Efficient Fourier-Based Linear Combination of Unitaries and Applications in Quantum Optimization

TL;DR

This paper introduces Fourier-based linear combination of unitaries (LCU) techniques for efficient, hardware-friendly quantum circuit decompositions, enhancing near-term quantum optimization with rigorous guarantees.

Contribution

It develops Fourier-based LCU constructions for broad classes of unitaries, enabling simpler quantum circuits with controlled sampling overhead in optimization algorithms.

Findings

Efficient decomposition of complex unitaries into single-qubit gates.

Maintains performance guarantees compared to fully coherent implementations.

Validated on 12-qubit simulations and 106-qubit experiments.

Abstract

We investigate ancilla-free linear combination of unitaries (LCU) as a framework for approximating complex quantum circuits. This is particularly effective for quantum optimization algorithms, where candidate solutions can be evaluated classically and the task is to sample high-quality bitstrings rather than reproduce the full output distribution. We show that Fourier-based LCU constructions efficiently decompose broad classes of diagonal and non-diagonal unitaries, replacing highly connected qubit interactions with single-qubit gate layers or significantly simpler structures at the cost of a polynomial sampling overhead. Applied to algorithms such as QAOA, this yields efficient, hardware-friendly decompositions of, for instance, cardinality-constraint penalties and the fully connected XY-mixer, while maintaining rigorous performance guarantees compared to fully coherent implementations. Furthermore, we establish a formal connection between Fourier-based quantum penalties and Lagrangian relaxation, offering a unified perspective on constraint handling. We validate our approach using exact statevector simulations of 12-qubit circuits and large-scale experiments on 106 superconducting qubits. Our results illustrate how approximate sampling via an LCU systematically trades circuit complexity for sampling overhead, extending the practical reach of near-term quantum optimization.