The XYZ$^2$ hexagonal stabilizer code

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 [1], with a specific implementation of the boundary. It utilizes weight-six (XY ZXY Z) 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.

Hexagonal matching codes with two-body measurements

“…We consider a noise model in which errors occur on each qubit with probability p between rounds, and for which the measurements of stabilizer operators report an incorrect result with probability q. It is known that there is a finite threshold for p the case of q = 0 [20]. We assume the same for q = p, and denote the threshold p c .…”

A cellular automaton decoder for a noise-bias tailored color code

“…Certain qubits have even been designed to maintain their bias as they undergo unitary operations [21][22][23][24][25] . As such, considerable work has been invested in producing tailored quantum error-correcting codes together with specialized decoding algorithms that concentrate on correcting common types of error [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] . In the limit of very high bias, certain codes have been shown to reproduce the restricted dynamics of the quasiparticle excitations of fracton topological codes in lowerdimensional systems 37,38,42 .…”

“…As such, considerable work has been invested in producing tailored quantum error-correcting codes together with specialized decoding algorithms that concentrate on correcting common types of error [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] . In the limit of very high bias, certain codes have been shown to reproduce the restricted dynamics of the quasiparticle excitations of fracton topological codes in lowerdimensional systems 37,38,42 . These dynamics can be understood in terms of the materialized symmetries 6,15 , or more generally the system symmetries of a code together with its noise model 37 .…”