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arXiv:2503.12142 · Physical Review A

Designing quantum error correction codes for practical spin qudit systems

Sumin Lim, Arzhang Ardavan

Published Jul 24, 2025• 5 citations
arXivHTMLPDF

Abstract

The implementation of practical error correction protocols is essential for deployment of quantum information technologies. Ways of exploiting high-spin nuclei, which have multilevel quantum resources, have attracted interest in this context because they offer additional Hilbert space dimensions in a spatially compact and theoretically efficient structure. We present a quantitative analysis of the performance of a spin-qudit-based error-correctable quantum memory, with reference to the actual Hamiltonians of several potential candidate systems. First, the ideal code-word implemented on a spin-7/2 nucleus, which provides first-order Pauli- X , Y and Z error correction, has intrinsic infidelity due to mixed eigenstates under realistic conditions. We confirm that expansion to a spin-9/2 system with tailored code words can compensate for this infidelity. Second, we claim that electric field fluctuations, which are inevitable in real systems, should also be considered as a noise source, and we illustrate an encoding and decoding scheme for a multi-spin-qudit-based error correction code that can simultaneously compensate for both electric- and magnetic-field perturbations. Such strategies are important as we move beyond the current noisy-intermediate quantum era, and fidelities above two or three nines become crucial for the implementation of quantum technologies.

DOI
10.1103/7q6l-d4qh

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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.
We construct a fault-tolerant quantum error-correcting protocol based on a qubit encoded in a large spin qudit using a spin-cat code, analogous to the continuous-variable cat encoding. With this, we can correct the dominant error sources, namely processes that can be expressed as error operators that are linear or quadratic in the components of angular momentum. Such codes tailored to dominant error sources can exhibit superior thresholds and lower resource overheads when compared to those designed for unstructured noise models. A key component is the gate that preserves the rank of spherical tensor operators. Categorizing the dominant errors as phase and amplitude errors, we demonstrate how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Furthermore, we propose a measurement-free error-correction scheme to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical gate errors, we establish that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. We consider a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of 87 Sr , and show how to generate the universal gate set, including the rank-preserving gate, using quantum control and the Rydberg blockade. These findings pave the way for encoding a qubit in a large spin with the potential to achieve fault tolerance, high threshold, and reduced resource overhead in quantum information processing. Published by the American Physical Society 2024
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Abstract Molecular Nanomagnets may enable the implementation of qudit-based quantum error-correction codes which exploit the many spin levels naturally embedded in a single molecule, a promising step towards scalable quantum processors. To fully realize the potential of this approach, a microscopic understanding of the errors corrupting the quantum information encoded in a molecular qudit is essential, together with the development of tailor-made quantum error correction strategies. We address these central points by first studying dephasing effects on the molecular spin qudit produced by the interaction with surrounding nuclear spins, which are the dominant source of errors at low temperatures. Numerical quantum error correction codes are then constructed, by means of a systematic optimization procedure based on simulations of the coupled system-bath dynamics, that provide a striking enhancement of the coherence time of the molecular computational unit. The sequence of pulses needed for the experimental implementation of the codes is finally proposed.
This paper presents the design and theoretical framework for a single-bit quantum error correction memory unit based on the principles of quantum error correction. The core claim is to construct a fault-tolerant memory unit for storing quantum information reliably. The proposed unit utilizes a multi-party quantum error correction coding scheme coupled with an error decoding system to mitigate the effects of decoherence and noise on individual qubits. The key innovation lies in the efficient implementation of quantum error correction at the single-qubit level, addressing a critical bottleneck in quantum computing. The theoretical framework details the encoding and decoding processes, emphasizing the real-time feedback mechanism for sustained quantum state preservation. Mathematical formulations are presented to illustrate the operation of the unit, including the error correction codes and the decoding algorithms. The unit's design offers a pathway toward scalable quantum memory solutions, crucial for advanced quantum computation and quantum communication protocols.
Quantum computers are highly sensitive to noise due to interactions with the environment, which severely limits the utility of near-term devices. Quantum error correction addresses this challenge by encoding logical information over multiple qubits in order to preserve information in the presence of noise. This facilitates the realization of the theoretical advantages of quantum computing such as exponential speed-up compared to classical computers for certain algorithms. Physical realizations of qubits may have different noise profiles, including biased noise, where phase-flip errors may be more likely than bit-flip errors. We propose a quantum error-correcting code, the XYZ² code, which is tailored to biased noise. The code has high error thresholds against biased noise, below which logical errors are exponentially suppressed with the linear dimension of the code. The important classical task of deriving corrections given partial information about errors is called decoding. We investigate several decoding strategies tailored to specific challenges, ranging from a novel maximum-likelihood sampling decoder, to a model-free, data-driven, neural-network-based decoder, to an efficient multi-step decoder for concatenated codes. We also study the link between quantum error-correcting codes and their statistical-mechanical counterparts by mapping to generalized random-bond Ising models and subsequently deriving the exact results for finite-size corrections under biased noise for moderate code sizes. Quantum error correction is still in its developing stages, and this work provides a useful contribution to the field by bridging a gap between theory and practice. By adapting well-studied topological codes to error models that are closer to reality, this work paves the way for the realization of a fault-tolerant quantum architecture.