Topological error correcting processes from fixed-point path integrals

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.

arXiv commentversion accepted for publication in Quantum

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Cited by in corpus (12)

• Constructions and performance of hyperbolic and semi-hyperbolic Floquet codes
• Quantum computation from dynamic automorphism codes
• Perturbative stability and error correction thresholds of quantum codes
• Dynamical Logical Qubits in the Bacon-Shor Code
• XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime
• Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases
• Error Correction in Dynamical Codes
• Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma
• Competing automorphisms and disordered Floquet codes
• Universal fault tolerant quantum computation in 2D without getting tied in knots
• Logical operations with a dynamical qubit in Floquet-Bacon-Shor code
• Planar fault-tolerant circuits for non-Clifford gates on the 2D color code