A polynomial size model with implicit SWAP gate counting for exact qubit reordering

TL;DR

This paper presents a polynomial-size integer linear programming model for the qubit reordering problem in quantum circuits, enabling optimal solutions for larger instances and facilitating heuristic development.

Contribution

It introduces a scalable ILP model for the Nearest Neighbor Compliance problem, solving larger benchmark instances to optimality and aiding heuristic design.

Findings

Successfully solved 131 benchmark instances to optimality.

Handled circuits with up to 18 qubits and over 100 gates.

Model is suitable for heuristic development due to quick solution discovery.

Abstract

Due to the physics behind quantum computing, quantum circuit designers must adhere to the constraints posed by the limited interaction distance of qubits. Existing circuits need therefore to be modified via the insertion of SWAP gates, which alter the qubit order by interchanging the location of two qubits' quantum states. We consider the Nearest Neighbor Compliance problem on a linear array, where the number of required SWAP gates is to be minimized. We introduce an Integer Linear Programming model of the problem of which the size scales polynomially in the number of qubits and gates. Furthermore, we solve $131$ benchmark instances to optimality using the commercial solver CPLEX. The benchmark instances are substantially larger in comparison to those evaluated with exact methods before. The largest circuits contain up to $18$ qubits or over $100$ quantum gates. This formulation also seems to be suitable for developing heuristic methods since (near) optimal solutions are discovered quickly in the search process.