Decoder-Consistent Hamiltonians for POVM-Based Quantum Relaxations

Abstract

In compression-based quantum relaxations like QRAO, classical variables are encoded into qubits and decoded after optimization. We formalize that the choice of the quantum Hamiltonian is fundamentally determined by this decoder. By representing the decoder as a POVM, we define a unique decoder-consistent Hamiltonian via the pullback of the post-decoding expected objective value. Using this framework, we reveal that standard QRAO Hamiltonians are inconsistent for certain mixed-degree quadratic functions, and we provide new approximation guarantees for the MaxCut problem based directly on POVM decoder design.