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Fault-tolerant resource comparison of qudit and qubit encodings for diagonal quadratic operators

Samuel Godwood1,*, Dog̃a Murat Kürkçüog̃lu2, Gabriel N. Perdue2, Marina Maneyro1, and Alessandro Roggero3,4

  • *Contact author: s.godwood@liverpool.ac.uk
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Phys. Rev. A 114, 022427 – Published 12 August, 2026

DOI: https://doi.org/10.1103/832q-bqlx

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−itϕx2 in a uniform field-amplitude discretization of a real scalar field, using either one logical d-level qudit or nb=⌈log2d⌉ logical qubits. We analyze two standard settings: product-formula simulation and linear combination of unitaries (LCU) per 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.

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Phys. Rev. A 114, 022427– Published 12 August, 2026
  • Received 27 May 2026
  • Revised 20 July 2026
  • Accepted 23 July 2026
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