Numerical estimate of the threshold for quantum expander codes

Hao Chen

Souvik Dihidar, S.W. McLaughlin, Prasad Tetali

This paper estimates threshold values of the message-passing iterative decoder for expander codes on discrete-input additive white Gaussian noise (AWGN) channels. We give a new upper bound on the rate of these codes and argue that expander codes are good codes at high rates on this channel.

Markus Grassl

Mauricio Gutiérrez, Kenneth R. Brown

Classical simulations of noisy stabilizer circuits are often used to estimate the threshold of a quantum error-correcting code. Physical noise sources are efficiently approximated by random insertions of Pauli operators. For a single qubit, more accurate approximations that still allow for efficient simulation can be obtained by including Clifford operators and Pauli operators conditional on measurement. We examine the feasibility of employing these expanded error approximations to obtain better threshold estimates. We calculate the level-1 pseudothreshold for the Steane [[7,1,3]] code for amplitude damping and dephasing along a non-Clifford axis. The expanded channels estimate the actual channel action more accurately than the Pauli channels before error correction. However, after error correction, the Pauli twirling approximation yields very accurate estimates of the performance of quantum error-correcting protocols in the presence of the actual noise channel.

Rohit Mukherjee, Simon Trebst

In quantum error correction, the error threshold provides essential quantitative guidance for the ability to bring about fault-tolerance through decoding the effects of incoherent noise, weak measurement or inference. However, the numerical value of an error threshold is typically only accessible through large-scale numerical simulations of the underlying noise model. Here we introduce an analytical estimate of error thresholds falling into the Nishimori universality class via a Fourier--Walsh projection scheme that maps the critical point of the underlying disorder-free statistical-mechanics model to the Born-disordered Nishimori critical point. Using a minimal replica theory approach, this closed-form estimate is obtained from a projection of the exact replicated single-bond weight which we find to reproduce (within a percentage point) the known numerical thresholds of random-bond and random-plaquette Ising models / $\mathbb Z_2$ stabilizer codes in spatial dimensions $d=2-5$, and extends to Potts variables with $q\le4$. The main application of our projection scheme is to $\mathbb Z_q$ surface codes, whose decoding problem maps to the disordered $q$-state clock model. For $q\ge5$ the clean clock model has \textit{two} Berezinskii--Kosterlitz--Thouless transitions, which the projection maps to two Nishimori temperatures that bound an intermediate information-critical phase. The resulting threshold values not only accurately agree with recent decohered-$\mathbb Z_q$-toric-code numerics, but are found to satisfy the Gilbert--Varshamov self-dual entropy relation $\ln q \simeq H_q(T_1^\ast)+H_q(T_2^\ast),$ although no duality condition is imposed in the construction. Our approach thereby points to a deeper connection between the clean and Born-disordered models, while allowing for instant analytical estimates of error thresholds for a variety of stabilizer codes.

Antoine Grospellier, Anirudh Krishna

Hypergraph product codes introduced by Tillich and Zémor are a class of quantum LDPC codes with constant rate and distance scaling with the square-root of the block size. Quantum expander codes, a subclass of these codes, can be decoded using the linear time small-set-flip algorithm of Leverrier, Tillich and Zémor. In this paper, we numerically estimate the performance for the hypergraph product codes under independent bit and phase flip noise. We focus on two families of hypergraph product codes. The first family has rate $1/61 \sim 1.6\%$, has qubits of weight $10$ or $12$ and stabilizers of weight $11$. We report a threshold near $4.6\%$ for the small-set-flip decoder. We also show that for similar rate, the performance of the hypergraph product is better than the performance of the toric code as soon as we deal with more than $500$ logical qubits and that for $3600$ logical qubits, the logical error rate for the hypergraph product code is several orders of magnitude smaller. The second family has rate $0.2$, qubits of weight $10$ and $20$ and stabilizers of weight $15$. We report a threshold near $2\%$ for the small-set-flip decoder.