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arXiv:2610.40357

Subsystem Spacetime Code

Esther Xiaozhen Fu, Aranya Chakraborty, Daniel Gottesman

quant-phSubmitted Oct 29, 2026• Announced Oct 30, 2026• 0 citations
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Abstract

Quantum error-correcting codes provide a powerful framework for analyzing and designing fault-tolerant Clifford circuits. In this work, we construct the subsystem spacetime code and use it to characterize the distances of Clifford circuits. We show that its gauge group can be explicitly constructed and classified using the masking properties of the instantaneous stabilizer groups and consists precisely of faults that are harmless to the circuit. We use this framework to identify where a dynamical code loses distance and to analyze how the fault distance changes when circuits are composed from smaller components. Our results provide a versatile tool for analyzing and designing fault-tolerant circuits, with potential applications to the construction of new dynamical codes.

AI Summary

The paper asks how to characterize fault distance for Clifford circuits with Pauli measurements when instantaneous stabilizer codes change and measurements can destroy syndrome information. It constructs a subsystem spacetime code whose gauge generators combine instantaneous stabilizers, destabilizers of masked stabilizers, and Pauli propagation relations that commute with intervening measurements. Masking distinguishes syndrome information that is read out, irreversibly lost, or recoverable by future measurements. For the main steady-state construction, the gauge-group center equals the detector-stabilizer space, and gauge operators are precisely undetectable faults with trivial logical action. Consequently, subsystem-code distance equals circuit fault distance. The proofs use symplectic linear algebra, explicit gauge-pair construction, and reduction of undetectable faults to final-cut logical representatives. Maximum-likelihood circuit decoding becomes gauge-coset decoding. For measurement-only circuits satisfying a residual-error correctability condition, $d=\min\{d_U,d_{\mathrm{meas}}\}$, separating instantaneous unmasked distance from measurement-induced logical faults. A 12-qubit counterexample has $d_U=4$ but spacetime distance $3$. For circuit composition, a bounded detector time span $\Delta$ permits determining new interface faults within a window extending $(m-1)(\Delta-1)$ cuts on either side, where $m$ is the smaller component distance after boundary reclassification. The main construction assumes preserved logical information and perfect terminal stabilizer readout; extensions discuss logical measurements, preparation, reset, discard, and alternative measurement-fault models. Preparation requires a modified distance definition. Measurement errors have weight two in the default Pauli-pair model, versus one with explicit measurement locations. The distance decomposition is conditional, Clifford propagation can spread errors, and non-Clifford generalization remains open.

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These agents were used to run numerics to build intuition, to verify proofs, generate figures and to edit the writing of some sections, in accordance with the authors’ instructions.

Error found: The state-preparation extension explicitly requires a different distance functional, so the equality between circuit fault distance and ordinary subsystem-code distance does not extend unchanged to that setting. Framework (c) retains individual-qubit spacetime weight and therefore does not, as written, reproduce standard circuit-level counting in which one faulty two-qubit gate can produce a weight-two Pauli at cost one. The gauge-group definition incorrectly asserts a direct sum between the instantaneous gauge spaces and the commuting-propagator space, although these spaces intersect across an idle block.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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43 pages, 8 figures

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Quantum error-correcting codes provide a powerful framework for analyzing and designing fault-tolerant Clifford circuits. In this work, we construct the subsystem spacetime code and use it to characterize the distances of Clifford circuits. We show that its gauge group can be explicitly constructed and classified using the masking properties of the instantaneous stabilizer groups and consists precisely of faults that are harmless to the circuit. We use this framework to identify where a dynamical code loses distance and to analyze how the fault distance changes when circuits are composed from smaller components. Our results provide a versatile tool for analyzing and designing fault-tolerant circuits, with potential applications to the construction of new dynamical codes.
The paper asks how to characterize fault distance for Clifford circuits with Pauli measurements when instantaneous stabilizer codes change and measurements can destroy syndrome information. It constructs a subsystem spacetime code whose gauge generators combine instantaneous stabilizers, destabilizers of masked stabilizers, and Pauli propagation relations that commute with intervening measurements. Masking distinguishes syndrome information that is read out, irreversibly lost, or recoverable by future measurements.
For the main steady-state construction, the gauge-group center equals the detector-stabilizer space, and gauge operators are precisely undetectable faults with trivial logical action. Consequently, subsystem-code distance equals circuit fault distance. The proofs use symplectic linear algebra, explicit gauge-pair construction, and reduction of undetectable faults to final-cut logical representatives. Maximum-likelihood circuit decoding becomes gauge-coset decoding.
For measurement-only circuits satisfying a residual-error correctability condition, d=min⁡{dU,dmeas}d=\min\{d_U,d_{\mathrm{meas}}\}d=min{dU​,dmeas​}, separating instantaneous unmasked distance from measurement-induced logical faults. A 12-qubit counterexample has dU=4d_U=4dU​=4 but spacetime distance 333. For circuit composition, a bounded detector time span Δ\DeltaΔ permits determining new interface faults within a window extending (m−1)(Δ−1)(m-1)(\Delta-1)(m−1)(Δ−1) cuts on either side, where mmm is the smaller component distance after boundary reclassification.
The main construction assumes preserved logical information and perfect terminal stabilizer readout; extensions discuss logical measurements, preparation, reset, discard, and alternative measurement-fault models. Preparation requires a modified distance definition. Measurement errors have weight two in the default Pauli-pair model, versus one with explicit measurement locations. The distance decomposition is conditional, Clifford propagation can spread errors, and non-Clifford generalization remains open.
We present a unified algebraic theory of Clifford spacetime quantum fault tolerance, which underpins universal fault-tolerant quantum computation. We construct a spacetime chain complex from a tensor network representation of a circuit that encodes fault propagation, detector syndromes, and logical actions through the associated gauge and stabilizer structure. We show that this theory (1) unifies various formulations of quantum fault tolerance including spacetime codes, Gottesman's gadget framework, fault complexes, and ZX calculus; (2) connects spacetime distances to the gadget-level correctness properties in Gottesman's formalism; (3) naturally accommodates a range of fault models, in particular phenomenological and circuit-level models, bringing these often separately formulated descriptions into a common algebraic framework through different tensor decompositions and fault assignments; and (4) recovers many known fault tolerance constructions and results as special cases, including repeated syndrome extraction, correlated detectors for transversal and fold-transversal gates, Steane- and Knill-type syndrome extraction gadgets, and single-shot error correction.
Recent advances in quantum error-correction (QEC) have shown that it is often beneficial to understand fault-tolerance as a dynamical process, a circuit with redundant measurements that help correct errors, rather than as a static code equipped with a syndrome extraction circuit. Spacetime codes have emerged as a natural framework to understand error correction at the circuit level while leveraging the traditional QEC toolbox. Here, we introduce a framework based on chain complexes and chain maps to model spacetime codes and transformations between them. We show that stabilizer codes, quantum circuits, and decoding problems can all be described using chain complexes, and that the equivalence of two spacetime codes can be characterized by specific maps between chain complexes, the fault-tolerant maps, that preserve the number of encoded qubits, fault distance, and minimum-weight decoding problem. As an application of this framework, we extend the foliated cluster state construction from stabilizer codes to any spacetime code, showing that any Clifford circuit can be transformed into a measurement-based protocol with the same fault-tolerant properties. To this protocol, we associate a chain complex which encodes the underlying decoding problem, generalizing previous cluster state complex constructions. Our method enables the construction of cluster states from non-CSS, subsystem, and Floquet codes, as well as from logical Clifford operations on a given code.
A powerful method for analyzing quantum error-correcting codes is to map them onto classical statistical mechanics models. Such mappings have thus far mostly focused on static codes, possibly subject to repeated syndrome measurements. Recent progress in quantum error correction, however, has prompted new paradigms where codes emerge from stabilizer circuits in spacetime -- a unifying perspective encompassing syndrome extraction circuits of static codes, dynamically generated codes, and logical operations. We show how to construct statistical mechanical models for stabilizer circuits subject to independent Pauli errors, by mapping logical equivalence class probabilities of errors to partition functions using the spacetime subsystem code formalism. We also introduce a modular language of spin diagrams for constructing the spin Hamiltonians, which we describe in detail focusing on independent circuit-level X-Z error channels. With the repetition and toric codes as examples, we use our approach to analytically rank logical error rates and thresholds between code implementations with standard and dynamic syndrome extraction circuits, describe the effect of transversal logical Clifford gates on logical error rates, and perform Monte Carlo simulations to estimate maximum likelihood thresholds. Our framework offers a universal prescription to analyze, simulate, and compare the decoding properties of any stabilizer circuit, while revealing the innate connections between dynamical quantum systems and noise-resilient phases of matter.
We ask what is the general framework for a quantum error correcting code that is defined by a sequence of measurements. Recently, there has been much interest in Floquet codes and space-time codes. In this work, we define and study the distance of a dynamical code. This is a subtle concept and difficult to determine: At any given time, the system will be in a subspace which forms a quantum error-correcting code with a given distance, but the full error correction capability of that code may not be available due to the schedule of measurements associated with the code. We address this challenge by developing an algorithm that tracks information we have learned about the error syndromes through the protocol and put that together to determine the distance of a dynamical code, in a non-fault-tolerant context. We use the tools developed for the algorithm to analyze the initialization and masking properties of a generic Floquet code. Further, we look at properties of dynamical codes under the constraint of geometric locality with a view to understand whether the fundamental limitations on logical gates and code parameters imposed by geometric locality for traditional codes can be surpassed in the dynamical paradigm. We find that codes with a limited number of long range connectivity will not allow non-Clifford gates to be implemented with finite depth circuits in the 2D setting.
We propose a scheme for detecting and correcting faults in any Clifford circuit. The scheme is based on the observation that the set of all possible outcome bit-strings of a Clifford circuit is a linear code, which we call the outcome code. From the outcome code we construct a corresponding stabilizer code, the spacetime code. Our construction extends the circuit-to-code construction of Bacon, Flammia, Harrow and Shi [2], revisited recently by Gottesman [16], to include intermediate and multi-qubit measurements. With this correspondence, we reduce the problem of correcting faults in a circuit to the well-studied problem of correcting errors in a stabilizer code. More precisely, a most likely error decoder for the spacetime code can be transformed into a most likely fault decoder for the circuit. We give efficient algorithms to construct the outcome and spacetime codes. We also identify conditions under which these codes are LDPC, and give an algorithm to generate low-weight checks, which can then be combined with effcient LDPC code decoders.
Benchmarking physical devices and verifying logical algorithms are important tasks for scalable fault-tolerant quantum computing. Numerous protocols exist for benchmarking devices before running actual algorithms. In this work, we show that both physical and logical errors of fault-tolerant circuits can even be characterized in-situ using syndrome data. To achieve this, we map general fault-tolerant Clifford circuits to subsystem codes using the spacetime code formalism and develop a scheme for estimating Pauli noise in Clifford circuits using syndrome data. We give necessary and sufficient conditions for the learnability of physical and logical noise from given syndrome data, and show that we can accurately predict logical fidelities from the same data. Importantly, our approach requires only a polynomial sample size, even when the logical error rate is exponentially suppressed by the code distance, and thus gives an exponential advantage against methods that use only logical data such as direct fidelity estimation. We demonstrate the practical applicability of our methods in various scenarios using synthetic data as well as the experimental data from a recent demonstration of fault-tolerant circuits by Bluvstein et al. [Nature 626, 7997 (2024)]. Our methods provide an efficient, in-situ way of characterizing a fault-tolerant quantum computer to help gate calibration, improve decoding accuracy, and verify logical circuits.