The Complexity of Quantum Circuit Mapping with Fixed Parameters

TL;DR

This paper investigates the computational complexity of quantum circuit mapping (QCM) with fixed parameters, providing algorithms and complexity classifications for various scenarios relevant to NISQ device implementation.

Contribution

It introduces an exact polynomial-time algorithm for QCM with fixed architecture and analyzes the problem's complexity under different fixed parameters, including qubit count and circuit depth.

Findings

Exact polynomial-time algorithm for fixed-architecture QCM

QCM is NL-complete when qubit count is fixed

QCM is W[1]-hard when parameterized by the number of qubits

Abstract

A quantum circuit must be preprocessed before implementing on NISQ devices due to the connectivity constraint. Quantum circuit mapping (QCM) transforms the circuit into an equivalent one that is compliant with the NISQ device's architecture constraint by adding SWAP gates. The QCM problem asks the minimal number of auxiliary SWAP gates, and is NP-complete. The complexity of QCM with fixed parameters is studied in the paper. We give an exact algorithm for QCM, and show that the algorithm runs in polynomial time if the NISQ device's architecture is fixed. If the number of qubits of the quantum circuit is fixed, we show that the QCM problem is NL-complete by a reduction from the undirected shortest path problem. Moreover, the fixed-parameter complexity of QCM is W[1]-hard when parameterized by the number of qubits of the quantum circuit. We prove the result by a reduction from the clique problem. If taking the depth of the quantum circuits and the coupling graphs as parameters, we show that the QCM problem is still NP-complete over shallow quantum circuits, and planar, bipartite and degree bounded coupling graphs.