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arXiv:2509.09603

Fault-tolerant transformations of spacetime codes

Arthur Chalom Pesah, Austin K. Daniel, Ilan Tzitrin, Michael Vasmer

Published Sep 11, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2509.09603

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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.
Fault-tolerant quantum computation is inherently a spacetime problem, requiring not merely good static quantum error-correcting codes but also low-overhead protocols for protecting and manipulating encoded quantum information over time. Fault complexes provide a homological formalism for treating such protocols as single spacetime objects. Here we initiate the study of low-overhead fault complexes by introducing \emph{spacetime lifting}, a method that constructs fault complexes from symmetry-reduced product structures beyond standard foliation. We show that spacetime lifting yields fault complexes and in particular memory experiments with almost-linear fault distance in the total spacetime cost, which substantially outperforms existing constructions. We further endow fault complexes with the operational interpretation of measurement-based cluster state protocols and identify general conditions under which they realize fault-tolerant logical teleportation, showing that spacetime-lifted constructions combine favorable parameter scaling with operational realizations. Our work establishes a systematic route towards more efficient quantum fault tolerance through general complex constructions.
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.
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