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arXiv:2502.05992 · Physical Review A

Qudit vs. qubit: Simulated performance of error-correction codes in higher dimensions

James Keppens, Quinten Eggerickx, Vukan Levajac, George Simion, Bart Sorée

Published Sep 3, 2025• 4 citations
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Abstract

Qudits can be described by a state vector in a $q$-dimensional Hilbert space, enabling a more extensive encoding and manipulation of information compared to qubits. This implies that conducting fault-tolerant quantum computations using qudits rather than qubits might entail less overhead. In this work, we investigate the viability of qudits in error correction codes by creating and simulating the quantum circuitry for the smallest qudit error correction code with a multidimensional circuit-level noise model and specifically adapted decoders. After introducing a flag qudit to protect the code from hook errors, comparable error thresholds of the order of $10^{-4}$ are obtained for qudits of dimensions $2$, $3$ and $5$.

DOI
10.1103/2w52-qd2j

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Qudits can be described by a state vector in a qqq-dimensional Hilbert space, enabling a more extensive encoding and manipulation of information compared to qubits. This implies that conducting fault-tolerant quantum computations using qudits rather than qubits might entail less overhead. In this work, we investigate the viability of qudits in error correction codes by creating and simulating the quantum circuitry for the smallest qudit error correction code with a multidimensional circuit-level noise model and specifically adapted decoders. After introducing a flag qudit to protect the code from hook errors, comparable error thresholds of the order of 10−410^{-4}10−4 are obtained for qudits of dimensions 222, 333 and 555.
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.
Noise rates in quantum computing experiments have dropped dramatically, but reliable qubits remain precious. Fault-tolerance schemes with minimal qubit overhead are therefore essential. We introduce fault-tolerant error-correction procedures that use only two extra qubits. The procedures are based on adding "flags" to catch the faults that can lead to correlated errors on the data. They work for various distance-three codes. In particular, our scheme allows one to test the ⟦5,1,3⟧ code, the smallest error-correcting code, using only seven qubits total. Our techniques also apply to the ⟦7,1,3⟧ and ⟦15,7,3⟧ Hamming codes, thus allowing us to protect seven encoded qubits on a device with only 17 physical qubits.
Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multi-level physical systems. Here, we present a general framework and working examples for encoding fault-tolerant logical qudits in finite-dimensional spin systems. We construct distance-333, distance-555 codewords, and general 2t+12t+12t+1-distance codes that can be implemented using a single physical qudit or a small number of coupled qudits for higher distances, while requiring a Hilbert space dimension significantly smaller than conventional constructions based on multiple logical qubits. Logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations. We further discuss schematic designs for physical implementation and required single-gate fidelities, which are compatible with current spin qudit platforms. This strategy provides a resource-efficient path toward realizing fault-tolerant logical qudits in finite multi-level physical systems.
Quantum computers have steadily improved over the last decade, but developing fault-tolerant quantum computing (FTQC) techniques, required for useful, universal computation remains an ongoing effort. Key elements of FTQC such as error-correcting codes and decoding are supported by a rich bed of stabilizer simulation software such as Stim and CHP, which are essential for numerically characterizing these protocols at realistic scales. Recently, experimental groups have built nascent high-dimensional quantum hardware, known as qudits, which have a myriad of attractive properties for algorithms and FTQC. Despite this, there are no widely available qudit stabilizer simulators. We introduce the first open-source realization of such a simulator for all dimensions. We demonstrate its correctness against existing state vector simulations and benchmark its performance in evaluating and sampling quantum circuits. This simulator is the essential computational infrastructure to explore novel qudit error correction as earlier stabilizer simulators have been for qubits.
Quantum computers have the potential to solve several interesting problems in polynomial \ntime for which no polynomial time classical algorithms have been found. However, \none of the major challenges in building quantum devices is that quantum systems are very \nsensitive to noise arising from undesired interactions with the environment. Noise can lead \nto errors which can corrupt the results of the computation. Quantum error correction is \none way to mitigate the effects of noise arising in quantum devices. \nWith a plethora of quantum error correcting codes that can be used in various settings, \none of the main challenges of quantum error correction is understanding how well various \ncodes perform under more realistic noise models that can be observed in experiments. \nThis thesis proposes a new decoding algorithm which can optimize threshold values of \nerror correcting codes under different noise models. The algorithm can be applied to \nany Markovian noise model. Further, it is shown that for certain noise models, logical \nClifford corrections can further improve a code's threshold value if the code obeys certain \nsymmetries. \n \nSince gates and measurements cannot in general be performed with perfect precision, \nthe operations required to perform quantum error correction can introduce more errors \ninto the system thus negating the benefits of error correction. Fault-tolerant quantum \ncomputing is a way to perform quantum error correction with imperfect operations while \nretaining the ability to suppress errors as long as the noise is below a code's threshold. \nOne of the main challenges in performing fault-tolerant error correction is the high resource \nrequirements that are needed to obtain very low logical noise rates. With the use of \nflag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable \nto arbitrary distance codes. Various code families are shown to satisfy the requirements \nof flag fault-tolerant error correction. We also provide circuits using a constant number of \nqubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer \nqubits than previous fault-tolerant error correction protocols. \n \nIt is often the case that the noise afflicting a quantum device cannot be fully characterized. \nFurther, even with some knowledge of the noise, it can be very challenging to use \nanalytic decoding methods to improve the performance of a fault-tolerant scheme. This \nthesis presents decoding schemes using several state of the art machine learning techniques \nwith a focus on fault-tolerant quantum error correction in regimes that are relevant to near \nterm experiments. It is shown that even in low noise rate regimes and with no knowledge \nof the noise, noise can be further suppressed for small distance codes. Limitations of machine \nlearning decoders as well as the classical resources required to perform active error \ncorrection are discussed. \n \nIn many cases, gate times can be much shorter than typical measurement times of \nquantum states. Further, classical decoding of the syndrome information used in quantum \nerror correction to compute recovery operators can also be much slower than gate times. \nFor these reasons, schemes where error correction can be implemented in a frame (known \nas the Pauli frame) have been developed to avoid active error correction. In this thesis, we \ngeneralize previous Pauli frame schemes and show how Clifford frame error correction can \nbe implemented with minimal overhead. Clifford frame error correction is necessary if the \nlogical component of recovery operators were chosen from the Clifford group, but could \nalso be used in randomized benchmarking schemes.
The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111, 200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are subject to generalized uncorrelated bit-flip noise (and the higher-dimensional equivalent). We show that, with appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more than 88%8 can be achieved.