Tailoring Surface Codes for Highly Biased Noise

Abstract

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 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 and smooth open boundaries, it is controlled by the parameter g = gcd(j, k), where j and k are dimensions of the surface code lattice. We demonstrate a significant improvement in logical failure rate with pure dephasing for coprime codes that have g = 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 O( √ n) 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 logical failure rate against biased noise using a rotated surface code with approximately half the number of physical qubits.

Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code

“…Recently, there have been proposals for scaling up the cat codes by concatenating them with a repetition code [17] or a surface code [18] which are tailored to biased noise models [19][20][21]. These schemes take advantage of * kyungjoo.noh@yale.edu † christopher.chamberland@ibm.com the fact that the cat code can suppress bosonic dephasing (stochastic random rotation) errors exponentially in the size of the cat code, thereby yielding a qubit with a biased noise predominated either by bit-flip or phaseflip errors.…”

Fault-Tolerant Thresholds for the Surface Code in Excess of 5% Under Biased Noise

“…Of particular note is noise that is biased towards dephasing: a common feature of many architectures. With biased noise and reliable measurements, it is known that the surface code can be tailored to accentuate commonly occurring errors and that an appropriate decoder will give substantially increased thresholds [28,29]. However, these high thresholds were obtained using decoders with no known efficient implementation in the realistic setting where measurements are unreliable.Here we propose an efficient decoder for the surface code that is tailored to correct for local noise biased towards dephasing, demonstrating exceptional fault-tolerant thresholds.…”

“…While the tailored surface code exhibits an exceptional robustness to Z errors, its error correction thresholds have been obtained using approximate maximum likelihood (ML) decoders based on a tensor network contraction [18,28,29]. In the experimentally relevant fault-tolerant regime where measurement errors occur, such decoders are inefficient due to the difficulty in contracting higher-dimensional tensor networks.…”

“…An efficient decoder for infinite bias (only Z errors) presented in Ref. [29] does not extend to finite bias. Our main contribution here is a practical and efficient decoder that performs well for both finite bias and noisy measurements, demonstrating that the exceptional gains of this tailored code under biased noise extend to the fault-tolerant regime.…”

Triangular color codes on trivalent graphs with flag qubits

“…Thus, one should not expect the inferior error-correction performance of the color code. Indeed, this was confirmed with color code decoders matching the performance of the toric code decoders [29][30][31] assuming perfect syndrome measurement circuits.…”