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

TL;DR

The paper addresses permutation-based combinatorial optimization on quantum devices, focusing on how to efficiently generate and explore permutation sets with limited qubits. It introduces QuPer, a variational algorithm whose circuit P(\theta) outputs a doubly-stochastic matrix by combining rotations and parametrized CX gates, leveraging Bruhat decomposition to span a large subset of permutations with $q=\log_2 n$ qubits, and using ancilla qubits to enlarge the span toward near-full permutation coverage. The authors establish theoretical bounds showing the span scales as $2^{O(q^2)}$ without ancilla and as $2^{O((q+m)^2)}$ with $m$ ancilla, implying $m \gtrsim \sqrt{n \log n}$ is necessary for full coverage; they apply QuPer to quadratic assignment and graph isomorphism, reporting competitive results against classical heuristics on small-to-moderate instances and demonstrating simulations up to $n=256$ with $20$ qubits. The approach offers a flexible, hardware-conscious quantum framework for permutation optimization, enabling cost functions to be evaluated classically while maintaining quantum-generated proposal distributions, and highlighting a tunable trade-off between circuit depth, ancilla, and optimizer performance. Overall, the work provides a general, scalable pathway toward quantum-accelerated permutation problems with practical relevance for near-term quantum devices.

Abstract

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.

Figures (23)

• Figure 1: Pictorial description of the proposed variational approach, labeled QuPer, with the reference to all the sections where each element is discussed.
• Figure 2: Relations between $\mathtt{cx}$ and $\mathtt{x}$ introduced in the new group.
• Figure 3: Variational circuit $\mathscr{B}(\theta)$ spanning the Borel subgroup $B_6$.
• Figure 4: Construction of a parameterized $\mathtt{swap}$ gate.
• Figure 5: Variational circuit $\mathscr{W}(\phi)$ spanning the Weyl subgroup $W$.

Theorems & Definitions (34)

• Theorem 1: Informal
• Example 1: Symmetric group $S_n$
• Proposition 1: bataille2022quantum
• Proposition 2: steinberg2016lecturesbataille2022quantum
• Theorem 2: The Bruhat decomposition bruhat1972groupesbourbaki1968groupes
• Proposition 3
• proof
• Definition 1: Semi-direct product dummit2004abstract
• Proposition 4: johnson1997presentations
• Proposition 5