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

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 $U=e<sup>{-itφ_x<sup>2}$ 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.

• The paper systematically compares non-Clifford resource costs for qudit and qubit encodings of diagonal quadratic evolutions using product-formula and LCU constructions.
• Product-formula methods offer no scalable qudit advantage because qudit synthesis would need an exponentially smaller per-rotation cost, although dimensions d=3 and d=5 can provide constant-factor savings.
• LCU methods favor qubits asymptotically, but finite-dimension qudit benefits appear for physically relevant cutoffs, reaching roughly d≤19 without switching and d≤21 with idealized code conversion.

This paper presents a systematic fault-tolerant resource comparison between single logical $d$-level qudit encodings and registers of $n_b=\lceil\log_2 d\rceil$ logical qubits for implementing onsite diagonal quadratic evolutions, exemplified by $U=e^{-it\phi_x^2}$ in a uniform field-amplitude (Jordan–Lee–Preskill) digitization of a real scalar field (2604.26792). The resource metric throughout is the non-Clifford gate count after synthesis into a discrete fault-tolerant gate set, and the analysis is carried out in two simulation regimes: product-formula (Trotter) steps and LCU/block-encoding constructions. The central finding is deliberately qualified: within the constructive models studied, qudits do not provide an asymptotic advantage for this operator class, but explicit finite-$d$ break-even conditions identify low-dimensional windows in which constant-factor savings are possible.

Problem setup

The digitized field operator $\phi_x$ acts diagonally on a symmetric grid of $d=2M+1$ levels with eigenvalues $\lambda_n=-\phi_{\max}+n\,\delta_\phi$. The target unitary $e^{-it\phi_x^2}$ is representative of a broader class of onsite quadratic diagonal evolutions $e^{-it\,g(N)}$ arising in lattice Hamiltonians, gauge theories, condensed matter, and quantum chemistry. Qubit implementations use binary or signed-binary embeddings on $n_b$ qubits; qudit implementations act natively on the $n_b=\lceil\log_2 d\rceil$0-level truncation. Because tight synthesis bounds for general single-qudit rotations are not known uniformly in $n_b=\lceil\log_2 d\rceil$1, the authors express all qudit costs in terms of embedded two-level $n_b=\lceil\log_2 d\rceil$2 rotations $n_b=\lceil\log_2 d\rceil$3 and parameterize their synthesis cost as $n_b=\lceil\log_2 d\rceil$4, against the qubit baseline $n_b=\lceil\log_2 d\rceil$5. The resulting thresholds are framed explicitly as compiler targets rather than closed-form costs, and the finite-$n_b=\lceil\log_2 d\rceil$6 crossover analyses restrict to prime local dimensions, reflecting the state of the qudit Clifford+$n_b=\lceil\log_2 d\rceil$7/magic-state-distillation literature.

Product-formula regime

On the qubit side, a Trotter step of $n_b=\lceil\log_2 d\rceil$8 requires at most $n_b=\lceil\log_2 d\rceil$9 synthesized $U=e^{-it\phi_x^2}$0 gates: squaring the binary expansion of $U=e^{-it\phi_x^2}$1 produces only one- and two-body $U=e^{-it\phi_x^2}$2 and $U=e^{-it\phi_x^2}$3 Pauli terms, each contributing one rotation. On the qudit side, any diagonal unitary decomposes into exactly $U=e^{-it\phi_x^2}$4 adjacent two-level $U=e^{-it\phi_x^2}$5 rotations via a Givens-rotation argument, with unique angles modulo $U=e^{-it\phi_x^2}$6; for the quadratic phase these angles are generically all nontrivial.

The asymptotic consequence is stark. For the qudit construction to beat the qubit baseline asymptotically, the per-primitive synthesis cost of embedded $U=e^{-it\phi_x^2}$7 rotations would need to be smaller than qubit-optimal $U=e^{-it\phi_x^2}$8 synthesis by a factor $U=e^{-it\phi_x^2}$9 — an exponentially strong advantage in $d$0. Since volume-counting arguments impose $d$1 lower bounds and known number-theoretic synthesis procedures achieve matching $d$2 scaling, the authors conclude this exponential per-rotation advantage is incompatible with the standard synthesis landscape. Phase kickback does not evade the issue, since a diagonal phase function must still be synthesized after coherently computing $d$3.

The finite-$d$4 crossover analysis nonetheless identifies a narrow window. At representative precision $d$5, the break-even prefactor satisfies $d$6 ($d$7), $d$8 ($d$9), and drops to $\phi_x$0 ($\phi_x$1). Only for $\phi_x$2 and $\phi_x$3 does the threshold exceed the effective qubit-$\phi_x$4 prefactor at matched primitive precision, meaning Regime 1 offers at most a low-$\phi_x$5 opportunity rather than scalable savings.

LCU/block-encoding regime

In Regime 2 the motivation for qudits is structurally stronger: the entangling part of SELECT, $\phi_x$6, is a generalized controlled-$\phi_x$7 and hence Clifford on qudits, whereas incrementers and qudit Fourier transforms are also Clifford. The paper develops two comparisons.

Qubit baseline. Using the signed-binary projector-LCU construction of Su et al. and Spagnoli et al., $\phi_x$8 admits an exact block encoding with normalization $\phi_x$9 and per-call $d=2M+1$0 count $d=2M+1$1, where $d=2M+1$2. This cost grows only as $d=2M+1$3.

Qudit construction. The generalized-Pauli expansion $d=2M+1$4 has irreducible length $d=2M+1$5: all Fourier coefficients $d=2M+1$6 are nonzero for odd $d=2M+1$7, so PREP amounts to generic $d=2M+1$8-dimensional state preparation requiring $d=2M+1$9 embedded two-level rotations. Under the idealized code-switching model — where the $\lambda_n=-\phi_{\max}+n\,\delta_\phi$0-qubit index register converts freely to a single $\lambda_n=-\phi_{\max}+n\,\delta_\phi$1-level qudit for the controlled-$\lambda_n=-\phi_{\max}+n\,\delta_\phi$2 interaction — the qudit side costs $\lambda_n=-\phi_{\max}+n\,\delta_\phi$3 $\lambda_n=-\phi_{\max}+n\,\delta_\phi$4 gates per call due to the $\lambda_n=-\phi_{\max}+n\,\delta_\phi$5-synthesis of the PREP cascade, versus $\lambda_n=-\phi_{\max}+n\,\delta_\phi$6 for the qubit baseline. Theorem 3 establishes that the ratio of qubit to qudit total cost tends to zero as $\lambda_n=-\phi_{\max}+n\,\delta_\phi$7: the qubit encoding is asymptotically cheaper, driven entirely by per-call cost growth since both query counts are $\lambda_n=-\phi_{\max}+n\,\delta_\phi$8 in $\lambda_n=-\phi_{\max}+n\,\delta_\phi$9 at fixed $e^{-it\phi_x^2}$0 and $e^{-it\phi_x^2}$1.

Finite-d results and code-switching budgets

Weighting per-call costs by the qubitization query count $e^{-it\phi_x^2}$2 yields end-to-end comparisons at precision-dominated ($e^{-it\phi_x^2}$3) and time-dominated ($e^{-it\phi_x^2}$4) points, with $e^{-it\phi_x^2}$5 and $e^{-it\phi_x^2}$6.

In the fixed-encoding model, the condition that the break-even prefactor exceeds the same-precision qubit-$e^{-it\phi_x^2}$7 reference holds only for $e^{-it\phi_x^2}$8 ($e^{-it\phi_x^2}$9) and $e^{-it\,g(N)}$0 ($e^{-it\,g(N)}$1) in the precision-dominated regime, but extends to all prime dimensions up to $e^{-it\,g(N)}$2 in the time-dominated regime, with the largest value $e^{-it\,g(N)}$3 against a reference of $e^{-it\,g(N)}$4. For context, existing qutrit Clifford+$e^{-it\,g(N)}$5 synthesis reports effective prefactors around $e^{-it\,g(N)}$6–$e^{-it\,g(N)}$7, suggesting the favorable $e^{-it\,g(N)}$8 window may be practically reachable, though a direct comparison requires a gate-set conversion not generally available.

Under the code-switching model, the qudit construction wins only for $e^{-it\,g(N)}$9 and $n_b$0 in the precision-dominated regime (crossover between $n_b$1 and $n_b$2, with $n_b$3 and absolute savings of roughly $n_b$4 $n_b$5 gates), while in the time-dominated regime it wins for $n_b$6, with the largest relative advantage $n_b$7 and largest absolute savings $n_b$8 $n_b$9 gates at $n_b=\lceil\log_2 d\rceil$00. These savings translate into per-switch overhead budgets: assuming $n_b=\lceil\log_2 d\rceil$01 directional switches per query, qudits remain advantageous provided each switch costs fewer than $n_b=\lceil\log_2 d\rceil$02 non-Cliffords, yielding budgets up to $n_b=\lceil\log_2 d\rceil$03 $n_b=\lceil\log_2 d\rceil$04 gates near the most favorable dimensions (e.g., $n_b=\lceil\log_2 d\rceil$05 at $n_b=\lceil\log_2 d\rceil$06).

Physical relevance of the favorable windows

The identified windows overlap meaningfully with physically motivated truncations. Klco and Savage estimate $n_b=\lceil\log_2 d\rceil$07–$n_b=\lceil\log_2 d\rceil$08 qubits per site before digitization error falls below typical Trotter error; only the lower end ($n_b=\lceil\log_2 d\rceil$09–$n_b=\lceil\log_2 d\rceil$10) overlaps the time-dominated break-even region, while $n_b=\lceil\log_2 d\rceil$11 lies outside it in both models. By contrast, gauge-theory applications often require explicitly small cutoffs: $n_b=\lceil\log_2 d\rceil$12D scalar QED appears sufficient at $n_b=\lceil\log_2 d\rceil$13, pure $n_b=\lceil\log_2 d\rceil$14 gauge theory in $n_b=\lceil\log_2 d\rceil$15D achieves per-mille plaquette accuracy at $n_b=\lceil\log_2 d\rceil$16, and trapped-ion qudit simulations of 2D lattice QED operate at $n_b=\lceil\log_2 d\rceil$17 and $n_b=\lceil\log_2 d\rceil$18. These cases fall squarely within the favorable regions, motivating fault-tolerant qudit investigations at small prime dimensions.

Limitations and open questions

Several caveats bound the scope of these conclusions. First, the absence of tight, gate-set-dependent synthesis bounds for general single-qudit diagonal operations forces the parameterization by the prefactor $n_b=\lceil\log_2 d\rceil$19; deriving closed-form costs for arbitrary odd $n_b=\lceil\log_2 d\rceil$20, or proving whether $n_b=\lceil\log_2 d\rceil$21 saturates worst-case lower bounds, remains open and is related to NP-complete nearest-unitary decision problems. Second, the Regime 2 comparisons are construction-specific: they compare the native qudit LCU against one standard signed-binary qubit projector-LCU baseline, not optimal lower bounds over all qubit encodings, so alternative qubit constructions could shift the comparison in either direction. Third, the code-switching model assumes free isometric conversion between the index register and the qudit, with invalid computational states unpopulated; the actual fault-tolerant overhead of qubit–qudit code conversion is architecture-dependent and unresolved, though qubit fusion/qudit fission via non-stabilizer resource states offers one possible route whose distillation costs remain unquantified. Fourth, the query-count proxy introduces $n_b=\lceil\log_2 d\rceil$22 shifts near break-even, and the restriction to prime $n_b=\lceil\log_2 d\rceil$23 reflects the current synthesis literature rather than a fundamental constraint.

Conclusion

The paper reframes the qudit-versus-qubit question for quadratic diagonal evolutions away from asymptotic claims and toward explicit, finite-$n_b=\lceil\log_2 d\rceil$24 compiler targets. Within the studied constructions, product-formula implementations would require an exponentially strong per-primitive synthesis advantage for qudits to win, and the LCU setting favors qubits asymptotically in $n_b=\lceil\log_2 d\rceil$25; nevertheless, meaningful constant-factor savings arise at low dimensions — up to $n_b=\lceil\log_2 d\rceil$26 in the fixed-encoding time-dominated regime and $n_b=\lceil\log_2 d\rceil$27 under idealized code-switching — precisely where several lattice gauge theory applications require modest truncations. A definitive verdict awaits dimension-specific synthesis results for embedded $n_b=\lceil\log_2 d\rceil$28 primitives and architecture-level accounting of code conversion, injection, and distillation overheads (2604.26792).