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arXiv:2410.18265

Floquet Codes from Coupled Spin Chains

Bowen Yan, Penghua Chen, Shawn Xingshan Cui

Published Oct 23, 2024• 0 citations
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Abstract

We propose a novel construction of the Floquet 3D toric code and Floquet $X$-cube code through the coupling of spin chains. This approach not only recovers the coupling layer construction on foliated lattices in three dimensions but also avoids the complexity of coupling layers in higher dimensions, offering a more localized and easily generalizable framework. Our method extends the Floquet 3D toric code to a broader class of lattices, aligning with its topological phase properties. Furthermore, we generalize the Floquet $X$-cube model to arbitrary manifolds, provided the lattice is locally cubic, consistent with its Fractonic phases. We also introduce a unified error-correction paradigm for Floquet codes by defining a subgroup, the Steady Stabilizer Group (SSG), of the Instantaneous Stabilizer Group (ISG), emphasizing that not all terms in the ISG contribute to error correction, but only those terms that can be referred to at least twice before being removed from the ISG. We show that correctable Floquet codes naturally require the SSG to form a classical error-correcting code, and we present a simple 2-step Bacon-Shor Floquet code as an example, where SSG forms instantaneous repetition codes. Finally, our construction intrinsically supports the extension to $n$-dimensional Floquet $(n,1)$ toric codes and generalized $n$-dimensional Floquet $X$-cube codes.

DOI
10.48550/arxiv.2410.18265

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We propose a novel construction of the Floquet 3D toric code and Floquet XXX-cube code through the coupling of spin chains. This approach not only recovers the coupling layer construction on foliated lattices in three dimensions but also avoids the complexity of coupling layers in higher dimensions, offering a more localized and easily generalizable framework. Our method extends the Floquet 3D toric code to a broader class of lattices, aligning with its topological phase properties. Furthermore, we generalize the Floquet XXX-cube model to arbitrary manifolds, provided the lattice is locally cubic, consistent with its Fractonic phases. We also introduce a unified error-correction paradigm for Floquet codes by defining a subgroup, the Steady Stabilizer Group (SSG), of the Instantaneous Stabilizer Group (ISG), emphasizing that not all terms in the ISG contribute to error correction, but only those terms that can be referred to at least twice before being removed from the ISG. We show that correctable Floquet codes naturally require the SSG to form a classical error-correcting code, and we present a simple 2-step Bacon-Shor Floquet code as an example, where SSG forms instantaneous repetition codes. Finally, our construction intrinsically supports the extension to nnn-dimensional Floquet (n,1)(n,1)(n,1) toric codes and generalized nnn-dimensional Floquet XXX-cube codes.
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
Floquet quantum error-correcting codes provide an operationally economical route to fault tolerance by dynamically generating stabilizer structures using only two-body Pauli measurements. But while it is well established that stabilizer codes in higher spatial dimensions gain additional levels of intrinsic robustness, higher-dimensional Floquet codes have hitherto been explored only in limited scope. Here we introduce a 3d generalization of a Floquet code whose instantaneous stabilizer group realizes a 3d fermionic toric code, while crucially preserving all three logical qubits throughout the entire measurement sequence. One central ingredient is the identification of a 3d lattice geometry that generalizes the features of the Kekulé lattice underlying the 2d Hastings-Haah code - specifically, a structure where deleting any one edge color yields a two-color subgraph that decomposes into short, closed loops rather than homologically nontrivial chains. This loop property avoids the collapse of logical information that plagues naive sequential two-color measurement schedules on many 3d lattices. Although, for our lattice geometry, a simple 3-round cycle that sequentially measures the three types of parity checks does not expose the full error syndrome set, we show that one can append a measurement sequence to extract the missing syndromes without disturbing the logical subspace. Beyond code design, 3d tricoordinated lattice geometries define a family of 3d monitored Kitaev models, in which random measurements of the non-commuting parity checks give rise to dynamically created entangled phases with nontrivial topology. In discussing the general structure of their underlying phase diagrams and, in particular, the existence of certain quantum critical points, we again make a connection to the general preservation of logical information in time-ordered Floquet protocols.
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.
Floquet codes define fault-tolerant protocols through periodic measurement sequences that drive a dynamically evolving stabilizer group. They provide a natural framework for hardware supporting two-qubit parity measurements but no unitary entangling gates. However, few known constructions achieve both high encoding rates and high thresholds. We close this gap by introducing Stairway codes, a family of high-rate Floquet protocols obtained by Floquetifying Abelian two-block group algebra codes, a class that includes the bivariate bicycle codes. By representing the static code as a foliated ZX-calculus network within a (w−1)(w{-}1)(w−1)-dimensional space-time lattice and rotating the time axis, we decompose its weight-www stabilizers into a periodic sequence of pairwise measurements. This reduces the design of new codes within this family to the selection of favorable periodic boundary conditions. We identify instances with competitive parameters, analyze their distance under circuit-level noise, and demonstrate logical error rates surpassing those of other Floquet codes at comparable encoding rates. Remarkably, our construction requires fewer than 300 physical qubits to match the distance and encoding rate of semi-hyperbolic Floquet codes that use over 1300 qubits.
We propose a new class of error-correcting dynamic codes in two and three dimensions that has no explicit connection to any parent subsystem code.The two-dimensional code, which we call the CSS (Calderbank-Shor-Steane) honeycomb code, is geometrically similar to that of the honeycomb code by Hastings and Haah and also dynamically embeds an instantaneous toric code.However, unlike the honeycomb code, it possesses an explicit CSS structure and its gauge checks do not form a subsystem code.Nevertheless, we show that our dynamic protocol conserves logical information and possesses a threshold for error correction.We generalize this construction to three dimensions and obtain a code that fault tolerantly alternates between realizing two type-I fracton models, the checkerboard and the X-cube model.Finally, we show the compatibility of our CSS honeycomb-code protocol and the honeycomb code by showing the possibility of randomly switching between the two protocols without information loss while still measuring error syndromes.We call this more general aperiodic structure "dynamic tree codes," which we also generalize to three dimensions.We construct a probabilistic finite automaton prescription that generates dynamic tree codes correcting any single-qubit Pauli errors and can be viewed as a step toward the development of practical fault-tolerant random codes.
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.