Abstract
The color code is remarkable for its ability to perform fault-tolerant logic gates. This motivates the design of practical decoders that minimize the resource cost of color-code quantum computation. Here we propose a decoder for the planar color code with a triangular boundary where we match syndrome defects on a nontrivial manifold that has the topology of a Möbius strip. A basic implementation of our decoder used on the color code with hexagonal lattice geometry demonstrates a logical failure rate that is competitive with the optimal performance of the surface code. The logical failure rate scales approximately like , with , error rate , and the code length. Furthermore, by exhaustively testing over five billion error configurations, we find that a modification of our decoder that manually compares inequivalent recovery operators can correct all errors of weight for codes with distance . Our decoder is derived using relations among the stabilizers that preserve global conservation laws at the lattice boundary. We present generalizations of our method to depolarizing noise and fault-tolerant error correction, as well as to Majorana surface codes, higher-dimensional color codes, and single-shot error correction.
20 More- Received 4 October 2021
- Accepted 9 December 2021
DOI:https://doi.org/10.1103/PRXQuantum.3.010310
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Quantum error-correcting codes are many-body quantum systems that protect information that is processed during a quantum computation. To design a useful, large-scale quantum computer, we need to develop practical quantum error-correcting codes that can be realized with modern technology. The color code offers a number of advantages for two-dimensional fault-tolerant quantum computation. In particular, it has very efficient ways of performing quantum logic gates. To exploit its favorable properties, we need to design better decoding algorithms to improve its tolerance to errors. These are classical algorithms that determine how to correct errors in code experiences.
Here we design a practical decoder with high performance by careful examination of the underlying physics of the color code. We can view quantum error-correcting codes as phases of matter where errors are excitations that give rise to quasiparticles. These quasiparticles respect conservation laws whereby errors create quasiparticles in pairs with respect to some symmetry of the system. From this perspective, decoding then is simply a matching problem, where we aim to pair together quasiparticles to annihilate them, and reverse the effects of the error. In this work, we take a careful look at the conservation laws at the boundaries of the color code. Surprisingly, our examination reveals that the decoding problem for the color code on a planar lattice is mapped onto a manifold with nontrivial topology, namely that of a Möbius strip. Our new decoder demonstrates a measurable improvement in error-correcting performance compared with other matching decoders for the color code.