Grover Adaptive Search for the Higher-Order Formulation of Quadratic Assignment Problems

TL;DR

The Quadratic Assignment Problem (QAP) is NP-hard, and the authors address it with Grover Adaptive Search (GAS) by introducing permutation preparation operators (PPOs) for both QUBO and HUBO formulations. The QUBO approach uses $N^2$ binary variables, while the HUBO approach binary-encodes locations to require $N\lceil\log_2 N\rceil$ variables, showing substantial qubit savings. Algebraic and numerical analyses reveal that both formulations improve convergence over Hadamard-based GAS, with HUBO achieving similar depth and significantly fewer qubits, making it more scalable to larger instances. Collectively, the work demonstrates a practical pathway to exact quantum optimization for QAP on near-term and future quantum hardware, highlighting the trade-offs between qubit count, gate depth, and overall circuit complexity.

Abstract

We demonstrate that the search space of the quadratic assignment problem (QAP), known as an NP-hard combinatorial optimization problem, can be reduced using Grover adaptive search (GAS) with permutation preparation operator (PPO). To that end, we first revise the traditional quadratic unconstrained binary optimization (QUBO) formulation of the QAP into a higher-order unconstrained binary optimization (HUBO) formulation, introducing a binary encoding method. Algebraic analyses in terms of the number of qubits, quantum gates, circuit depth, and query complexity are performed, which indicate that our proposed approach significantly reduces the search space size, improving convergence performance to the optimal solution compared to the conventional one. Furthermore, although the PPO for HUBO has a greater circuit depth than the PPO for QUBO, when the analysis is extended to the entire state preparation operator, both HUBO and QUBO exhibit comparable depths. Therefore, owing to its smaller number of variables, HUBO can be concluded to be more effective.

Figures (12)

• Figure 1: An example of a quantum circuit for the GAS algorithm with $E(\mathrm{x})=1+2x_1-3x_1x_2x_3$ and $y_i=0$.
• Figure 2: Implementation of $\mathrm{U}_{\mathrm{W}}^6$.
• Figure 3: Construction of $\mathrm{U}_\mathrm{P,Q}^2$.
• Figure 4: Expansion from $\ket{P_{k-1}^{\mathrm{Q}}}$ to $\ket{P_{k}^{\mathrm{Q}}}$.
• Figure 5: Construction of $\mathrm{U}_\mathrm{P,Q}^3$.