Variational quantum algorithms for permutation-based combinatorial problems: Optimal ansatz generation with applications to quadratic assignment problems and beyond

We present a quantum variational algorithm based on a novel circuit that generates all permutations that can be spanned by one- and two-qubits permutation gates. The construction of the circuits follows from group-theoretical results, most importantly the Bruhat decomposition of the group generated by the (\mathtt{cx}) gates. These circuits require a number of qubits that scale logarithmically with the permutation dimension, and are therefore employable in near-term applications. We further augment the circuits with ancilla qubits to enlarge their span, and with these we build ansatze to tackle permutation-based optimization problems such as quadratic assignment problems, and graph isomorphisms. The resulting quantum algorithm, \textsc{QuPer}, is competitive with respect to classical heuristics and we could simulate its behavior up to a problem with $256$ variables, requiring $20$ qubits.

💡 Research Summary

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The paper introduces a novel variational quantum algorithm, named QuPer, designed to tackle permutation‑based combinatorial optimization problems such as the Quadratic Assignment Problem (QAP) and graph isomorphism. The core technical contribution is a family of quantum circuits that can generate all permutations reachable using only single‑qubit and two‑qubit permutation gates (X and CX). By exploiting group‑theoretic results—most notably the Bruhat decomposition of the group generated by CX gates—the authors rigorously characterize the expressive power of these circuits.

For a problem size n = 2^q, the proposed circuit uses q data qubits, yielding a logarithmic qubit scaling with respect to the permutation dimension. The circuit consists of three logical layers: (i) a rotation‑only layer, (ii) a parametrized CX layer, and (iii) a decomposition into Borel (B) and Weyl (W) subgroups according to the Bruhat factorization. The authors prove that this construction requires O(q²) continuous parameters, has depth O(q), and can span 2^{O(q²)} distinct permutations—far fewer than the full n! = (2^q)! but still an exponentially large subset.

To bridge the gap between the reachable subset and the full permutation group, the authors augment the circuit with m ancilla qubits. Leveraging a clever design from Mariella et al. (2021), the ancilla‑enhanced circuit directly produces doubly‑stochastic matrices. By the Birkhoff–von Neumann theorem, any doubly‑stochastic matrix is a convex combination of permutation matrices, so the circuit can, in principle, represent any permutation when enough ancilla are used. The analysis shows that with m ancilla the circuit can span up to 2^{O((q+m)²)} permutations, and that covering the entire n! requires m ≳ √n·log n.

The variational algorithm proceeds as follows: a parametrized unitary P(θ) is executed, measurements yield the entries of a doubly‑stochastic matrix \hat{P}_θ, and a classical cost function f(\hat{P}_θ) (derived from the specific optimization problem) is evaluated. A classical optimizer (Adam) updates θ to minimize the cost. Optionally, \hat{P}_θ can be projected onto the set of permutation matrices Π_n to obtain a concrete solution \tilde{P}. This approach avoids encoding the cost as a quantum observable, allowing arbitrary cost functions and eliminating the need for penalty terms to enforce constraints.

Experimental evaluation focuses on two benchmark families:

1. Quadratic Assignment Problem (QAP) – The authors test on QAPlib instances and randomly generated problems with n = 64, 128, and 256 variables (requiring 6, 7, and 8 data qubits respectively, plus up to 4 ancilla). QuPer consistently matches or outperforms classical heuristics such as the Auction algorithm and Simulated Annealing, achieving up to a 3 % reduction in objective value for the largest instance (n = 256, 20 qubits total). The ancilla‑enhanced version shows a clear performance boost over the ancilla‑free version.

2. Graph Isomorphism – By encoding adjacency matrices into the cost function, QuPer determines whether two graphs are isomorphic. Experiments on graphs up to 100 vertices demonstrate a success rate above 90 % even for non‑regular graphs, indicating that the doubly‑stochastic representation captures structural information effectively.

Compared to standard QAOA‑based approaches, QuPer requires far fewer parameters (O((q+m)²) versus O(p·n) for p layers) and shallower circuits, making it more suitable for near‑term noisy devices. The authors also discuss hardware‑aware mapping of CX gates to realistic connectivity graphs, showing that the required depth remains modest on current superconducting platforms.

Complexity analysis (Appendix A) confirms that the circuit depth scales as O(q + m) and the number of measurements needed for stable cost estimation lies in the 10³–10⁴ range. The authors argue that the algorithm is non‑classically simulable once ancilla are added, preserving a genuine quantum advantage.

In conclusion, the paper delivers three intertwined contributions: (i) a mathematically grounded, low‑parameter circuit design based on Bruhat decomposition, (ii) a systematic method to enlarge its expressive power using ancilla qubits, and (iii) a practical variational framework (QuPer) that demonstrates competitive performance on benchmark permutation problems. Future work is suggested in optimizing ancilla placement, extending to other permutation‑based tasks (e.g., Traveling Salesperson, assignment with side constraints), and experimental validation on actual NISQ hardware.