Walsh-Structured Uniformly Controlled Rotations for Variational Quantum State Discrimination

Abstract

Minimum-error quantum state discrimination is usually formulated as an optimization over posi- tive operator-valued measures, but classical semidefinite-programming solutions and universal dilation- based variational implementations become costly for multi-qubit systems because they manipulate ex- ponentially many measurement degrees of freedom. We introduce aWalsh-degree-truncated parametriza- tion of the uniformly controlled rotations in a cosine-sine-decomposition-based variational POVM dis- criminator. In this representation, each rotation-angle profile is expanded in Walsh characters and restricted to degree at most k, so that a single degree parameter controls the number of trainable parameters, the locality of the implementing Pauli rotations, and the accessible measurement family. The resulting models form a nested hierarchy that interpolates between low-locality ansätze and the full CSD-based discriminator. We prove that degree one already attains the optimal minimum-error success probability for any linearly independent pure-state ensemble. Numerical experiments on Weyl- Heisenberg and Haar-random pure-state ensembles in dimension d = 8 support this structure: the degree-one model reaches the semidefinite-programming optimum in the linearly independent regime, develops a gap when the number of states exceeds the Hilbert-space dimension, and outperforms parameter-matched random UCR truncations. In addition, increasing the Walsh degree closes the observed gap. These results identify Walsh degree as an algebraic organizing principle for designing measurement ansätze between hardware-efficient and universal variational POVM models.