Multivariate Decoded Quantum Interferometry for Weighted Optimization

TL;DR

This paper introduces multivariate Decoded Quantum Interferometry (DQI) for weighted optimization, enhancing quantum algorithms' ability to handle weighted constraints and outperform classical methods in specific problems.

Contribution

The work develops multivariate DQI for weighted Max-LINSAT, providing explicit circuits, asymptotic analysis, and demonstrating superiority over classical algorithms in certain cases.

Findings

Multivariate DQI achieves higher expectation values than classical algorithms in weighted Max-LINSAT.

Explicit quantum circuits for state preparation are provided with a single decoder call.

Analysis includes imperfect decoding and extension to Hamiltonian DQI for block-structured Hamiltonians.

Abstract

Decoded Quantum Interferometry (DQI) is a recently introduced quantum algorithm that reduces discrete optimization to decoding with potential advantages over the best-known polynomial-time classical algorithms for certain Max-LINSAT problems. In its original formulation, however, DQI treats all constraints uniformly and cannot exploit the weight structure present in most optimization problems of interest. In this work, we develop multivariate Decoded Quantum Interferometry (multivariate DQI) for weighted optimization problems, focusing on the weighted Max-LINSAT problem over a prime field. Grouping constraints into $N$ blocks by distinct weights, we introduce multivariate DQI states built from $N$-variable polynomials of bounded total degree, and derive a closed-form asymptotic expression for both their optimal expectation value and their concentration behavior. We give an explicit preparation circuit using a single decoder call, and extend the analysis to imperfect decoding. We also show that, for certain weighted OPI problems, multivariate DQI outperforms a natural weighted analogue of Prange's algorithm, which serves as the weighted counterpart of the classical benchmark used in the unweighted setting. Finally, we extend the ideas to Hamiltonian DQI, obtaining approximate Gibbs states for commuting Pauli Hamiltonians with block structure.