Permutation-Preserving Ansatz

• The paper presents a quantum variational ansatz using group-theoretic decompositions that ensures the circuit generates only valid permutation configurations.
• The ansatz integrates exact encoding, feasibility-preserving dynamics, and symmetry-preserving parameter tying to directly embed combinatorial constraints.
• Empirical studies show reduced qubit overhead and competitive performance against classical heuristics in tasks like TSP and graph isomorphism.

Searching arXiv for the primary paper and closely related work on permutation-preserving ansätze. Permutation-preserving problem-inspired ansatz denotes a class of variational constructions in which the circuit architecture is tailored to permutation-structured feasible sets, so that the ansatz acts directly on valid permutations, tours, schedules, or closely related combinatorial objects rather than on a larger constrained space that must later be penalized. In the quantum-variational setting, the most explicit use of this phrase appears in QuPer, introduced for permutation-based combinatorial optimization such as quadratic assignment and graph isomorphism (Mermoud et al., 9 May 2025). Closely related constructions arise in compactly encoded traveling-salesman solvers (Lin et al., 1 May 2026, Fagiolo et al., 29 Aug 2025), graph-controlled feasibility-preserving mixers for scheduling (Palackal et al., 2023), and permutation-invariant circuit families exploiting symmetry reduction (Mansky et al., 2023). Across these works, the common design principle is that permutation structure is not treated as an afterthought: it determines the encoding, the elementary generators, the mixer or variational blocks, and often the post-processing map to the final discrete solution.

1. Concept and defining characteristics

In QuPer, the optimization target is written as

$\min_{P\in\Pi_n}\; f(P),$

where $\Pi_n$ is the set of all $n\times n$ permutation matrices and $f:\Pi_n\to\mathbb R$ is an arbitrary cost (Mermoud et al., 9 May 2025). The ansatz is problem-inspired because its circuit is derived from group-theoretical structure specific to permutations rather than from a generic hardware-efficient template. It is permutation-preserving in the sense that its elementary gates span a subgroup of permutation unitaries, and, with ancilla augmentation, generate convex combinations of such permutations represented as doubly-stochastic matrices before classical projection back to $\Pi_n$ (Mermoud et al., 9 May 2025).

A related but stricter meaning of permutation-preserving appears in compact TSP formulations. In the Lehmer-factorial encoding of tours, every computational-basis state corresponds to exactly one valid tour, so “there is no need for additional penalty terms or one-hot constraints” (Fagiolo et al., 29 Aug 2025). In the resource-efficient TSP framework, the ansatz is built from controlled register-swaps acting on whole city registers; because these blocks “never produce invalid codes or duplicate-city configurations,” the evolution remains confined to the valid permutation subspace (Lin et al., 1 May 2026).

For scheduling-type problems, the same principle is implemented at the mixer level rather than in the encoding. Graph-controlled Permutation Mixers in CM-QAOA perform small permutations of assignment bits only when a Boolean feasibility predicate certifies that the move remains in the feasible set (Palackal et al., 2023). This preserves constraints by construction rather than by adding energetic penalties.

These variants suggest a useful taxonomy. One branch preserves permutations by exact encoding, another by feasibility-preserving dynamics, and another by symmetry-preserving parameter tying. A plausible implication is that the phrase encompasses a family of techniques rather than a single circuit template.

2. Group-theoretical and encoding foundations

QuPer starts from amplitude encoding with $n=2^q$, where a $q$-qubit register in state $\sum_{i=0}^{2^q-1}\alpha_i\ket i$ encodes an $n$-dimensional vector (Mermoud et al., 9 May 2025). The relevant gate-generated group is

$LX_q=\langle X_q\cup CX_q\rangle,$

with

$\Pi_n$0

and every $\Pi_n$1 uniquely factorized as

$\Pi_n$2

The paper then invokes the Bruhat factorization

$\Pi_n$3

where $\Pi_n$4 is the Borel subgroup of invertible upper-triangular matrices and $\Pi_n$5 is the Weyl subgroup of permutation matrices (Mermoud et al., 9 May 2025). This decomposition directly induces the three-block circuit structure of the ansatz.

In the TSP compact-encoding literature, the foundation is different but analogous. The Lehmer-factorial map sends a tour $\Pi_n$6 to an integer

$\Pi_n$7

represented on

$\Pi_n$8

qubits (Fagiolo et al., 29 Aug 2025). Since each basis state decodes to a unique tour, the encoding itself implements the combinatorial admissibility condition.

The resource-efficient TSP framework instead fixes one city and stores the remaining $\Pi_n$9 tour positions in $n\times n$0-qubit registers: $n\times n$1 Adjacent transpositions are realized by register-swap unitaries $n\times n$2, which act as generators of $n\times n$3 on valid codes (Lin et al., 1 May 2026). Here, the group action is explicit at the circuit level.

For permutation-invariant quantum circuits, the symmetric group $n\times n$4 is identified with SWAPs on $n\times n$5 qubits, and the associated Lie-algebra generators $n\times n$6 yield continuous families

$n\times n$7

The parameter count of the fully permutation-invariant subalgebra scales as

$n\times n$8

which quantifies symmetry-induced reduction in variational degrees of freedom (Mansky et al., 2023).

3. Canonical circuit constructions

The clearest canonical form is the QuPer system ansatz

$n\times n$9

where the $f:\Pi_n\to\mathbb R$0-block is built from parallel single-qubit $f:\Pi_n\to\mathbb R$1 gates, the $f:\Pi_n\to\mathbb R$2-blocks from parametrized $f:\Pi_n\to\mathbb R$3 gates corresponding to upper-triangular transvections, and the $f:\Pi_n\to\mathbb R$4-block from parametrized adjacent $f:\Pi_n\to\mathbb R$5 gates implementing simple reflections (Mermoud et al., 9 May 2025). The resulting circuit has depth $f:\Pi_n\to\mathbb R$6 and parameter count

$f:\Pi_n\to\mathbb R$7

and spans exactly

$f:\Pi_n\to\mathbb R$8

permutation unitaries without ancillas (Mermoud et al., 9 May 2025).

The same work augments this base circuit with $f:\Pi_n\to\mathbb R$9 ancilla qubits. After Hadamards on the ancillas, ancilla-controlled applications of $\Pi_n$0, and uncomputation, the output matrix becomes

$\Pi_n$1

so the ansatz represents a convex combination of base permutations rather than a single one (Mermoud et al., 9 May 2025). This is the “quantum boost” mechanism that enlarges the reachable region of the permutation polytope.

In the resource-efficient TSP formulation, a single adjacent-link variational block is

$\Pi_n$2

with $\Pi_n$3 the ancilla-controlled swap of all $\Pi_n$4 qubits in two neighboring registers (Lin et al., 1 May 2026). A layer applies these blocks in ascending order: $\Pi_n$5 and the full ansatz of depth $\Pi_n$6 is

$\Pi_n$7

with $\Pi_n$8 parameters (Lin et al., 1 May 2026).

In “Freeze and Conquer,” the ansatz is not derived from explicit group decompositions but from a discrete search over 675 topologies of the form

$\Pi_n$9

with rotation axes $n=2^q$0 and entanglement patterns $n=2^q$1, under the alternating $n=2^q$2–$n=2^q$3–$n=2^q$4–$n=2^q$5–$n=2^q$6 constraint (Fagiolo et al., 29 Aug 2025). The fixed five-block circuit

$n=2^q$7

is applied directly to the compact permutation encoding, so the gates “preserve the permutation encoding automatically” (Fagiolo et al., 29 Aug 2025).

4. Feasibility preservation, invariance, and expressivity

The central technical question is whether the ansatz preserves the feasible combinatorial subspace and whether it can adequately traverse that subspace.

For graph-controlled permutation mixers, feasibility is formalized with a constraint graph $n=2^q$8 whose edges encode assignment-clash, precedence-clash, and machine-clash relations in the flexible job-shop problem (Palackal et al., 2023). A permutation $n=2^q$9 is allowed on a bit string $q$0 only when the Boolean predicate

$q$1

holds, with

$q$2

The controlled mixer

$q$3

moves amplitude from $q$4 to $q$5 only when $q$6, and Theorem 4.1 shows that the full CM-QAOA state remains inside the feasible subspace $q$7 at every step (Palackal et al., 2023). Theorem 4.2 then establishes connectivity by showing that any two feasible schedules can be linked by a sequence of allowed transpositions.

In the resource-efficient TSP ansatz, feasibility is preserved more simply. Because the building blocks are controlled swaps of whole registers containing valid city codes, they “never produce invalid codes or duplicate-city configurations” and therefore keep the state within the valid permutation subspace (Lin et al., 1 May 2026). The state can be expanded as

$q$8

with marginal tour probabilities

$q$9

This is exact feasible-subspace evolution rather than approximate penalty suppression (Lin et al., 1 May 2026).

Permutation-invariant circuits address a different property: invariance under relabeling. The invariance condition

$\sum_{i=0}^{2^q-1}\alpha_i\ket i$0

is enforced by symmetrizing parameters over orbits of $\sum_{i=0}^{2^q-1}\alpha_i\ket i$1, for example through blocks such as

$\sum_{i=0}^{2^q-1}\alpha_i\ket i$2

A two-layer example interleaves global $\sum_{i=0}^{2^q-1}\alpha_i\ket i$3-rotations and all-to-all SWAP-family generators (Mansky et al., 2023). This does not encode permutations as solutions; instead it restricts the variational search to the $\sum_{i=0}^{2^q-1}\alpha_i\ket i$4-invariant sector.

A common misconception is that any ansatz acting on permutation-labeled data is automatically permutation-preserving. The literature distinguishes at least three non-equivalent properties: preserving the validity of encoded permutations (Lin et al., 1 May 2026, Fagiolo et al., 29 Aug 2025), preserving feasibility under constrained moves (Palackal et al., 2023), and being invariant under qubit or label permutations (Mansky et al., 2023).

5. Optimization workflows and cost evaluation

The optimization loop depends strongly on how the ansatz represents solutions.

QuPer does not encode the objective as a Hamiltonian. Instead, it samples a doubly-stochastic matrix $\sum_{i=0}^{2^q-1}\alpha_i\ket i$5, optionally projects it to a permutation $\sum_{i=0}^{2^q-1}\alpha_i\ket i$6, and evaluates the cost classically (Mermoud et al., 9 May 2025). For quadratic assignment,

$\sum_{i=0}^{2^q-1}\alpha_i\ket i$7

and for graph isomorphism,

$\sum_{i=0}^{2^q-1}\alpha_i\ket i$8

The cost is computed after projection in $\sum_{i=0}^{2^q-1}\alpha_i\ket i$9 time, while gradients are estimated by the parameter-shift rule plus classical postprocessing (Mermoud et al., 9 May 2025).

The TSP resource-efficient framework uses a diagonal distance Hamiltonian on the feasible subspace: $n$0 In the divide-and-conquer formulation, the full Hamiltonian is expanded as

$n$1

and, under a product-state approximation,

$n$2

A single basis measurement yields one feasible tour sample $n$3, and after $n$4 shots the empirical estimator is

$n$5

with $n$6 the tour length (Lin et al., 1 May 2026).

In “Freeze and Conquer,” the optimize–freeze–reuse workflow separates structural search from parameter search. Training uses simulated annealing over the discrete topology space, with fitness

$n$7

the probability that a measurement yields the optimal tour of the training instance (Fagiolo et al., 29 Aug 2025). The simulated-annealing loop uses $n$8, cooling rate $n$9, $LX_q=\langle X_q\cup CX_q\rangle,$0, and iteration cap $LX_q=\langle X_q\cup CX_q\rangle,$1; each candidate topology is evaluated by a local VQE in which 100 random parameter vectors are sampled, the best 10 are refined by Powell, and final $LX_q=\langle X_q\cup CX_q\rangle,$2 is estimated from $LX_q=\langle X_q\cup CX_q\rangle,$3 shots (Fagiolo et al., 29 Aug 2025). After training, the best topology is frozen and reused on new instances, with only Powell re-optimization of parameters from the trained initialization.

These workflows illustrate two contrasting strategies. QuPer and the divide-and-conquer TSP formulation rely on classically reconstructed costs from sampled feasible outputs; Freeze and Conquer amortizes architectural search across instances. This suggests that problem-inspired permutation ansätze can support both instance-specific and reusable regimes.

6. Resources, empirical behavior, and limitations

The principal resource advantage emphasized across the literature is qubit compression.

Selected constructions and reported scaling

Construction	Qubit / variable scaling	Key circuit or parameter scaling
QuPer system register	$LX_q=\langle X_q\cup CX_q\rangle,$4 plus $LX_q=\langle X_q\cup CX_q\rangle,$5 ancillas	$LX_q=\langle X_q\cup CX_q\rangle,$6, depth $LX_q=\langle X_q\cup CX_q\rangle,$7 (Mermoud et al., 9 May 2025)
Lehmer-encoded TSP	$LX_q=\langle X_q\cup CX_q\rangle,$8	5-block ansatz, 675 candidate topologies (Fagiolo et al., 29 Aug 2025)
Register-swap TSP ansatz	$LX_q=\langle X_q\cup CX_q\rangle,$9 data qubits plus one ancilla	$\Pi_n$00 (Lin et al., 1 May 2026)
GCPM for FJSP	$\Pi_n$01 main qubits plus ancilla registers	Worst-case Toffoli count $\Pi_n$02, depth roughly $\Pi_n$03 (Palackal et al., 2023)
Permutation-invariant circuits	symmetry-reduced parameter space	$\Pi_n$04 parameters (Mansky et al., 2023)

QuPer is explicitly intended for near-term use because the number of system qubits scales logarithmically with permutation dimension (Mermoud et al., 9 May 2025). The paper reports simulation up to $\Pi_n$05, requiring $\Pi_n$06 qubits, and states that QuPer is competitive with classical heuristics, with typical optimality gaps of a few percent on QAPlib instances (Mermoud et al., 9 May 2025).

For TSP, the resource-efficient framework reports best average success rates of $\Pi_n$07, $\Pi_n$08, and $\Pi_n$09 for 4-, 5-, and 6-city instances, respectively, in numerical simulation (Lin et al., 1 May 2026). The related Freeze and Conquer study reports average optimal-trip sampling probabilities of $\Pi_n$10 for 4-city cases, $\Pi_n$11 for 5-city cases, $\Pi_n$12 for 6-city cases, and approximately $\Pi_n$13 for 7-city cases after reuse, indicating a marked onset of scalability limitations at 7 cities (Fagiolo et al., 29 Aug 2025).

GCPMs provide strong formal guarantees but with substantial hardware overhead. Implementing the sequential mixer requires the main register, an auxiliary copy register, three further $\Pi_n$14-qubit ancilla registers $\Pi_n$15, two single-qubit ancillae including $\Pi_n$16, and a global-AND qubit; for $\Pi_n$17, the total Toffoli count in one mixer layer is $\Pi_n$18, with gate depth roughly $\Pi_n$19 (Palackal et al., 2023). The work explicitly notes that this is demanding for current NISQ devices.

Permutation-invariant circuits reduce parameter complexity to $\Pi_n$20, but the same paper presents this as “an indication that symmetry restricts the applicability of quantum computing” (Mansky et al., 2023). This caution is important: symmetry reduction improves tractability only when the target low-energy sector or feasible set genuinely respects the imposed symmetry.

7. Related directions and broader significance

The phrase “permutation-preserving problem-inspired ansatz” is most directly associated with QuPer (Mermoud et al., 9 May 2025) and with the resource-efficient TSP framework (Lin et al., 1 May 2026), but the surrounding literature shows that the underlying idea has broader methodological significance.

One extension concerns reusable ansätze. Freeze and Conquer demonstrates that a permutation-preserving compact encoding can be paired with an optimize–freeze–reuse pipeline in which all heavy structural search is performed once and only parameter re-optimization is repeated on new instances (Fagiolo et al., 29 Aug 2025). This suggests that structural priors derived from permutation geometry may transfer across instances of the same problem family.

A second extension concerns constraint graphs and local feasibility checks. GCPMs show that problems whose feasible solutions are independent sets or fixed-size vertex subsets of a graph can be treated by redefining the edge set $\Pi_n$21 and the control function $\Pi_n$22 (Palackal et al., 2023). A plausible implication is that permutation-preserving ansätze are part of a larger class of feasibility-preserving variational constructions for structured combinatorial spaces.

A third extension concerns symmetry-adapted variational design beyond combinatorial optimization. Permutation-invariant circuits provide a Lie-algebraic recipe for enforcing full $\Pi_n$23 symmetry in quantum circuits and for symmetrizing an existing ansatz by orbit-averaging parameters (Mansky et al., 2023). This is conceptually adjacent to permutation-preserving combinatorial ansätze, though the operational goal is different.

Finally, there is an instructive contrast with qubit permutation for layout optimization. PermVQE adds an outer optimization loop that permutes qubits to minimize a mutual-information-based cost

$\Pi_n$24

thereby reducing ansatz depth for chemistry problems (Tkachenko et al., 2020). This work is about permuting the representation of a problem on hardware rather than preserving combinatorial permutations as feasible solutions. The contrast clarifies that “permutation-preserving” in the problem-inspired-ansatz literature refers primarily to solution-space structure, not merely to relabeling qubits.

Taken together, these works define a technically coherent paradigm: exploit the algebra, encoding geometry, and feasibility structure of permutation-based problems to design ansätze whose state space is already adapted to the discrete objects being optimized. The resulting circuits may preserve valid tours exactly (Lin et al., 1 May 2026, Fagiolo et al., 29 Aug 2025), preserve constrained schedules under controlled local permutations (Palackal et al., 2023), or generate rich subsets and mixtures of permutation matrices from group-theoretic building blocks (Mermoud et al., 9 May 2025). The main benefits are reduced qubit overhead, elimination or reduction of penalty terms, and a variational search space better aligned with the combinatorial target; the main limitations are expressivity bottlenecks, classical post-processing overhead, and, in some constructions, substantial ancilla and gate costs.