauses. For the code capacity of color codes on a hexagonal lattice, our decoder has a higher threshold and superior scaling in noise suppression compared to belief propagation with ordered statistics post-processing (BP-OSD), while showing similar scaling in computational cost. Further, we decode surface codes and recently proposed bivariate quantum low-density parity check (QLDPC) codes where we find lower error rates than BP-OSD. Finally, we connect the complexity of MaxSAT decoding to a computational phase transition controlled by the clause density of the MaxSAT problem, where we show that our mapping is always in the computationally ''easy`` phase. Our MaxSAT decoder can be further parallelised or implemented on ASICs and FPGAs, promising potential further speedups of several orders of magnitude. Our work provides a flexible platform towards practical applications on quantum computers.

Figures (7)

• Figure 1: Performance of our MaxSAT decoder and BP-OSD decoder against physical error rate $p$ for the Color code. a) Logical error rate $p_\text{L}$ against $p$. Dashed line is fit with \ref{['eq:heuristic_fit']} with $d_\text{fit}^{\text{BP-OSD}}=7.3$ and $d_\text{fit}^{\text{MaxSAT}}=9.6$ for $d=9$, and $d_\text{fit}^{\text{BP-OSD}}=9.2$ and $d_\text{fit}^{\text{MaxSAT}}=13.8$ for $d=13$. We find pseudo-thresholds $p_\text{p-th}^{\text{BP-OSD}}=0.109$ and $p_\text{p-th}^{\text{MaxSAT}}=0.122$ for $d=9$, while $p_\text{p-th}^{\text{BP-OSD}}=0.112$ and $p_\text{p-th}^{\text{MaxSAT}}=0.130$ for $d=13$. b) Decoding time $T_\text{decode}$ against $p$.
• Figure 2: Scaling of MaxSAT and BP-OSD decoder for the Color code with error probability $p=0.1$. a) Logical error rate $p_\text{L}$ against distance $d$. Dashed line is fit with $p_\text{L}\propto \exp(-\gamma d)$ where we find $\gamma^{\text{MaxSAT}}\approx0.14$ and $\gamma^{\text{BP-OSD}}\approx0.06$. b) Decoding time $T_\text{decode}$ against number of data qubits $n$. Dashed line is fit with $T_\text{decode}\propto n^{\beta}$ with $\beta^{\text{MaxSAT}}\approx 1.46$ and $\beta^{\text{BP-OSD}}\approx 1.41$.
• Figure 3: Performance for decoding the QLDPC codes introduced by IBM Bravyi_2024 MaxSAT decoders and BP-OSD decoder. We show logical error rate $p_\text{L}$ against physical error rate $p$. Dashed line is fit with \ref{['eq:heuristic_fit']}. a)$\llbracket108,8,10\rrbracket$ QLDPC code with $d_\text{fit}^{\text{BP-OSD}}=10.7$, $d_\text{fit}^{\text{BP-OSD}}=9.8$. b)$\llbracket144,12,12\rrbracket$ QLDPC code with $d_\text{fit}^{\text{BP-OSD}}=14.2$, $d_\text{fit}^{\text{BP-OSD}}=13.7$.
• Figure 4: Threshold of Color code for a,b) MaxSAT and c,d) BP-OSD decoder for depolarizing noise. a,c) We show logical error $p_\text{L}$ against physical error $p$ for different distances $d$ with fitted threshold $p_\text{th}=15.20\pm0.05\%$ for MaxSAT and $p_\text{th}=13.23\pm0.04\%$ for BP-OSD, indicated as dashed vertical line. b,d) Rescaled error around threshold $p_\text{th}$ of fit with exponent $\nu=0.71$ for MaxSAT and $\nu=0.65$ for BP-OSD, where dashed line is the fitted critical scaling ansatz.
• Figure 5: Logical error rate $p_\text{L}$ against physical error rate $p$ for the rotated surface code for different distances $d$. Dots are MaxSAT solver results, while dashed line is BP-OSD.