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  • Open Access

Fault-tolerant encoding of logical qudits in spin systems

Sumin Lim*

  • Graduate School of Quantum Science and Technology, KAIST, Daejeon 34141, Republic of Korea and CAESR, The Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

  • *Contact author: suminlim@kaist.ac.kr
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Phys. Rev. A 113, 042439 – Published 16 April, 2026

DOI: https://doi.org/10.1103/n28j-r8bg

Abstract

Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multilevel 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-3 codewords, distance-5 codewords, and general (2t+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 multilevel physical systems.

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Phys. Rev. A 113, 042439– Published 16 April, 2026
  • Received 11 November 2025
  • Revised 14 February 2026
  • Accepted 26 March 2026
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