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
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
Applying single-qubit Clifford unitaries to a Pauli stabilizer code produces a Clifford-deformed variant whose stabilizers remain Pauli operators, but with locally rotated Pauli axes. Such deformations provide a simple way to tailor a fixed code to anisotropic noise, and have enabled unusually high thresholds under strongly biased dephasing. In this work, we discuss zero-rate quantum low-density parity-check (LDPC) codes, for which there exist Clifford-deformed variants where the number of biased logical operators scales slower than the distance, or there exists a basis of logical operators whose overlap satisfies certain scaling conditions; in this case, the code-capacity threshold for the Clifford-deformed variant under i.i.d. pure dephasing noise approaches 50%. This property provably explains previously known code examples with 50% biased noise thresholds, such as XY surface code, XZZX surface code, color code, as well as some 3D Clifford-deformed codes. As a concrete new example, we study Clifford deformations of the tile codes of Ref. [1]. Similar to the phase diagram of 50% thresholds for random Clifford deformations of the surface code in Ref. [2], we find a similar phase diagram for the tile codes. We also construct several translationally invariant deformations of the tile code with 50% thresholds, and present numerical evidence for improved performance at finite bias and under circuit-level noise. In the circuit-level setting, performance is governed by the residual bias after a full syndrome-extraction cycle, linking our simulations to phenomenological models commonly used to study Clifford-deformed codes. We estimate this residual bias for different qubit platforms by modeling microscopic implementations of tile-code syndrome extraction.
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.
The aim of this bachelor’s thesis is to investigate the approach of using the Ising model on a Clifford-deformed surface code (CDSC). For this purpose an original CDSC, which has been given the name G81, has been constructed and analyzed using the Ising model. A partial analytical solution has been calculated for the G81 code using the Ising model at a specific point, which has been compared with numerical calculations obtained using a tensor network-based decoder. The tensor network-based decoder has been used on the G81 code to find properties dependent on the size of the code. The partial analytical solution did expectedly not correspond to the numerical calculations, which confirms that the solution is not fully complete. However, the numerical calculations obtained for the G81 code are similar to the analytical solution for the XZZX code in the special point, which opens for the questions if the two are equivalent. The threshold for the physical error probability for the G81 code is approximately p ≈ 0.3 for η = 30.
Practical fault-tolerant quantum computing requires qubit-efficient quantum error correction (QEC). Clifford deformations (CDs) modify the stabilizers of a QEC code to lower its logical error rate at no additional resource cost. However, the number of possible CDs grows exponentially with the number of qubits, making exhaustive search infeasible beyond small codes. In this work, we further show that under circuit level noise the optimal CD depends strongly on both the noise model and decoder. To make systematic CD optimization possible, we introduce Evolution-Based Best-arm Identification (EBBI), a shot-efficient search algorithm that treats the circuit, noise model and decoder as a black box. We benchmark EBBI's performance on the rotated surface code, under two strongly Z-biased circuit-level noise models with the PyMatching decoder. At distance 3, EBBI finds the optimal CD using up to ∼1100× fewer shots than random search. At distance 5, when run at the target error rate, the median EBBI run converges to the CDs we believe to be optimal, while random search never converges. Under a Z-biased SI1000-like noise model, EBBI discovers a new CD, the B-XZZX code, which we generalized to larger distances. It differs from XZZX on only a fraction O(d−1) of the qubits at code distance d, yet its relative advantage grows from ≈5% at d=3 to ≈12% at d=11. We thus show that the B-XZZX pattern found by EBBI can be systematically scaled to larger code distances. Our work represents a first step towards scalable CD optimization for general QEC codes.
We can design efficient quantum error-correcting (QEC) codes by tailoring them to our choice of quantum architecture. Useful tools for constructing such codes include Clifford deformations and appropriate gauge fixings of compass codes. In this work, we find Clifford deformations that can be applied to elongated compass codes resulting in QEC codes with improved performance under noise models with errors biased towards dephasing commonly seen in quantum computing architectures. These Clifford deformations enhance decoder performance by introducing symmetries, while the stabilizers of compass codes can be selected to obtain more information on high-rate errors. As a result, the codes exhibit thresholds that increase with bias and lower logical error rates under both code capacity and phenomenological noise models. One of the Clifford deformations we explore yields QEC codes with better thresholds and logical error rates than those of the XZZX surface code at moderate biases under code capacity noise.
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.