Stabilizer Formalism For Operator Algebra Quantum Error Correction

Guillaume Dauphinais¹, David W. Kribs^1,2, and Michael Vasmer^1,3,4

¹Xanadu, Toronto, ON M5G 2C8, Canada
²Department of Mathematics & Statistics, University of Guelph, Guelph, ON N1G 2W1, Canada
³Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
⁴Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada

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Abstract

We introduce a stabilizer formalism for the general quantum error correction framework called operator algebra quantum error correction (OAQEC), which generalizes Gottesman’s formulation for traditional quantum error correcting codes (QEC) and Poulin’s for operator quantum error correction and subsystem codes (OQEC). The construction generates hybrid classical-quantum stabilizer codes and we formulate a theorem that fully characterizes the Pauli errors that are correctable for a given code, generalizing the fundamental theorems for the QEC and OQEC stabilizer formalisms. We discover hybrid versions of the Bacon-Shor subsystem codes motivated by the formalism, and we apply the theorem to derive a result that gives the distance of such codes. We show how some recent hybrid subspace code constructions are captured by the formalism, and we also indicate how it extends to qudits.

Popular summary

Quantum error correction is a central topic in the development of new quantum technologies, with origins as an independent field of study going back almost three decades, and now touching on almost every aspect of quantum information science. More recent developments included the introduction of a framework called ‘operator algebra quantum error correction’ (OAQEC) that generalized previous approaches, while additionally enabling extensions to full blown infinite-dimensional error correction and providing an error correction framework for hybrid codes used for the simultaneous encoding of classical and quantum information. The last few years have witnessed significant renewed interest in OAQEC from a few different directions, including hybrid classical-quantum coding theory, experimental quantum computing, and, somewhat unexpectedly, from black hole theory.

The ‘stabilizer formalism’ is a bedrock of quantum error correction. With its initial formulation introduced in the early days of the field and a subsequent generalization obtained for important ‘subsystem codes’, it provides a toolbox for the construction and characterization of codes for the central class of Pauli error models. In this paper, we introduce a stabilizer formalism for finite-dimensional OAQEC which generalizes the previous formulations. The resulting codes constructed include hybrid classical-quantum stabilizer codes, and motivated by this, we discover hybrid versions of an important class of subsystem codes. We prove a theorem that fully characterizes the error sets that are correctable for a given stabilizer code, generalizing the fundamental theorems from previous settings. We also present several examples and show how some recent hybrid code constructions are captured by the formalism.

► BibTeX data

@article{Dauphinais2024stabilizerformalism, doi = {10.22331/q-2024-02-21-1261}, url = {https://doi.org/10.22331/q-2024-02-21-1261}, title = {Stabilizer {F}ormalism for {O}perator {A}lgebra {Q}uantum {E}rror {C}orrection}, author = {Dauphinais, Guillaume and Kribs, David W. and Vasmer, Michael}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1261}, month = feb, year = {2024} }

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[2] ChunJun Cao, “Stabilizer Codes Have Trivial Area Operators”, arXiv:2306.14996, (2023).

[3] Abhijeet Alase, Kevin D. Stubbs, Barry C. Sanders, and David L. Feder, “Exponential suppression of Pauli errors in Majorana qubits via quasiparticle detection”, arXiv:2307.08896, (2023).

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Stabilizer Formalism for Operator Algebra Quantum Error Correction

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