The XYZ$^2$ hexagonal stabilizer code

Basudha Srivastava1, Anton Frisk Kockum2, and Mats Granath1

1Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden
2Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Gothenburg, Sweden

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Abstract

We consider a topological stabilizer code on a honeycomb grid, the "XYZ$^2$" code. The code is inspired by the Kitaev honeycomb model and is a simple realization of a "matching code" discussed by Wootton [J. Phys. A: Math. Theor. 48, 215302 (2015)], with a specific implementation of the boundary. It utilizes weight-six ($XYZXYZ$) plaquette stabilizers and weight-two ($XX$) link stabilizers on a planar hexagonal grid composed of $2d^2$ qubits for code distance $d$, with weight-three stabilizers at the boundary, stabilizing one logical qubit. We study the properties of the code using maximum-likelihood decoding, assuming perfect stabilizer measurements. For pure $X$, $Y$, or $Z$ noise, we can solve for the logical failure rate analytically, giving a threshold of 50%. In contrast to the rotated surface code and the XZZX code, which have code distance $d^2$ only for pure $Y$ noise, here the code distance is $2d^2$ for both pure $Z$ and pure $Y$ noise. Thresholds for noise with finite $Z$ bias are similar to the XZZX code, but with markedly lower sub-threshold logical failure rates. The code possesses distinctive syndrome properties with unidirectional pairs of plaquette defects along the three directions of the triangular lattice for isolated errors, which may be useful for efficient matching-based or other approximate decoding.

Quantum computers are more sensitive to noise than their classical digital counterparts. For the latter, errors are bit-flip errors (0 flipped to 1, or vice versa), whereas for the former, phase-flip errors (e.g. 0+1 flipped to 0-1) are also an issue since quantum bits (qubits) can be in a superposition of 0 and 1. Topological stabilizer codes are quantum error-correcting codes that store quantum information in logical qubits, consisting of groups of physical qubits, and protect against errors by repeated local measurements. For biased noise, for example, if phase-flip errors are more likely than bit-flip errors, these stabilizer measurements can be modified to increase the threshold of the code, that is, the physical error rate below which the logical error rate decreases by increasing the number of physical qubits in the code. We propose a stabilizer code implemented on a hexagonal (honeycomb) grid of physical qubits and show that, under the assumption of perfect measurements, the code possesses high threshold values for highly biased noise. Building a quantum computer with a high connectivity that allows for measuring the six qubits of a hexagon may thus provide an advantage over lower-connectivity structures such as a square lattice.

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Cited by

[1] Jonathan F. San Miguel, Dominic J. Williamson, and Benjamin J. Brown, "A cellular automaton decoder for a noise-bias tailored color code", arXiv:2203.16534.

[2] Karl Hammar, Alexei Orekhov, Patrik Wallin Hybelius, Anna Katariina Wisakanto, Basudha Srivastava, Anton Frisk Kockum, and Mats Granath, "Error-rate-agnostic decoding of topological stabilizer codes", Physical Review A 105 4, 042616 (2022).

[3] James R. Wootton, "Hexagonal matching codes with two-body measurements", Journal of Physics A Mathematical General 55 29, 295302 (2022).

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