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

• URL: http://arxiv.org/abs/2604.26792v1
• Date: Wed, 29 Apr 2026 15:25:08 GMT
• Title: Fault-Tolerant Resource Comparison of Qudit and Qubit Encodings for Diagonal Quadratic Operators
• Abstract summary: Finite local Hilbert-space truncations arise naturally in quantum simulations of lattice field theories and motivate qudit encodings.<n>We compare the non-Clifford cost of implementing diagonal evolutions, exemplified by $U=e-it_x2$ in a uniform field-amplitude discretization of a real scalar field, using either one logical $d$-level qudit or $n_b=eil log drceil$ logical qubits.
• Score: 36.94429692322632
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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