• Open Access

Morphing Quantum Codes

Michael Vasmer and Aleksander Kubica
PRX Quantum 3, 030319 – Published 8 August 2022

Abstract

We introduce a morphing procedure that can be used to generate new quantum codes from existing quantum codes. In particular, we morph the 15-qubit Reed-Muller code to obtain a [[10,1,2]] code that is the smallest-known stabilizer code with a fault-tolerant logical T gate. In addition, we construct a family of hybrid color-toric codes by morphing the color code. Our code family inherits the fault-tolerant gates of the original color code, implemented via constant-depth local unitaries. As a special case of this construction, we obtain toric codes with fault-tolerant multiqubit control-Z gates. We also provide an efficient decoding algorithm for hybrid color-toric codes in two dimensions and numerically benchmark its performance for phase-flip noise. We expect that morphing may also be a useful technique for modifying other code families such as triorthogonal codes.

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  • Received 29 December 2021
  • Accepted 13 July 2022

DOI:https://doi.org/10.1103/PRXQuantum.3.030319

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)

Quantum Information, Science & Technology

Authors & Affiliations

Michael Vasmer1,2,* and Aleksander Kubica1,2,3,4

  • 1Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
  • 2Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • 3AWS Center for Quantum Computing, Pasadena, California 91125, USA
  • 4California Institute of Technology, Pasadena, California 91125, USA

  • *mvasmer@perimeterinstitute.ca

Popular Summary

Current quantum computers are on the cusp of being more powerful than their classical counterparts but environmental noise limits the depth of the quantum circuits that can be reliably implemented on them. One solution to this problem is to use quantum error-correcting codes (QECCs), which protect a logical qubit by encoding its state into the state of multiple physical qubits. Using QECCs, one can design fault-tolerant implementations of quantum algorithms, which still give the correct output when executed using error-prone hardware. Therefore, constructing QECCs with good performance and easy implementations of quantum logic gates is an important part of designing fault-tolerant quantum computers.

In our work, we introduce a systematic procedure for modifying QECCs called morphing, which allows us to construct new QECCs from existing ones. We morph the 15-qubit Reed-Muller code to obtain a ten-qubit code with a fault-tolerant implementation of the T gate, a quantum gate that is often difficult to implement using QECCs. Our code is the smallest-known code with a fault-tolerant implementation of this gate. We also morph the color code, obtaining a family of hybrid codes that interpolate between the color code and the toric code. We propose a decoding algorithm for our hybrid codes and observe good performance in numerical simulations.

In the future, we anticipate that morphing could be used to generate interesting new QECCs starting from, for example, triorthogonal codes or pin codes. Also, morphing may help us to understand the connection between topological subsystem codes such as the gauge color code and the subsystem toric code.

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Vol. 3, Iss. 3 — August - October 2022

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It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

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