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arXiv:2607.13784

Foliated Quantum Error Correction for Qudits

Gözde Üstün, Simon J. Devitt, Jason Saied

quant-phSubmitted Jul 15, 2026• Announced Jul 16, 2026• 0 citations
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Abstract

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 $d$, such as the qudit toric code (stabilizer, CSS), the $d$-dimensional perfect $[[5,1,3]]$ code (stabilizer, non-CSS), and a straightforward $d$-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.

Comments
15 Figures in total (5 in the main text and appendices, 10 in Supplementary Material.) and 1 table

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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.
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.
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.
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.
The main focus of this thesis is to explore the advantages of using higher-dimensional quantum systems (qudits) as building blocks for fault-tolerant quantum computation. In particular, we investigate the two main essential ingredients of many state-of-the-art fault-tolerant schemes [133], which are magic state distillation and topological error correction. The theory for both of these components is well established for the qubit case, but little has been known for the generalised qudit case. For magic state distillation, we first present a general numerical approach that can be used to investigate the distillation properties of any stabilizer code. We use this approach to study small threedimensional (qutrit) codes and classify, for the first time, new types of qutrit magic states. We then provide an analytic study of a family of distillation protocols based on the quantum Reed-Muller codes. We discover a particular five-dimensional code that, by many measures, outperforms all known qubit codes. For the topological error correction, we study the qudit toric code serving as a quantum memory. For this purpose we examine an efficient renormalization group decoder to estimate the error correction threshold. We find that when the qudit toric code is subject to a generalised bit-flip noise, and for a sufficiently high dimension, a threshold of 30% can be obtained under perfect decoding.
Foliated quantum codes are a resource for fault-tolerant measurement-based quantum error correction for quantum repeaters and for quantum computation. They represent a general approach to integrating a range of possible quantum error correcting codes into larger fault-tolerant networks. Here, we present an efficient heuristic decoding scheme for foliated quantum codes, based on message passing between primal and dual code ``sheets.'' We test this decoder on two different families of sparse quantum error correcting code: turbo codes and bicycle codes, and show reasonably high numerical performance thresholds. We also present a construction schedule for building such code states.
Abstract Fault-tolerant cluster states form the basis for scalable measurement-based quantum computation. Recently, new stabilizer codes for scalable circuit-based quantum computation have been introduced that have very high thresholds under biased noise where the qubit predominantly suffers from one type of error, e.g. dephasing. However, extending these advances in stabilizer codes to generate high-threshold cluster states for biased noise has been a challenge, as the standard method for foliating stabilizer codes to generate fault-tolerant cluster states does not preserve the noise bias. In this work, we overcome this barrier by introducing a generalization of the cluster state that allows us to foliate stabilizer codes in a bias-preserving way. As an example of our approach, we construct a foliated version of the XZZX code which we call the XZZX cluster state. We demonstrate that under a circuit-level-noise model, our XZZX cluster state has a threshold more than double the usual cluster state when dephasing errors are more likely than errors that cause bit flips by a factor of order ~100 or more.