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.
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.
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.
We show that a simple modification of the surface code can exhibit an enormous gain in the error correction threshold for a noise model in which Pauli Z errors occur more frequently than X or Y errors. Such biased noise, where dephasing dominates, is ubiquitous in many quantum architectures. In the limit of pure dephasing noise we find a threshold of 43.7(1)% using a tensor network decoder proposed by Bravyi, Suchara, and Vargo. The threshold remains surprisingly large in the regime of realistic noise bias ratios, for example 28.2(2)% at a bias of 10. The performance is, in fact, at or near the hashing bound for all values of the bias. The modified surface code still uses only weight-4 stabilizers on a square lattice, but merely requires measuring products of Y instead of Z around the faces, as this doubles the number of useful syndrome bits associated with the dominant Z errors. Our results demonstrate that large efficiency gains can be found by appropriately tailoring codes and decoders to realistic noise models, even under the locality constraints of topological codes.
Abstract To date, the surface code has become a promising candidate for quantum error correcting codes because it achieves a high threshold and is composed of only the nearest gate operations and low-weight stabilizers. Here, we have exhibited that the logical failure rate can be enhanced by manipulating the lattice size of surface codes that they can show an enormous improvement in the number of physical qubits for a noise model where dephasing errors dominate over relaxation errors. We estimated the logical error rate in terms of the lattice size and physical error rate. When the physical error rate was high, the parameter estimation method was applied, and when it was low, the most frequently occurring logical error cases were considered. By using the minimum weight perfect matching decoding algorithm, we obtained the optimal lattice size by minimizing the number of qubits to achieve the required failure rates when physical error rates and bias are provided .
Noise in quantum computing is countered with quantum error correction. Achieving optimal performance will require tailoring codes and decoding algorithms to account for features of realistic noise, such as the common situation where the noise is biased towards dephasing. Here we introduce an efficient high-threshold decoder for a noise-tailored surface code based on minimum-weight perfect matching. The decoder exploits the symmetries of its syndrome under the action of biased noise and generalizes to the fault-tolerant regime where measurements are unreliable. Using this decoder, we obtain fault-tolerant thresholds in excess of 6% for a phenomenological noise model in the limit where dephasing dominates. These gains persist even for modest noise biases: we find a threshold of ∼5% in an experimentally relevant regime where dephasing errors occur at a rate 100 times greater than bit-flip errors.
Studying the threshold behavior of surface codes under biased noise models is an active area of research. Previous work Tuckett et al. (2018), using an optimal tensor-network decoder, demonstrated that replacing Z-type stabilizers with Y-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a ZY surface code by replacing the X-type stabilizers with Y-type stabilizers. We compare it with the standard ZX surface code under circuit-level Pauli-X biased noise, with and without an additional gate-based XX crosstalk noise. We find that for the ZX surface code, the X-memory threshold increases monotonically with bias while the Z-memory threshold decreases and saturates. For the ZY surface code, the Y-memory threshold is nearly constant across all bias values. The Z-memory thresholds of the ZX and ZY codes are consistent within the uncertainty. Adding XX crosstalk reduces the Z-memory threshold beyond the fitting uncertainty while leaving the X memory threshold largely unaffected. The choice of CNOT ordering redistributes threshold performance between the two logical memories. Our work extends prior observations from code-capacity level noise to circuit-level noise. It also indicates the need for decoders capable of jointly reasoning over correlated syndrome information so that tailored stabilizer structures could be fully utilized for quantum error correction.
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.