The XYZ<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi/><mml:mn>2</mml:mn></mml:msup></mml:math> hexagonal stabilizer code

Abstract

We consider a topological stabilizer code on a honeycomb grid, the "XYZ<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi/><mml:mn>2</mml:mn></mml:msup></mml:math>" 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 (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mi>Z</mml:mi><mml:mi>X</mml:mi><mml:mi>Y</mml:mi><mml:mi>Z</mml:mi></mml:math>) plaquette stabilizers and weight-two (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi><mml:mi>X</mml:mi></mml:math>) link stabilizers on a planar hexagonal grid composed of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> qubits for code distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>, 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math>, or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> only for pure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math> noise, here the code distance is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> for both pure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> and pure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Y</mml:mi></mml:math> noise. Thresholds for noise with finite <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> 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.

Topics & Concepts