Topological Error Correcting Processes From Fixed-point Path Integrals

Andreas Bauer

Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

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Abstract

We propose a unifying paradigm for analyzing and constructing topological quantum error correcting codes as dynamical circuits of geometrically local channels and measurements. To this end, we relate such circuits to discrete fixed-point path integrals in Euclidean spacetime, which describe the underlying topological order: If we fix a history of measurement outcomes, we obtain a fixed-point path integral carrying a pattern of topological defects. As an example, we show that the stabilizer toric code, subsystem toric code, and CSS Floquet code can be viewed as one and the same code on different spacetime lattices, and the honeycomb Floquet code is equivalent to the CSS Floquet code under a change of basis. We also use our formalism to derive two new error-correcting codes, namely a Floquet version of the $3+1$-dimensional toric code using only 2-body measurements, as well as a dynamic code based on the double-semion string-net path integral.

Featured image: Toric-code tensor-network path integral as $ZX$ diagram (in black/gray) on cubic lattice (in orange). With time $t$ going upwards, patches shaded in blue become $XXXX$ and $ZZZZ$ measurements. When choosing time to go in the $(1,1,1)$-direction, we get the CSS honeycomb Floquet code where each individual 4-index tensor becomes a $XX$ or $ZZ$ measurement.

Popular summary

Since quantum information is sensitive to noise, scalable quantum computation requires error correction, where the information of a few logical qubits is encoded non-locally in a larger number of physical qubits. A particularly appealing flavor of quantum error correction is topological, where the configurations of physical qubits look like closed-loop pattern. Then, logical quantum information is encoded globally in the homology class, that is, the winding numbers of these loops around non-contractible paths. Traditionally, the codes used for topological error correction are stabilizer codes such as the toric code, consisting of a set of operators that detect errors on the physical qubits. To achieve robustness to noise, these operators are measured over and over again. However, viewing error-correction as a dynamic circuit in spacetime rather than a static stabilizer code offers much richer possibilities for constructing fault-tolerant protocols. This has become apparent especially since the recent discovery so so-called Floquet codes. In this paper, we present a systematic framework to analyze such dynamic fault-tolerant protocols in a unified way and construct new ones. We do this by directly relating error-correcting circuits to discrete path integrals representing the underlying topological phases of matter in spacetime.

► BibTeX data

@article{Bauer2024topologicalerror, doi = {10.22331/q-2024-03-20-1288}, url = {https://doi.org/10.22331/q-2024-03-20-1288}, title = {Topological error correcting processes from fixed-point path integrals}, author = {Bauer, Andreas}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1288}, month = mar, year = {2024} }

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Cited by

[1] Oscar Higgott and Nikolas P. Breuckmann, “Constructions and performance of hyperbolic and semi-hyperbolic Floquet codes”, arXiv:2308.03750, (2023).

[2] Tyler D. Ellison, Joseph Sullivan, and Arpit Dua, “Floquet codes with a twist”, arXiv:2306.08027, (2023).

[3] Arpit Dua, Nathanan Tantivasadakarn, Joseph Sullivan, and Tyler D. Ellison, “Engineering Floquet codes by rewinding”, arXiv:2307.13668, (2023).

[4] Michael Liaofan Liu, Nathanan Tantivasadakarn, and Victor V. Albert, “Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat’s Lemma”, arXiv:2311.18003, (2023).

[5] Margarita Davydova, Nathanan Tantivasadakarn, Shankar Balasubramanian, and David Aasen, “Quantum computation from dynamic automorphism codes”, arXiv:2307.10353, (2023).

[6] Hector Bombin, Chris Dawson, Terry Farrelly, Yehua Liu, Naomi Nickerson, Mihir Pant, Fernando Pastawski, and Sam Roberts, “Fault-tolerant complexes”, arXiv:2308.07844, (2023).

[7] Brenden Roberts, Sagar Vijay, and Arpit Dua, “Geometric phases in generalized radical Floquet dynamics”, arXiv:2312.04500, (2023).

[8] Alex Townsend-Teague, Julio Magdalena de la Fuente, and Markus Kesselring, “Floquetifying the Colour Code”, arXiv:2307.11136, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-03-21 01:37:20). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2024-03-21 01:37:19).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Topological error correcting processes from fixed-point path integrals

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Topological error correcting processes from fixed-point path integrals

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