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arXiv:2604.26792 · Physical Review A

Fault-Tolerant Resource Comparison of Qudit and Qubit Encodings for Diagonal Quadratic Operators

Samuel Godwood, Doğa Murat Kürkçüoğlu, Gabriel N. Perdue, Marina Maneyro, Alessandro Roggero

quant-phSubmitted Apr 29, 2026• Announced Apr 30, 2026• 0 citations
v1 · Apr 29, 2026v2 · May 20, 2026v3 · Jul 29, 2026
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Abstract

Finite local Hilbert-space truncations arise naturally in quantum simulations of lattice field theories and motivate qudit encodings, but their fault-tolerant advantage over qubit encodings remains unclear. We compare the non-Clifford cost of implementing quadratic diagonal evolutions, exemplified by $U=e^{-itφ_x^2}$ in a uniform field-amplitude discretization of a real scalar field, using either one logical $d$-level qudit or $n_b=\lceil \log_2 d\rceil$ logical qubits. We analyze two standard settings: product-formula simulation and LCU/block encoding, taking the resource metric to be the number of non-Clifford gates after synthesis into a discrete logical gate set. Because tight synthesis bounds for general single-qudit rotations are not known, we express the qudit constructions in terms of embedded two-level $SU(2)$ rotations and derive explicit finite-$d$ break-even conditions for their synthesis cost; these serve as compiler targets for when qudit encodings can outperform the qubit baseline. Within the constructive models studied here, product-formula implementations would require an exponentially stronger per-primitive synthesis advantage for qudits to win asymptotically, while in the LCU setting the qubit encoding is asymptotically cheaper in $d$. Nevertheless, the finite-$d$ threshold analysis identifies low dimensional regions in which qudits can yield meaningful constant-factor savings, particularly for LCU-based implementations. As a secondary analysis of the LCU construction, we use an idealized negligible-overhead qubit-qudit code-switching model to give an absolute $T$-count comparison, and reinterpret the savings as an allowable per-switch overhead budget.

DOI
10.1103/832q-bqlx

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Finite local Hilbert-space truncations arise naturally in quantum simulations of lattice field theories and motivate qudit encodings, but their fault-tolerant advantage over qubit encodings remains unclear. We compare the non-Clifford cost of implementing quadratic diagonal evolutions, exemplified by U=e−itφx2U=e^{-itφ_x^2}U=e−itφx2​ in a uniform field-amplitude discretization of a real scalar field, using either one logical ddd-level qudit or nb=⌈log⁡2d⌉n_b=\lceil \log_2 d\rceilnb​=⌈log2​d⌉ logical qubits. We analyze two standard settings: product-formula simulation and LCU/block encoding, taking the resource metric to be the number of non-Clifford gates after synthesis into a discrete logical gate set. Because tight synthesis bounds for general single-qudit rotations are not known, we express the qudit constructions in terms of embedded two-level SU(2)SU(2)SU(2) rotations and derive explicit finite-ddd break-even conditions for their synthesis cost; these serve as compiler targets for when qudit encodings can outperform the qubit baseline. Within the constructive models studied here, product-formula implementations would require an exponentially stronger per-primitive synthesis advantage for qudits to win asymptotically, while in the LCU setting the qubit encoding is asymptotically cheaper in ddd. Nevertheless, the finite-ddd threshold analysis identifies low dimensional regions in which qudits can yield meaningful constant-factor savings, particularly for LCU-based implementations. As a secondary analysis of the LCU construction, we use an idealized negligible-overhead qubit-qudit code-switching model to give an absolute TTT-count comparison, and reinterpret the savings as an allowable per-switch overhead budget.
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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.
Universal fault-tolerant quantum computing combines versatile but expensive operations with specialized but cheap ones. Its efficiency depends on how much computation can be pushed onto the cheap operations and on the size of the code needed to do so. Automorphism gates provide such cheap operations using only physical single-qubit Clifford gates and qubit permutations. Yet no general theory characterizes their maximum logical power or the minimum code size needed to attain it. We develop such a theory. For stabilizer codes encoding k≥3k\geq3k≥3 logical qubits, we show that the largest logical group attainable by automorphisms is generated by all addressable SSS and CX\mathrm{CX}CX gates, and we construct codes attaining it. While this group contains exponentially fewer gates than the full Clifford group, adding one suitable non-Clifford gate yields universality. We further classify the largest logical groups attainable using qubit permutations, physical single-qubit Cliffords, or both across general stabilizer and CSS codes, and derive refined bounds for self-dual CSS subclasses. Achieving the maximum-size logical group through automorphisms requires n=Θ(2k)n=Θ(2^k)n=Θ(2k) physical qubits. By contrast, all addressable diagonal Clifford gates, generated by SSS and CZ\mathrm{CZ}CZ, require only n=Θ(k2)n=Θ(k^2)n=Θ(k2) physical qubits when implemented using physical single-qubit Cliffords alone. Both bounds are tight. This polynomial qubit cost extends beyond Cliffords to all addressable diagonal gates at any fixed level of the Clifford hierarchy, using physical single-qubit diagonal gates. Thus, for full addressability, the sharpest physical-qubit cost divide lies between diagonal and CX\mathrm{CX}CX-type gates, not between Clifford and non-Clifford gates.
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, I investigate the viability of qudits across three key components of a fault-tolerant quantum computer: hardware control, quantum error-correction codes, and logical gate operations. This includes benchmarking physically decomposed qudit gates against qubit-based implementations for a specific donor-spin system, threshold simulations of qudit error-correction codes using tailored simulators and decoders, and simulations of magic state distillation circuits with qudits. At the hardware level, a numerical leakage-aware randomized benchmarking protocol is applied to a native ququart encoding in a donor-spin system and compared against an equivalent pair of encoded qubits. The ququart implementation is found to reduce the lower-bound gate infidelity by 40 – 50%, a consequence of the fewer ESR and EDSR pulses and correspondingly lower cumulative noise exposure. This provides a concrete, platform-specific demonstration of reduced physical-gate overhead prior to the introduction of any error-correction redundancy. At the level of quantum error correction, the viability of qudits is investigated through the construction and simulation of the quantum circuits for the five-qudit perfect code. The code is evaluated under a multidimensional circuit-level noise model using decoders specifically adapted to the qudit setting. This yields the first threshold estimates reported for qutrit and ququint codes under circuit-level noise. 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. These simulations were subsequently performed using Sdim, a stabilizer tableau simulator that enabled threshold estimation up to dimension q = 7, a regime inaccessible to state-vector simulators such as Cirq. At the level of logical operations, magic state distillation protocols of the Bravyi-Kitaev type are simulated under logical circuit-level noise, to my knowledge for the first time for this protocol family. Once the Clifford operations implementing the distillation circuit are subject to a nonzero logical error rate, the achievable output fidelity saturates within a finite usable distillation range rather than an arbitrarily improvable one: roughly 0.9-15.6% for the |T⟩₂-type protocol, 3.7-5% for the |H⟩₂-type protocol and 5.4-7.4% for the |H+⟩₃-type protocol at a representative logical circuit-level noise level of pCL = 10–3. The qutrit protocol achieves a more favorable distillation threshold but suffers from a substantially lower yield due to its reduced syndrome acceptance probability. This highlights a trade-off between threshold performance and yield that is intrinsic to the higher-dimensional construction. Further investigation with more advanced and larger-scale protocols is required to better understand and characterize this behavior. Overall, these results show that qudits can provide real efficiency benefits: higher-dimensional control can improve physical gate fidelity, and qutrit and ququint error-correction thresholds can match those of qubits when hook errors are properly handled. However, logical circuit-level noise limits the maximum magic state fidelity, showing that the benefits of qudits are not yet fully realized at the logical-operation level. This thesis provides the first broad numerical study of these effects and introduces the simulation tools needed to explore larger code families, decoders, and distillation protocols.
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
Abstract Fault-tolerant quantum computation (FTQC) schemes using large block codes that encode k > 1 qubits in n physical qubits can potentially reduce the resource overhead to a great extent because of their high encoding rate. However, the fault-tolerant (FT) logical operations for the encoded qubits are difficult to find and implement, which usually takes not only a very large resource overhead but also long in situ computation time. In this paper, we focus on Calderbank–Shor–Steane [[ n , k , d ]] (CSS) codes and their logical FT Clifford circuits. We show that the depth of an arbitrary logical Clifford circuit can be implemented fault-tolerantly in O (1) steps in situ via either Knill or Steane syndrome measurement circuit, with the qualified ancilla states efficiently prepared. Particularly, for those codes satisfying k / n ∼ Θ(1), the resource scaling for Clifford circuits implementation on the logical level can be the same as on the physical level up to a constant, which is independent of code distance d . With a suitable pipeline to produce ancilla states, our scheme requires only a modest resource cost in physical qubits, physical gates, and computation time for very large scale FTQC.