Clifford-Deformed Surface Codes

Abstract

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mn>3</a:mn><a:mo>×</a:mo><a:mn>3</a:mn></a:math> 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:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:msup><d:mi>d</d:mi><d:mo>′</d:mo></d:msup></d:math>, 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mn>50</g:mn><g:mi mathvariant="normal">%</g:mi></g:math> 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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><k:mi>X</k:mi><k:mi>Z</k:mi><k:mi>Z</k:mi><k:mi>X</k:mi></k:math> and <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><n:mi>X</n:mi><n:mi>Y</n:mi></n:math> codes. Published by the American Physical Society 2024

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