Abstract
The surface code, with a simple modification, exhibits ultra-high error correction thresholds when the noise is biased towards dephasing. Here, we identify features of the surface code responsible for these ultra-high 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 sub-threshold 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/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 co-prime 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 co-prime or rotated surface code using only $O(\sqrt{n})$ physical qubits. Finally, we use approximate maximum likelihood decoding to demonstrate that this improvement persists for a general Pauli noise biased towards 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.
Highlights
Quantum error-correcting codes are expected to play a fundamental role in enabling quantum computers to operate at a large scale in the presence of noise
We describe the structure of the surface code with pure Y noise and show that this structure implies a 50% error threshold and a significant performance advantage in terms of the logical failure rate with coprime and rotated codes compared to square codes
It is important to note that our results apply to pure Z noise, i.e., dephasing noise, and the Z-biased noise prevalent in many quantum architectures, through the simple modification [7] of the surface code that exchanges the roles of Z and Y operators in stabilizer and logical operator definitions
Summary
Quantum error-correcting codes are expected to play a fundamental role in enabling quantum computers to operate at a large scale in the presence of noise. We perform numerical simulations, using strongly converged approximate maximum-likelihood decoding, to demonstrate the aforementioned significant reduction in the logical failure rate against biased noise that is achieved using a rotated j × j code, containing n 1⁄4 j2 physical qubits, compared to a square j × j code, containing n 1⁄4 2j2 − 2j þ 1 physical qubits. For a given bias, the relative advantage of (odd) rotated codes over square codes increases with the code size, until lowrate errors become the dominant source of logical failure and high-rate errors are effectively suppressed, motivating the search for efficient near-optimal biased-noise decoders for rotated codes Note that this performance with biased noise is not shared by all topological codes; in stark contrast, the triangular 6.6.6 color code [10] exhibits a decrease in the threshold with bias; see the Appendix A. Appendix A gives comparative results for color codes, and Appendix B defines the exact maximum-likelihood decoder used in simulations of pure Y noise on square and coprime surface codes
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