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Physical Review A

Dynamical logical qubits in the Bacon-Shor code

M. Sohaib Alam, Eleanor Rieffel

Published Jul 18, 2025• 9 citations
Publisher

Abstract

Floquet error-correcting codes usually involve non-trivial geometries and encode a small number of logical qubits, but here the authors introduce a Floquet code construction based on the simple Bacon-Shor code, defined on a square lattice, encoding many more logical qubits than previously thought. This opens a new door for dynamical code design.

DOI
10.1103/nfxv-3dp7

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Floquet error-correcting codes usually involve non-trivial geometries and encode a small number of logical qubits, but here the authors introduce a Floquet code construction based on the simple Bacon-Shor code, defined on a square lattice, encoding many more logical qubits than previously thought. This opens a new door for dynamical code design.
The Bacon-Shor code is a quantum error correcting subsystem code composed of weight 2 check operators that admits a single logical qubit, and has distance ddd on a d×dd \times dd×d square lattice. We show that when viewed as a Floquet code, by choosing an appropriate measurement schedule of the check operators, it can additionally host several dynamical logical qubits. Specifically, we identify a period 4 measurement schedule of the check operators that preserves logical information between the instantaneous stabilizer groups. Such a schedule measures not only the usual stabilizers of the Bacon-Shor code, but also additional stabilizers that protect the dynamical logical qubits against errors. We show that the code distance of these Floquet-Bacon-Shor codes scales as Θ(d/k)Θ(d/\sqrt{k})Θ(d/k​) on an n=d×dn = d \times dn=d×d lattice with kkk dynamical logical qubits, along with the logical qubit of the parent subsystem code. Unlike the usual Bacon-Shor code, the Floquet-Bacon-Shor code family introduced here can therefore saturate the subsystem bound kd=O(n)kd = O(n)kd=O(n). Moreover, several errors are shown to be self-corrected purely by the measurement schedule itself. This work provides insights into the design space for dynamical codes and expands the known approaches for constructing Floquet codes.
Quantum error correction (QEC) protects quantum systems against inevitable noises and control inaccuracies, providing a pathway toward fault-tolerant (FT) quantum computation. Stabilizer codes, including surface code and color code, have long been the focus of research and have seen significant experimental progress in recent years. Recently proposed time-dynamical QEC, including Floquet codes and generalized time-dynamical code implementations, opens up new opportunities for FT quantum computation. By employing a periodic schedule of low-weight parity checks, Floquet codes can generate additional dynamical logical qubits, offering enhanced error correction capabilities and, potentially, higher code performance. Here, we experimentally implement the Floquet-Bacon-Shor code on a superconducting quantum processor. We encode a dynamical logical qubit within a 3×3 lattice of data qubits alongside a conventional static logical qubit. We demonstrate FT encoding and measurement of the two-qubit logical states and stabilize these states using repeated error detection. We showcase universal single-qubit logical gates on the dynamical qubit. Furthermore, by implementing a logical controlled-not (cnot) gate, we entangle the dynamical and static logical qubits, generating an error-detected logical Bell state with a fidelity of 75.9%. Our results highlight the potential of Floquet codes for resource-efficient FT quantum computation.
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.
Dynamical quantum error-correcting codes (QECC) offer wider possibilities in how one can protect logical quantum information from noise and perform fault-tolerant quantum computation compared to static QECCs. A family of dynamical QECCs called the ``Floquet codes'' consists of a periodic sequence of two-body measurements that enables error-correction on many-body systems, relaxing hardware implementation requirements and improving error-correction reliability. Existing results on Floquet codes has been focused on qubits, two-level quantum systems, with very little attention given on higher dimensional quantum systems, or qudits. We bridge this gap by proposing a simple, yet general construction of qudit Floquet codes based on a simple set of conditions on the sequence two-body measurements defining the code. Moreover, this construction applies to a large family of configurations of qudits on the vertices of a three-colorable lattice which connectivity represented by the edges. We show that this construction includes the existing constructions of both qubit and qudit Floquet codes as special cases. In addition, any qudit Floquet code obtained by our construction achieves a rate of encoded logical qudits over physical qudits approaching 12\frac{1}{2}21​ as the number of physical qudits in total and on the faces of the lattice grows larger, as opposed to vanishing rate in existing qudit Floquet code constructions.
A central goal in quantum error correction is to reduce the overhead of fault-tolerant quantum computing by increasing noise thresholds and reducing the number of physical qubits required to sustain a logical qubit. We introduce a potential path towards this goal based on a family of dynamically generated quantum error correcting codes that we call "hyperbolic Floquet codes.'' These codes are defined by a specific sequence of non-commuting two-body measurements arranged periodically in time that stabilize a topological code on a hyperbolic manifold with negative curvature. We focus on a family of lattices for n qubits that, according to our prescription that defines the code, provably achieve a finite encoding rate ( 1 / 8 + 2 / n ) while still requiring only two-body measurements. Similar to hyperbolic surface codes, the distance of the code at each time-step scales at most logarithmically in n . The family of lattices we choose indicates that this scaling is achievable in practice. We develop and benchmark an efficient matching-based decoder that provides evidence of a threshold near 0.1% in a phenomenological noise model and 0.25% in an entangling measurements noise model. Utilizing weight-two check operators and a qubit connectivity of 3, one of our hyperbolic Floquet codes uses 400 physical qubits to encode 52 logical qubits with a code distance of 8, i.e., it is a [ [ 400 , 52 , 8 ] ] code. At small error rates, comparable logical error suppression to this code requires 5x as many physical qubits (1924) when using the honeycomb Floquet code with the same noise model and decoder.
A Bacon-Shor code is a subsystem quantum error-correcting code on an $L\ifmmode\times\else\texttimes\fi{}L$ lattice where the $2(L\ensuremath{-}1)$ weight-2L2L2L stabilizers are usually inferred from the measurements of $2L(L\ensuremath{-}1)$ weight-2 gauge operators. Here, we show that the stabilizers can be measured directly and fault tolerantly with bare ancillary qubits by constructing circuits that follow the pattern of gauge operators. We then examine the implications of this method for small quantum error-correcting codes by comparing distance-3 versions of the rotated surface code and the Bacon-Shor code with the standard depolarizing model and in the context of a trapped-ion quantum computer. We find that for a simple circuit of prepare, error correct, and measure, the Bacon-Shor code outperforms the surface code by requiring fewer qubits, taking less time, and having a lower error rate.