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arXiv:2410.12963 · Physical Review A

Single-shot and measurement-based quantum error correction via fault complexes

Timo Hillmann, Guillaume Dauphinais, Ilan Tzitrin, Michael Vasmer

Published Sep 16, 2025• 5 citations
arXivHTMLPDF

Abstract

Photonics provides a viable path to a scalable fault-tolerant quantum computer. The natural framework for this platform is measurement-based quantum computation, where fault-tolerant graph states supersede traditional quantum error-correcting codes. However, the existing formalism for foliation—the construction of fault-tolerant graph states—does not reveal how certain properties, such as single-shot error correction, manifest in the measurement-based setting. We introduce the fault complex, a representation of dynamic quantum error-correction protocols particularly well suited to describe foliation. Our approach enables precise computation of fault tolerance properties of foliated codes and provides insights into circuit-based quantum computation. Analyzing the fault complex yields improved thresholds for three- and four-dimensional toric codes, a generalization of stability experiments, and the existence of single-shot lattice surgery with higher-dimensional topological codes.

DOI
10.1103/cjb4-l57n

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Photonics provides a viable path to a scalable fault-tolerant quantum computer. The natural framework for this platform is measurement-based quantum computation, where fault-tolerant graph states supersede traditional quantum error-correcting codes. However, the existing formalism for foliation—the construction of fault-tolerant graph states—does not reveal how certain properties, such as single-shot error correction, manifest in the measurement-based setting. We introduce the fault complex, a representation of dynamic quantum error-correction protocols particularly well suited to describe foliation. Our approach enables precise computation of fault tolerance properties of foliated codes and provides insights into circuit-based quantum computation. Analyzing the fault complex yields improved thresholds for three- and four-dimensional toric codes, a generalization of stability experiments, and the existence of single-shot lattice surgery with higher-dimensional topological codes.
We present a framework for foliating any Pauli-based quantum error-correcting code over prime-dimensional qudits. For any such code, we obtain a qudit graph state that can be measured to perform fault-tolerant measurement-based quantum computing. Such a paradigm is of interest in platforms such as photonics, where measurement-based protocols are natural and high-dimensional states are readily available. We discuss several examples for arbitrary prime dimension ddd, such as the qudit toric code (stabilizer, CSS), the ddd-dimensional perfect [[5,1,3]][[5,1,3]][[5,1,3]] code (stabilizer, non-CSS), and a straightforward ddd-dimensional generalization of the CSS honeycomb code (dynamical, CSS). Under a simple error model, we numerically calculate thresholds for the foliated qudit toric code and demonstrate that they are comparable to the non-foliated version.
In order to build a scalable quantum computer error correction will be required to reduce the impact of errors. Implementing error correction in the framework of measurement based computation manifests itself as the construction of fault tolerant cluster states (FTCSs). While any 2-dimensional stabilizer code can be used to construct a FTCS through the process of foliation, here we find FTCSs that cannot be constructed through foliation of a stabilizer code, and identify new examples of self-dual codes that can still be implemented in a 2-dimensional physical architecture.
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.
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.
Quantum communication and computation is an emerging field in quantum mechanics. There are many promising applications including secure communications, simulation of complex quantum systems or quantum algorithms which perform faster than classical methods. The main barrier to realising these possibilities is coping with the inevitable errors induced by noise and experimental imperfections. Quantum error correction is a process which allows for the storage and retrieval of encoded qubits with high fidelity even in an imperfect environment. Error correction operates by encoding qubit states into a subspace of a larger system of qubits. The presence of errors is detected by performing stabilizer measurements. These measurements do not act on the encoding subspace and the measurement results, or syndrome, can be used to infer errors. The decoding problem is to use syndrome information to find the most likely errors which have occurred. This is performed using classical computing resources. Once the errors have been inferred operations can be applied to the system to counteract their effect. While many quantum error correcting codes are known not all are suitable in a practical setting. When choosing a potential code we must consider the following. Does the construction use simple interactions that can be physically realised? How high is the physical qubit to encoded qubit overhead? Does an efficient decoder exist? Is the encoding scheme scalable and do the error correcting properties improve with scaling? All of these properties are desirable for a practical scheme, though some compromise may be required. In this thesis I show how cluster states can be used as a resource for implementing CSS quantum codes. Additionally I show how these cluster codes can be linked together in a layered fashion to create foliated codes. These foliated codes can be used with any constituent CSS code as a basis for fault tolerant error correction. This process is an extension of Raussendorf’s toric code measurement based processing scheme. I first generalise this measurement based scheme to all CSS quantum codes, which is presented in chapter 3. I then present a novel decoding scheme for foliated turbo codes, chapter 4, and present the decoding performance of some turbo code families in chapter 5. These turbo codes are a class of finite rate codes whose overhead and complexity are flexible. Chapter 6 presents the decoding performance of foliated bicycle codes. The low overhead of physical resources to logical qubits of both these code families make these foliated schemes an attractive alternative to the toric code when considering practical measurement based computing designs with large numbers of encoded qubits.
Abstract Fault-tolerant protocols and quantum error correction (QEC) are essential to building reliable quantum computers from imperfect components that are vulnerable to errors. Optimizing the resource and time overheads needed to implement QEC is one of the most pressing challenges. Here, we introduce a new topological quantum error-correcting code, the three-dimensional subsystem toric code (3D STC). The 3D STC can be realized with geometrically-local parity checks of weight at most three on the cubic lattice with open boundary conditions. We prove that one round of parity-check measurements suffices to perform reliable QEC with the 3D STC even in the presence of measurement errors. We also propose an efficient single-shot QEC decoding strategy for the 3D STC and numerically estimate the resulting storage threshold against independent bit-flip, phase-flip and measurement errors to bepSTC ≈ 1.045%. Such a high threshold together with local parity-check measurements make the 3D STC particularly appealing for realizing fault-tolerant quantum computing.