Near-frustration-free electronic structure Hamiltonian representations and lower bound certificates

TL;DR

This paper introduces a unified framework linking sum-of-squares (SOS) Hamiltonian representations with variational two-particle reduced density matrix (v2RDM) theory, enabling improved lower bounds and symmetry enforcement for electronic structure problems.

Contribution

It establishes a connection between SOS decompositions and v2RDM, providing explicit constructions for electronic Hamiltonians and demonstrating their effectiveness through numerical benchmarks.

Findings

Validated representations improve spectral gap amplification.

Reduced block encoding costs in quantum algorithms.

Demonstrated applicability to molecular systems and clusters.

Abstract

Hamiltonian representations based on the sum-of-squares (SOS) hierarchy provide rigorous lower bounds on ground-state energies and facilitate the design of efficient classical and quantum simulation algorithms. This work presents a unified framework connecting SOS decompositions with variational two-particle reduced density matrix (v2RDM) theory. We demonstrate that the ``weighted'' SOS ansatz naturally recovers the dual of the v2RDM program, enabling the strict enforcement of symmetry constraints such as particle number and spin. We provide explicit SOS constructions for the Hubbard model and electronic structure Hamiltonians, ranging from spin-free approximations to full rank-2 expansions. We also highlight theoretical connections to block-invariant symmetry shifts. Numerical benchmarks on molecular systems and Iron-Sulfur clusters validate these near frustration-free representations, demonstrating their utility in improving spectral gap amplification and reducing block encoding costs in quantum algorithms.