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This research establishes a formal link between decoder design and Hamiltonian construction in compression-based quantum relaxations, identifying inconsistencies in standard approaches for mixed-degree quadratic functions. By defining a unique decoder-consistent Hamiltonian via POVM pullback, the work provides improved approximation guarantees for problems like MaxCut. The findings offer a critical theoretical refinement for practitioners developing quantum algorithms that rely on encoding and decoding classical variables.
Decoder-Consistent Hamiltonians for POVM-Based Quantum Relaxations
arXiv:2606.05604v1 Announce Type: new 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.
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.
arXiv:2606.05604 [quant-ph]
(or arXiv:2606.05604v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2606.05604 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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