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arXiv:2211.02116

Tailoring Three-Dimensional Topological Codes for Biased Noise

Huang, Eric, Arthur Chalom Pesah, Christopher T. Chubb, Michael Vasmer, Arpit Dua

Published Nov 3, 2022• 4 citations
arXivHTMLPDF

Abstract

Tailored topological stabilizer codes in two dimensions have been shown to exhibit high storage threshold error rates and improved subthreshold performance under biased Pauli noise. Three-dimensional (3D) topological codes can allow for several advantages including a transversal implementation of non-Clifford logical gates, single-shot decoding strategies, parallelized decoding in the case of fracton codes as well as construction of fractal lattice codes. Motivated by this, we tailor 3D topological codes for enhanced storage performance under biased Pauli noise. We present Clifford deformations of various 3D topological codes, such that they exhibit a threshold error rate of $50\%$ under infinitely biased Pauli noise. Our examples include the 3D surface code on the cubic lattice, the 3D surface code on a checkerboard lattice that lends itself to a subsystem code with a single-shot decoder, the 3D color code, as well as fracton models such as the X-cube model, the Sierpinski model and the Haah code. We use the belief propagation with ordered statistics decoder (BP-OSD) to study threshold error rates at finite bias. We also present a rotated layout for the 3D surface code, which uses roughly half the number of physical qubits for the same code distance under appropriate boundary conditions. Imposing coprime periodic dimensions on this rotated layout leads to logical operators of weight $O(n)$ at infinite bias and a corresponding $\exp[-O(n)]$ subthreshold scaling of the logical failure rate, where $n$ is the number of physical qubits in the code. Even though this scaling is unstable due to the existence of logical representations with $O(1)$ low-rate Pauli errors, the number of such representations scales only polynomially for the Clifford-deformed code, leading to an enhanced effective distance.

DOI
10.3929/ethz-b-000592781

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Various realizations of Kitaev’s surface code perform surprisingly well for biased Pauli noise. Attracted by these potential gains, we study the performance of Clifford-deformed surface codes (CDSCs) obtained from the surface code by the application of single-qubit Clifford operators. We first analyze CDSCs on the 3×3 square lattice and find that, depending on the noise bias, their logical error rates can differ by orders of magnitude. To explain the observed behavior, we introduce the effective distance d′ , which reduces to the standard distance for unbiased noise. To study CDSC performance in the thermodynamic limit, we focus on random CDSCs. Using the statistical mechanical mapping for quantum codes, we uncover a phase diagram that describes random CDSC families with 50% threshold at infinite bias. In the high-threshold region, we further demonstrate that typical code realizations outperform the thresholds and subthreshold logical error rates, at finite bias, of the best-known translationally invariant codes. We demonstrate the practical relevance of these random CDSC families by constructing a translation-invariant CDSC belonging to a high-performance random CDSC family. We also show that our translation-invariant CDSC outperforms well-known translation-invariant CDSCs, such as the XZZX and XY codes. Published by the American Physical Society 2024
For quantum computers to reach their full potential will require error correction. We study the surface code, one of the most promising quantum error correcting codes, in the context of predominantly dephasing (Z-biased) noise, as found in many quantum architectures. We find that the surface code is highly resilient to Y-biased noise, and tailor it to Z-biased noise, whilst retaining its practical features. We demonstrate ultrahigh thresholds for the tailored surface code: ~39% with a realistic bias of  = 100, and ~50% with pure Z noise, far exceeding known thresholds for the standard surface code: ~11% with pure Z noise, and ~19% with depolarizing noise. Furthermore, we provide strong evidence that the threshold of the tailored surface code tracks the hashing bound for all biases. We reveal the hidden structure of the tailored surface code with pure Z noise that is responsible for these ultrahigh thresholds. As a consequence, we prove that its threshold with pure Z noise is 50%, and we show that its distance to Z errors, and the number of failure modes, can be tuned by modifying its boundary. For codes with appropriately modified boundaries, the distance to Z errors is O(n) compared to O(n1/2) for square codes, where n is the number of physical qubits. We demonstrate that these characteristics yield a significant improvement in logical error rate with pure Z and Z-biased noise. Finally, we introduce an efficient approach to decoding that exploits code symmetries with respect to a given noise model, and extends readily to the fault-tolerant context, where measurements are unreliable. We use this approach to define a decoder for the tailored surface code with Z-biased noise. Although the decoder is suboptimal, we observe exceptionally high fault-tolerant thresholds of ~5% with bias  = 100 and exceeding 6% with pure Z noise. Our results open up many avenues of research and, recent developments in bias-preserving gates, highlight their direct relevance to experiment.
Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit XXX corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of ≈2.5 %\approx 2.5\,\%≈2.5%, close to the ≈2.9 %\approx 2.9\,\%≈2.9% of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for ZZZ errors is required and the non-Clifford gates in the circuit reduce the threshold from ≈2.9 %\approx 2.9\,\%≈2.9% to ≈1.8 %\approx 1.8\,\%≈1.8% with a naive decoder. We show how ZZZ decoding can be improved using knowledge of the XXX corrections, pushing the threshold to ≈2.2 %\approx 2.2\,\%≈2.2%. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose XXX stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by CCZCCZCCZ, CSCSCS, and TTT gates.
The surface code, with a simple modification, exhibits ultrahigh error-correction thresholds when the noise is biased toward dephasing.Here, we identify features of the surface code responsible for these ultrahigh thresholds.We provide strong evidence that the threshold error rate of the surface code tracks the hashing bound exactly for all biases and show how to exploit these features to achieve significant improvement in the logical failure rate.First, we consider the infinite bias limit, meaning pure dephasing.We prove that the error threshold of the modified surface code for pure dephasing noise is 50%, i.e., that all qubits are fully dephased, and this threshold can be achieved by a polynomial time-decoding algorithm.We demonstrate that the subthreshold behavior of the code depends critically on the precise shape and boundary conditions of the code.That is, for rectangular surface codes with standard rough or smooth open boundaries, it is controlled by the parameter g ¼ gcdðj; kÞ, where j and k are dimensions of the surface code lattice.We demonstrate a significant improvement in the logical failure rate with pure dephasing for coprime codes that have g ¼ 1 and closely related rotated codes, which have a modified boundary.The effect is dramatic: The same logical failure rate achievable with a square surface code and n physical qubits can be obtained with a coprime or rotated surface code using only Oð ffiffiffi n p Þ physical qubits.Finally, we use approximate maximum-likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased toward dephasing.In particular, comparing with a square surface code, we observe a significant improvement in the logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.
We introduce and analyze a family of Clifford-deformed bivariate bicycle codes that are tailored for biased noise. Our qLDPC codes are defined on a bipartite hexagonal lattice with limited-range gates and low-weight stabilizers. The code is non-CSS, featuring stabilizer generators that are each half X and half Z. We find small examples with high encoding rate that perform well for a large range of bias. In the limit of large noise bias, the code reduces to two independent classical cellular automaton codes, giving a distance scaling better than is possible with 2D topological quantum codes. Our construction combines two classical cellular automaton codes, LDPC codes that were recently proposed for use with noise-biased cat qubits, related to each other by a reflection. Each stabilizer in the quantum code is obtained by multiplying an all-X stabilizer from the first code with an all-Z stabilizer from the second code, or the other way around. The result is a self-dual quantum code with a number of qubits equal to the sum of the input codes and stabilizer weight and locality determined by the input codes. Under strong noise bias, the effective distance of the quantum code approaches the distance of the input codes. We simulate the logical performance of our qLDPC codes under code-capacity noise and find strong suppression of the logical error rate.