• Open Access

Topological and Subsystem Codes on Low-Degree Graphs with Flag Qubits

Christopher Chamberland, Guanyu Zhu, Theodore J. Yoder, Jared B. Hertzberg, and Andrew W. Cross
Phys. Rev. X 10, 011022 – Published 31 January 2020

Abstract

In this work we introduce two code families, which we call the heavy-hexagon code and the heavy-square code. Both code families are implemented by assigning physical data and ancilla qubits to both vertices and edges of low-degree graphs. Such a layout is particularly suitable for superconducting qubit architectures to minimize frequency collisions and cross talk. In some cases, frequency collisions can be reduced by several orders of magnitude. The heavy-hexagon code is a hybrid surface and Bacon-Shor code mapped onto a (heavy-) hexagonal lattice, whereas the heavy-square code is the surface code mapped onto a (heavy-) square lattice. In both cases, the lattice includes all the ancilla qubits required for fault-tolerant error correction. Naively, the limited qubit connectivity might be thought to limit the error-correcting capability of the code to less than its full distance. Therefore, essential to our construction is the use of flag qubits. We modify minimum-weight perfect-matching decoding to efficiently and scalably incorporate information from measurements of the flag qubits and correct up to the full code distance while respecting the limited connectivity. Simulations show that high threshold values for both codes can be obtained using our decoding protocol. Further, our decoding scheme can be adapted to other topological code families.

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  • Received 30 July 2019
  • Revised 28 October 2019

DOI:https://doi.org/10.1103/PhysRevX.10.011022

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

Christopher Chamberland*, Guanyu Zhu, Theodore J. Yoder, Jared B. Hertzberg, and Andrew W. Cross

  • IBM T. J. Watson Research Center, Yorktown Heights, New York 10598, USA

  • *mathematicschris@gmail.com
  • guanyu.zhu@ibm.com

Popular Summary

Quantum information is very sensitive to its environment, which can easily corrupt the data of a computation. One way to mitigate the effects of noise is through fault-tolerant quantum-error correction, where redundancy is used to detect and correct errors when they occur. With the advent of new quantum devices, it is important to devise error-correction schemes that take realistic hardware architecture constraints into account. We propose new families of error-correcting codes and fault-tolerant methods to implement such codes in a scalable way, which are designed to be suitable for superconducting qubit architectures. We show that under realistic noise models, such codes exhibit competitive performance.

In particular, a significant challenge for practical designs of solid-state quantum computers (such as those based on superconducting qubits) is to reduce issues caused by crosstalk between different qubits. Crosstalk is problematic since it can easily lead to malignant errors that cannot be mitigated even with large distance codes.

We design two families of error-correcting codes where qubits are laid out on a low-degree graph; that is, we minimize the degree of connectivity between qubits. To make such schemes fault tolerant, we develop a new scalable and efficient decoding algorithm that incorporates information from flag qubits (additional qubits used to reduce the degree of the connectivity between data and syndrome measurement qubits) to correct correlated error patterns arising from a small number of faults.

The flag qubit schemes developed in our work have broad applications, and we believe it would be interesting to apply such schemes to other topological code families, in addition to code families with constant encoding rates.

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Vol. 10, Iss. 1 — January - March 2020

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