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

TL;DR

The paper presents a unified framework connecting weighted sum-of-squares (SOS) representations with variational two-particle reduced density matrix (v2RDM) theory to certify lower bounds on electronic-structure ground-state energies. It shows that the SOS dual recovers v2RDM lower bounds and develops explicit SOS constructions for Hubbard and electronic-structure Hamiltonians using spin-adapted and spin-free algebras, including weighted SOS terms to enforce particle-number and spin symmetries. Numerical demonstrations on small molecules and Iron–Sulfur clusters validate near-frustration-free representations and illuminate both the potential quantum-algorithm benefits (e.g., spectral-gap amplification and reduced block encoding costs) and the practical computational challenges (solver scalability, algebra selection). The work provides a solid theoretical linkage between SOS and v2RDM, practical SOS programs, and open-source code to enable further exploration and application in quantum and classical simulations. It suggests that careful choice of SOS generators and solver advances could render these certificates and representations broadly useful for scalable low-energy quantum simulations.

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.

Figures (5)

• Figure 1: Fock space ground-state energy of the six-site one-dimensional PBC Hubbard model (black) along with energy lower bound certificates (blue and red) computed with the Fock-SOS-nn algebra (blue) and the Fock-SOS-nn-dec algebra (red). The different red curves are labeled by a number which is the largest number of contiguous sites to include in the sum of SOS generators ($\delta$).
• Figure 2: $U=0$ one-dimensional periodic boundary six-site Hubbard model energies versus particle number $\eta$ and lower bounds using the Fock-SOS-nn algebra with the weighted SOS constraint of Eq. \ref{['eq:weight_in_sos']}. The yellow line is the Fock-SOS-nn plotted in blue of Fig. \ref{['fig:fock_space_hub']} with the additional ideal polynomial. The red curve adds the decoupled SOS block (Eq. \ref{['eq:o_number']}) to this algebra.
• Figure 3: The (a) potential energy curves for the dissociation of molecular nitrogen and (b) errors in the dissociation curves relative to full CI.
• Figure 4: The (a) potential energy curves for the symmetric double dissociation of water and (b) errors in the dissociation curves relative to full CI.
• Figure 5: Total semidefinite program solver run time for Hydrogen rings of different size. All calculations use the STO-3G basis and converge the SDP to primal feasibility error and primal-dual gap of $1\times 10^{-5}$. All calculations were performed on an Intel Xeon 2.0 GHz CPU. For both solvers six threads were used for each calculation. The BPSDP complexity is $\mathcal{O}(n^{6})$ for $n$ basis functions coming from eigen decomposing the Gram matrix for the level-2 SOS algebra. The reported scaling does not reflect this as the linear solve is the slowest part in the current libsdp implementation.