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arXiv:2504.19041 · Physical Review A

Phases of Floquet code under local decoherence

Yuchen Tang, Yimu Bao

Published Oct 27, 2025• 1 citation
arXivHTMLPDF

Abstract

Floquet code is a dynamical quantum memory with a periodically evolving logical space. As a defining feature, the code exhibits an anyon automorphism after each period, giving rise to a nontrivial evolution of each logical state. In this paper, we study the Floquet code under local decoherence and perfect measurements and demonstrate that, below the decoherence threshold, the code is in a robust phase characterized by the anyon automorphism. We first derive the three-dimensional (3D) statistical mechanics model for the maximum likelihood decoder of the two-dimensional (2D) Floquet code under local Pauli decoherence. We identify a class of two-qubit Pauli channels under which the 3D statistical mechanics model becomes decoupled 2D models and obtain the threshold for such decoherence channels. We then propose a diagnostic of the anyon automorphism in the presence of local decoherence. We analytically show that this diagnostic distinguishes the Floquet code from the toric code under repeated syndrome measurements and undergoes a phase transition at the threshold.

DOI
10.1103/sylh-bb3w

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Floquet code is a dynamical quantum memory with a periodically evolving logical space. As a defining feature, the code exhibits an anyon automorphism after each period, giving rise to a nontrivial evolution of each logical state. In this paper, we study the Floquet code under local decoherence and perfect measurements and demonstrate that, below the decoherence threshold, the code is in a robust phase characterized by the anyon automorphism. We first derive the three-dimensional (3D) statistical mechanics model for the maximum likelihood decoder of the two-dimensional (2D) Floquet code under local Pauli decoherence. We identify a class of two-qubit Pauli channels under which the 3D statistical mechanics model becomes decoupled 2D models and obtain the threshold for such decoherence channels. We then propose a diagnostic of the anyon automorphism in the presence of local decoherence. We analytically show that this diagnostic distinguishes the Floquet code from the toric code under repeated syndrome measurements and undergoes a phase transition at the threshold.
Topological order is a promising basis for quantum error correction, a key milestone towards large-scale quantum computing. Floquet codes provide a dynamical scheme for this while also exhibiting Floquet-enriched topological order where anyons periodically undergo a measurement-induced automorphism that acts uniformly in space. We study disordered Floquet codes where automorphisms have a spatiotemporally heterogeneous distribution: the automorphisms “compete.” We characterize the effect of this competition, showing how key features of the purification dynamics of mixed code states can be inferred from anyon and automorphism properties for any Abelian topological order. This perspective can explain the preservation or measurement of logical information in a dynamic automorphism (DA) code when subjected to a noise model of missing measurements. We demonstrate this using a DA color code with perturbed measurement sequences. The framework of competing automorphisms captures essential features of Floquet codes and robustness to noise, and may elucidate key mechanisms involving topological order, automorphisms, and fault tolerance.
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Floquet codes are a novel class of quantum error-correcting codes with dynamically generated logical qubits arising from a periodic schedule of noncommuting measurements. We utilize the interpretation of measurements in terms of condensation of topological excitations and the rewinding of measurement sequences to engineer new examples of Floquet codes. In particular, rewinding is advantageous for obtaining a desired set of instantaneous stabilizer groups on both toric and planar layouts. Our first example is a Floquet code with instantaneous stabilizer codes that have the same topological order as the three-dimensional (3D) toric code(s). This Floquet code also exhibits a splitting of the topological order of the 3D toric code under the associated sequence of measurements, i.e., an instantaneous stabilizer group of a single copy of the 3D toric code in one round transforms into an instantaneous stabilizer group of two copies of the 3D toric code up to nonlocal stabilizers in the following round. We further construct boundaries for this 3D code and argue that stacking it with two copies of the 3D subsystem toric code allows for a transversal implementation of the logical non-Clifford controlled-controlled- Z gate. We also show that the coupled-layer construction of the X-cube Floquet code can be modified by a rewinding schedule such that each of the instantaneous stabilizer codes is finite depth equivalent to the X-cube model up to toric codes; the X-cube Floquet code exhibits a splitting of the X-cube model into a copy of the X-cube model and toric codes under the measurement sequence. Our final 3D example is a generalization of the 2D Floquet toric code on the honeycomb lattice to three dimensions, which has instantaneous stabilizer codes with the same topological order as the 3D fermionic toric code. Published by the American Physical Society 2024
We study the protection of information in nearly critical topological quantum codes, constructed by perturbing topological stabilizer codes towards continuous quantum phase transitions. Our focus is on the transverse-field toric code subjected to local Pauli decoherence. Despite the strong quantum fluctuations of anyons when the transverse field is tuned infinitesimally close to the critical point, we show that a finite strength of Pauli decoherence remains necessary to irreversibly destroy information encoded in the ground-state manifold. Using a replica statistical physics mapping for the coherent information, we show that decoherence can be understood as introducing a two-dimensional inter-replica defect within a three-dimensional replica statistical physics model. A field theoretical analysis shows that this defect is perturbatively irrelevant to the bulk critical point, and cannot renormalize the transverse field strength, leading to a finite error threshold. We argue that a qualitatively similar conclusion can be drawn for a broad class of nearly critical topological codes, under a variety of decoherence channels.
Inspired by the coupled-layer construction of the X-Cube model, we introduce the X-Cube Floquet code, a dynamical quantum error-correcting code where the number of encoded logical qubits grows with system size. The X-Cube Floquet code is defined on a three-dimensional lattice, built from intersecting two-dimensional layers in the xyxyxy, yzyzyz, and xzxzxz directions, and consists of a periodic sequence of two-qubit measurements which couple the layers together. Within a single Floquet cycle, the codespace switches between that of the X-Cube fracton order and layers of entangled, two-dimensional toric codes. The encoded logical qubits' dynamics are analyzed, and we argue that the new code has a non-zero error threshold. We provide a new Hamiltonian realization of the X-Cube model and, more generally, explore the phase diagram related to the sequence of measurements that define the X-Cube Floquet code.
We introduce a classification of quantum phases of mixed states based on the ability to preserve and fault tolerantly transfer information. We formulate phase equivalence between sets of states that encode logical quantum information and are connected by shallow local channel circuits assisted by local measurements and global classical communication of measurement outcomes, where all quantum operations, including measurements, must be robust to noise. We show that the existence of a recovery threshold for local noise is a property of an entire phase. We prove that any two sets of states with nonzero thresholds that are related by finite-depth local channel circuits belong to the same fault tolerant phase. Our approach leads to a classification of subsystem codes without favoring any state of the gauge subsystem, and we demonstrate phase equivalence between 2D and 3D color codes using dimensional jumps within the 3D gauge color code. Lastly, we study how syndrome structure affects fault tolerance in circuits with measurements. We show that qudit stabilizer codes with extended syndromes permit single-shot state preparation, whereas a broad class of shallow circuits measuring point-like syndromes, including state preparation for 2D topological codes and non-Abelian topological orders with solvable anyon theories, is not fault tolerant.