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

• URL: http://arxiv.org/abs/2605.18985v1
• Date: Mon, 18 May 2026 18:05:08 GMT
• Title: Efficient Fourier-Based Linear Combination of Unitaries and Applications in Quantum Optimization
• Abstract summary: We investigate ancilla-free linear combination of unitaries (LCU) as a framework for approximating complex quantum circuits.<n>We show that Fourier-based LCU constructions efficiently decompose broad classes of diagonal and non-diagonal unitaries.
• Score: 0.7009487789080343
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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