Qudit SWAP Engine Overview

• Qudit SWAP Engine is a framework for implementing, optimizing, and routing SWAP operations in multilevel (d>2) quantum systems using diverse hardware primitives.
• It employs methods like three-gate decompositions, optimal-control waveforms, and SWAP-less routing to reduce resource cost and circuit depth.
• The engine extends to thermodynamic exchange, entanglement diagnostics, and algebraic graph formulations, revealing key dimension-dependent trade-offs.

Searching arXiv for the cited SWAP/qudit papers to ground the article in current arXiv records. A Qudit SWAP Engine denotes a family of architectures, constructions, and analytic frameworks centered on implementing, compiling, optimizing, routing, or exploiting SWAP operations in finite-dimensional quantum systems with local dimension $d>2$. Across the literature, the term encompasses at least four distinct but related meanings: exact two-qudit circuit synthesis of the SWAP permutation from generalized controlled operations; software-defined optimal-control synthesis of single-qudit level-selective exchanges such as $i\leftrightarrow j$ in multilevel hardware; SWAP-less routing schemes that use temporary higher-dimensional auxiliary levels to transport information without explicit SWAP insertion; and thermodynamic or many-body models in which swap operators are the central dynamical or Hamiltonian primitives (Wilmott, 2011, Wu et al., 2020, Özgüler et al., 2022, Saha et al., 2021, Shende et al., 2024, Sacchi, 2021, Klep et al., 26 Mar 2025). The unifying object is the qudit SWAP itself, which on a pair of $d$-level systems acts as $|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$, but the feasibility, resource cost, robustness, and interpretation of such an operation depend strongly on dimension, control model, and application domain (Wilmott, 2011, Garcia-Escartin et al., 2013).

1. Formal definition and gate-theoretic foundations

For a single $d$-level system with computational basis $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$, the two-qudit SWAP operator $S$ interchanges two subsystems according to

$$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$

In operator form,

$$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$

and, as a permutation of the $d^2$ basis states $i\leftrightarrow j$0, it maps $i\leftrightarrow j$1 (Wilmott, 2011, Garcia-Escartin et al., 2013). This permutation has $i\leftrightarrow j$2 fixed points $i\leftrightarrow j$3 and precisely $i\leftrightarrow j$4 disjoint $i\leftrightarrow j$5-cycles, one for each unordered pair $i\leftrightarrow j$6 with $i\leftrightarrow j$7 (Wilmott, 2011). Hence its signature is

$i\leftrightarrow j$8

so SWAP is even for $i\leftrightarrow j$9 and odd for $d$0 (Wilmott, 2011).

A second canonical formulation uses generalized Pauli and Fourier operators. The shift and phase operators are

$d$1

with $d$2 under the single-qudit Quantum Fourier Transform

$d$3

The controlled-shift gate is

$d$4

i.e.

$d$5

while the controlled-phase is

$d$6

These identities are the standard algebraic substrate for qudit SWAP synthesis (Garcia-Escartin et al., 2013).

The most direct three-gate qudit generalization of the qubit three-CNOT SWAP uses a SUM/UNSUM/SUM pattern,

$d$7

which reduces to the familiar qubit identity at $d$8 because $d$9 (Garcia-Escartin et al., 2013). A fully symmetric variant replaces these by three identical controlled gates,

$|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$0

so that

$|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$1

valid for arbitrary $|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$2 (Garcia-Escartin et al., 2013). Each $|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$3 may be realized as

$|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$4

and each $|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$5 as

$|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$6

so a three-gate SWAP uses $|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$7 and $|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$8 single-qudit QFTs (Garcia-Escartin et al., 2013).

2. Parity obstructions and dimension-dependent feasibility

A central result for two-qudit SWAP synthesis is that circuit feasibility depends sharply on $|i\rangle|j\rangle \mapsto |j\rangle|i\rangle$9 when the gate library is restricted to generalized controlled-NOT or controlled-SUM primitives with no ancillas or measurements (Wilmott, 2011). In the notation of that work, the allowed gates are

$d$0

with $d$1 denoting addition modulo $d$2 (Wilmott, 2011).

The argument proceeds entirely via permutation signatures. Every such CSUM gate induces a permutation of the $d$3 computational-basis states, and the determinant of its permutation matrix equals the permutation signature (Wilmott, 2011). For prime dimensions $d$4, the cycle analysis yields

$d$5

so $d$6 for $d$7 but $d$8 for odd primes (Wilmott, 2011). For qutrits, the paper gives an explicit worked example: CNOT1 has cycle type $d$9 and is even, while SWAP has cycle type $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$0 and is odd (Wilmott, 2011).

Because the signature of a composition is the product of the signatures of its factors, any CSUM-only circuit built from even CSUMs remains even. This yields the theorem that a two-qudit circuit composed entirely of generalized CNOT gates cannot implement SWAP when $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$1 (Wilmott, 2011). The proof is cleanest for $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$2 and, more generally, for odd prime $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$3, where CSUM is even while SWAP is odd. The paper states the impossibility under the explicit constraints of two-qudit-only gates, allowed primitives CSUM1 and CSUM2, and no ancillas or measurements (Wilmott, 2011).

The dimensional landscape is therefore nonuniform. For $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$4 or $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$5, SWAP has even parity, so parity does not forbid a CSUM-only implementation, although no explicit construction or gate count is given in that work (Wilmott, 2011). For $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$6, the situation depends on the parity of CSUM itself. The qubit case is the familiar positive example: SWAP decomposes into three CNOTs, and both SWAP and CNOT are odd, so parity is consistent (Wilmott, 2011). The paper notes that SWAP requires at least three CNOTs in the qubit case (Wilmott, 2011). For larger even composite dimensions, the paper does not provide a full classification; the illustrative $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$7 calculation gives an even CSUM permutation, so parity alone would forbid CSUM-only SWAP if SWAP were odd in that dimension, but the paper refrains from a general impossibility statement for all $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$8 (Wilmott, 2011).

This parity-based obstruction is conceptually distinct from constructive three-gate qudit SWAP identities such as the ones built from $\{|0\rangle,|1\rangle,\ldots,|d-1\rangle\}$9, $S$0, $S$1, QFT, and $S$2 (Garcia-Escartin et al., 2013). The two lines of work are compatible because they concern different primitive gate sets. A plausible implication is that “Qudit SWAP Engine” should be understood not as a single universal construction, but as a dimension- and primitive-dependent design problem.

3. Circuit constructions, routing, and SWAP-less transport

Beyond existence questions, qudit SWAP engines appear as explicit circuit templates for routing and state movement. One exact synthesis for dimension $S$3 uses three controlled additions and one local modular inversion:

$S$4

where $S$5 (Vezvaee et al., 2024). Acting on $S$6, the sequence maps

$S$7

thereby recovering SWAP exactly (Vezvaee et al., 2024). For $S$8, this was described as native to the fixed-frequency transmon ququart toolbox considered იქ (Vezvaee et al., 2024).

A more radical reinterpretation replaces SWAP insertion by SWAP-less routing through temporary promotion of intermediate nodes to higher-dimensional systems (Saha et al., 2021). In the binary case, intermediate qubits are treated as quaquads with levels $S$9. The source qubit conditionally raises the first intermediate into the auxiliary subspace, the “mark” is conditionally propagated through the chain, a destination action is triggered, and the route is uncomputed (Saha et al., 2021). For a three-node line $$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$0–$$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$1–$$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$2, the primitive sequence is

1. $$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$3,
2. $$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$4,
3. $$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$5, which yields the net map

$$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$6

i.e. a long-range CNOT from source to destination while restoring the intermediate exactly to its input (Saha et al., 2021). The same logic generalizes to longer chains and to arbitrary $$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$7-ary systems by promoting intermediates from $$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$8 to $$S(|i\rangle\otimes|j\rangle)=|j\rangle\otimes|i\rangle.$$9 levels with gates $$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$0, $$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$1, and $$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$2 (Saha et al., 2021).

This routing engine has explicit asymptotic and empirical resource advantages relative to SWAP-chain compilation. For a path of $$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$3 total nodes, the conventional SWAP-based gate count scales as

$$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$4

whereas the proposed qudit-routing scheme scales as

$$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$5

Depth scales as

$$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$6

compared with

$$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$7

for optimized SWAP insertion (Saha et al., 2021). The paper reports concrete comparisons: for a 3-qubit chain, the proposed method uses 3 gates versus 7 conventionally; for a 4-qubit chain, 5 gates versus 13, with depth 5 versus 7 under parallel SWAP optimization (Saha et al., 2021). It describes these reductions as a three times reduction in quantum cost with respect to gate count and approximately two times reduction with respect to circuit depth (Saha et al., 2021).

In ququart-based Fermi–Hubbard simulation, routing pressure is also reduced at the encoding level. The Qudit Fermionic Mapping uses one ququart per physical site to encode the four local states $$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$8, turning the hopping term into nearest-neighbor two-qudit interactions and the on-site interaction into local single-ququart gates (Vezvaee et al., 2024). This removes Jordan–Wigner string overhead and many SWAP-like layers. The reported gate-count reductions per Trotter step are 56 two-qudit gates versus 64 two-qubit gates for a $$S=\sum_{i,j=0}^{d-1}|j\rangle\langle i|\otimes|i\rangle\langle j|,$$9 layout, and 80 two-qudit gates versus 112 two-qubit gates for a $d^2$0 layout (Vezvaee et al., 2024). In that setting, SWAP is no longer merely a gate to synthesize; it becomes a routing cost to be eliminated by co-designing mapping, hardware, and two-qudit transpilation.

4. Software-defined optimal-control SWAPs in multilevel hardware

A distinct notion of qudit SWAP engine abandons discrete gate decomposition and instead synthesizes the target exchange directly as a single waveform (Wu et al., 2020). In a superconducting transmon, the relevant target is a single-qudit level-selective SWAP,

$d^2$1

which exchanges levels $d^2$2 and $d^2$3 while leaving all others unchanged (Wu et al., 2020, Özgüler et al., 2022). The demonstrated case is the $d^2$4 swap in the three-level subspace $d^2$5,

$d^2$6

with matrix

$d^2$7

in the ordered basis $d^2$8 (Wu et al., 2020).

The platform is a 3D-transmon qudit dispersively coupled to a 3D aluminum cavity for readout (Wu et al., 2020). The measured lowest four transition frequencies are

• $d^2$9 GHz,
• $i\leftrightarrow j$00 GHz,
• $i\leftrightarrow j$01 GHz, with coherence data including $i\leftrightarrow j$02s for $i\leftrightarrow j$03–$i\leftrightarrow j$04, $i\leftrightarrow j$05s for $i\leftrightarrow j$06–$i\leftrightarrow j$07, $i\leftrightarrow j$08s for $i\leftrightarrow j$09–$i\leftrightarrow j$10, and $i\leftrightarrow j$11s for $i\leftrightarrow j$12–$i\leftrightarrow j$13 (Wu et al., 2020). The optimization is carried out in a rotating frame with drive frequency $i\leftrightarrow j$14, using control Hamiltonians

$i\leftrightarrow j$15

with controls $i\leftrightarrow j$16 and $i\leftrightarrow j$17 (Wu et al., 2020). The objective is

$i\leftrightarrow j$18

where

$i\leftrightarrow j$19

and $i\leftrightarrow j$20 penalizes leakage into $i\leftrightarrow j$21 (Wu et al., 2020).

The reported implementation details are unusually concrete. The gate duration is $i\leftrightarrow j$22 ns, the time resolution is $i\leftrightarrow j$23 ns to match a 32 GSa/s AWG, the effective rotating-frame drive is approximately $i\leftrightarrow j$24 MHz, and the laboratory-frame drive is approximately $i\leftrightarrow j$25 MHz (Wu et al., 2020). Frequency content is concentrated near $i\leftrightarrow j$26 GHz and $i\leftrightarrow j$27 GHz, with a guard band suppressing spectral weight near $i\leftrightarrow j$28 GHz (Wu et al., 2020). The optimized control functions achieve simulated $i\leftrightarrow j$29, while experiment yields averaged entanglement fidelity $i\leftrightarrow j$30 and averaged gate fidelity $i\leftrightarrow j$31 (Wu et al., 2020). A simulation including only intrinsic $i\leftrightarrow j$32 errors gives approximately $i\leftrightarrow j$33 average gate fidelity, implying approximately $i\leftrightarrow j$34 coherent control error from hardware imperfections and residual calibration (Wu et al., 2020). Repeated-gate runs were performed up to 21 applications, with minimal population in the leakage level $i\leftrightarrow j$35 (Wu et al., 2020).

This software-defined approach is motivated by the compounding error of long primitive decompositions. The same paper notes that even single-qubit and two-qubit primitives with 99%–99.5% fidelity compound rapidly, with 20 gates at 99% each yielding approximately $i\leftrightarrow j$36 process fidelity (Wu et al., 2020). It further contrasts the single-shot waveform against a hypothetical 6–10-pulse decomposition with 99% primitives, which would cap fidelity near $i\leftrightarrow j$37, while deeper sequences in quantum simulation often drop below 85%–80% (Wu et al., 2020). Within this framework, a Qudit SWAP Engine is not a fixed circuit identity but a calibrated synthesis pipeline: define $i\leftrightarrow j$38, build the Hamiltonian model and transfer function, impose amplitude and bandwidth constraints, optimize for fidelity and leakage suppression, predistort the waveform, calibrate phases and amplitudes on device, and validate experimentally (Wu et al., 2020).

5. Spectator modes, robustness bounds, and compiler primitives

The usefulness of software-defined level-selective SWAPs depends on whether a pulse optimized in isolation remains accurate in a larger device. In circuit QED, populated spectator modes shift transition frequencies via cross-Kerr couplings and can detune the pulse away from the Hamiltonian for which it was optimized (Özgüler et al., 2022). The model is

$i\leftrightarrow j$39

and, for a target qudit in the presence of spectators in fixed Fock states $i\leftrightarrow j$40, the effective Hamiltonian reduces to

$i\leftrightarrow j$41

with $i\leftrightarrow j$42 and $i\leftrightarrow j$43 (Özgüler et al., 2022).

For a transition $i\leftrightarrow j$44, the spectator-induced detuning is

$i\leftrightarrow j$45

Pulses are synthesized via numerical quantum optimal control with B-spline parametrization in Juqbox.jl, following Petersson and Anders’ techniques, in a GRAPE-like workflow (Özgüler et al., 2022). Representative simulations consider SWAP$i\leftrightarrow j$46, SWAP$i\leftrightarrow j$47, SWAP$i\leftrightarrow j$48, and SWAP$i\leftrightarrow j$49 for a target oscillator with $i\leftrightarrow j$50 GHz and $i\leftrightarrow j$51 GHz, with gate durations 140 ns, 215 ns, 265 ns, and 425 ns respectively, one guard level, 200 optimization iterations, and no decoherence (Özgüler et al., 2022).

The main quantitative result is a small-detuning quadratic fidelity law. Writing the logical-frame propagator as $i\leftrightarrow j$52, the reported fidelity is

$i\leftrightarrow j$53

For small $i\leftrightarrow j$54,

$i\leftrightarrow j$55

and, equivalently,

$i\leftrightarrow j$56

with empirical log-log slope approximately 2 and strong agreement for $i\leftrightarrow j$57 (Özgüler et al., 2022). The paper’s central engineering rule is that spectator-induced shifts must be $i\leftrightarrow j$58 of the qudit nonlinearity in order to preserve high-fidelity single-qudit gates in the presence of populated spectator modes (Özgüler et al., 2022). Numerically, for $i\leftrightarrow j$59 MHz, this means $i\leftrightarrow j$60 MHz; in the simulations with $i\leftrightarrow j$61 GHz, 0.1% corresponds to $i\leftrightarrow j$62 MHz (Özgüler et al., 2022).

In compiler terms, the paper outlines a library model for a Qudit SWAP Engine: store calibrated SWAP$i\leftrightarrow j$63 pulses with metadata including duration, usable $i\leftrightarrow j$64 range, leakage bounds, and guard-level requirements; certify module reusability only when $i\leftrightarrow j$65; and trigger re-optimization, resets, or scheduling changes when predicted spectator shifts exceed the validated bound (Özgüler et al., 2022). This introduces an important distinction between synthesis and reuse. A pulse that is high-fidelity in isolation may not be a reliable compiler primitive unless spectator occupations, cross-Kerr couplings, and scheduling constraints are managed jointly.

6. Thermodynamic, algebraic, and diagnostic uses of swap operators

The term “Qudit SWAP Engine” also appears in contexts where SWAP is not merely a control primitive but the defining physical or mathematical operation.

In quantum thermodynamics, the working medium may consist of two multilevel systems exchanging energy through a partial-swap unitary (Sacchi, 2021). There the free Hamiltonians are

$i\leftrightarrow j$66

and the work stroke is governed by

$i\leftrightarrow j$67

where $i\leftrightarrow j$68 is the swap operator (Sacchi, 2021). Full swap corresponds to $i\leftrightarrow j$69, for which $i\leftrightarrow j$70 (Sacchi, 2021). The average work is

$i\leftrightarrow j$71

with $i\leftrightarrow j$72 the mean occupation number of qudit $i\leftrightarrow j$73 (Sacchi, 2021). The engine exhibits heat engine, refrigerator, and thermal accelerator regimes depending on $i\leftrightarrow j$74 relative to $i\leftrightarrow j$75 and 1, while the Otto efficiency

$i\leftrightarrow j$76

remains non-fluctuating for all $i\leftrightarrow j$77 (Sacchi, 2021). The same work derives exact joint work–heat statistics and a TUR bound

$i\leftrightarrow j$78

showing small violations of the standard TUR at low $i\leftrightarrow j$79 that shrink as $i\leftrightarrow j$80 increases and disappear in the bosonic limit (Sacchi, 2021).

A two-qubit NMR realization of a SWAP engine was subsequently used to test thermodynamic uncertainty relations experimentally (Shende et al., 2024). The cycle comprises initialization into Gibbs states, a SWAP work stroke implemented by an NMR pulse sequence, and relaxation (Shende et al., 2024). For the realized qubit SWAP,

$i\leftrightarrow j$81

the paper reports that the engine can work both as a heat engine and as a refrigerator (Shende et al., 2024). The generalized TUR is obeyed in all working regimes, while the tighter, more specific TUR is violated in certain regimes according to the abstract; the detailed block states that TUR-2 is always valid and TUR-1 is violated in parts of the heat engine regime (Shende et al., 2024). For qudits, the paper explicitly writes the general SWAP

$i\leftrightarrow j$82

and states that the heat and work formulas hold verbatim for $i\leftrightarrow j$83-level systems (Shende et al., 2024). This suggests a thermodynamic definition of “Qudit SWAP Engine” in which SWAP is the work-exchange stroke of an Otto-like cycle rather than a routing or synthesis primitive.

In a different direction, swap operators generate the Hamiltonians of Quantum Max $i\leftrightarrow j$84-Cut. For a graph $i\leftrightarrow j$85,

$i\leftrightarrow j$86

or, in the weighted form used in the paper,

$i\leftrightarrow j$87

with each $i\leftrightarrow j$88 unitary, Hermitian, and involutive (Klep et al., 26 Mar 2025). The algebra generated by these operators is presented as a quotient of a free *-algebra by symmetric-group relations and a single antisymmetrizer relation of degree $i\leftrightarrow j$89, equivalent to the vanishing of $i\leftrightarrow j$90 (Klep et al., 26 Mar 2025). This yields a tailored noncommutative polynomial optimization hierarchy for computing $i\leftrightarrow j$91 (Klep et al., 26 Mar 2025). In this setting, a “Qudit SWAP Engine” is an algebraic-computational framework built around the swap-generated operator algebra, rather than a hardware pulse sequence.

Finally, controlled-SWAP in qudit space is also an entanglement-diagnostic primitive. For two $i\leftrightarrow j$92-dimensional systems, the SWAP operator satisfies

$i\leftrightarrow j$93

and for $i\leftrightarrow j$94 reduces to the purity $i\leftrightarrow j$95 (Foulds et al., 2021). The controlled-SWAP test generalizes to qudits with a qubit ancilla controlling a SWAP on $i\leftrightarrow j$96-level subsystems, and the paper emphasizes that for odd $i\leftrightarrow j$97 the Bell-basis equivalence used at $i\leftrightarrow j$98 does not extend, making c-SWAP the preferred route (Foulds et al., 2021). Here again, the SWAP engine is a measurement module rather than a transport or synthesis layer.

7. Design principles, limitations, and open structure

Taken together, the literature defines several recurring design principles for any qudit SWAP engine.

First, dimension matters structurally. In CSUM-only two-qudit circuits, parity yields a clean impossibility for $i\leftrightarrow j$99 under strict resource restrictions (Wilmott, 2011). In exact three-gate constructions, arbitrary $d$00 is allowed, but only because the primitive set includes controlled-shift, controlled-phase, Fourier, or modular-negation resources that lie outside the restricted CSUM-only setting (Garcia-Escartin et al., 2013, Vezvaee et al., 2024).

Second, primitive count is not the only notion of cost. In software-defined multilevel control, the relevant objective is often to replace deep decompositions by a single calibrated waveform (Wu et al., 2020). In routing, the relevant comparison is often between explicit SWAP insertion and SWAP-less higher-dimensional transport (Saha et al., 2021). In ququart Fermi–Hubbard simulation, the decisive reduction comes from encoding and nearest-neighbor transpilation rather than from a standalone SWAP gate improvement (Vezvaee et al., 2024).

Third, spectator levels are both a resource and a liability. Temporary occupation of auxiliary levels enables SWAP-less routing (Saha et al., 2021), level-selective direct exchanges (Wu et al., 2020), and compact multilevel logical encodings (Vezvaee et al., 2024). At the same time, spectator-mode occupations generate cross-Kerr detunings that can spoil optimized pulses unless $d$01 (Özgüler et al., 2022). This tension is one of the defining engineering features of qudit architectures.

Fourth, additional resources are often decisive but not universally analyzed. The CSUM-impossibility work explicitly excludes ancillas, single-qudit gates, measurements, and classical control, and does not discuss whether those resources could circumvent the obstruction (Wilmott, 2011). The software-defined optimal-control work, by contrast, treats calibrated Hamiltonian knowledge and transfer-function compensation as the essential enabling resources (Wu et al., 2020). The SWAP-less routing work assumes availability of controlled increments conditioned on occupancy of auxiliary sectors (Saha et al., 2021). A plausible implication is that comparisons between qudit SWAP engines are only meaningful after fixing the full resource model.

Several open structural points remain visible even within the cited results. For $d$02 or $d$03, parity does not obstruct CSUM-only SWAP, but no explicit general construction or minimal gate count is given in the parity analysis (Wilmott, 2011). For larger even composite dimensions, the parity of CSUM can itself vary in ways that the paper does not fully classify (Wilmott, 2011). In optimal-control settings, robustness features such as detuning-averaged costs and composite pulses are identified as natural extensions but are not part of the reported baseline simulations (Özgüler et al., 2022). In thermodynamic settings, full-swap and partial-swap engines are exactly analyzable for equally spaced spectra, but non-proportional or strongly anharmonic spectra complicate performance formulas (Shende et al., 2024, Sacchi, 2021).

The common thread is that SWAP in qudit systems is no longer a trivial generalization of the qubit three-CNOT identity. It is a multi-faceted object: a permutation with nontrivial parity, a gate family with hardware-specific syntheses, a routing resource whose necessity can sometimes be eliminated, a source of thermodynamic work exchange, an entanglement probe, and an algebraic generator of graph Hamiltonians (Wilmott, 2011, Garcia-Escartin et al., 2013, Wu et al., 2020, Saha et al., 2021, Sacchi, 2021, Foulds et al., 2021, Shende et al., 2024, Klep et al., 26 Mar 2025). A Qudit SWAP Engine is therefore best understood as the organized set of methods that make those roles operational in finite-dimensional quantum systems.