New Quantum Simulations Promise Faster Routes to Designing Advanced Materials and Molecules

Scientists are continually seeking more efficient methods to represent complex quantum systems and accurately determine their ground-state energies. Nicholas C. Rubin, Guang Hao Low, and A. Eugene DePrince III, from Google Quantum AI and Florida State University respectively, have developed a novel framework linking sum-of-squares (SOS) decompositions with two-particle reduced density matrix (v2RDM) theory. Their research establishes a connection that allows for the rigorous enforcement of symmetry constraints, crucial for modelling realistic materials and molecules. This work presents near frustration-free Hamiltonian representations, validated through benchmarks on molecular systems and Iron-Sulfur clusters, and promises to improve spectral gap amplification and reduce computational costs in quantum simulations.

This work introduces a “weighted” SOS ansatz that naturally aligns with the dual of the v2RDM program, enabling the rigorous enforcement of symmetry constraints like particle number and spin.

The resulting near frustration-free representations demonstrably improve spectral gap amplification and reduce block encoding costs within quantum algorithms. Specifically, the study presents explicit SOS constructions for both the Hubbard model and electronic structure Hamiltonians, progressing from spin-free approximations to full rank-2 expansions.
These constructions leverage the inherent structure of these Hamiltonians to create more efficient and accurate representations. Theoretical connections to block-invariant symmetry shifts are also highlighted, further solidifying the framework’s potential for incorporating complex symmetries. Numerical benchmarks, conducted on molecular systems and Iron-Sulfur clusters, validate the efficacy of these near frustration-free representations.

These tests confirm the ability of the method to generate tighter lower bounds on ground-state energies, a crucial step in optimizing quantum simulations. The research demonstrates a unified approach to improving both p-positivity and locality considerations within the SOS representation, offering a versatile tool for Hamiltonian optimization.

The core innovation lies in the ability to express any Hermitian operator as a sum of squares plus a constant shift, establishing a non-negative representation and certifying a lower bound on ground-state energy. This approach allows for a hierarchy of operators, increasing in degree or spatial extent to generate increasingly precise lower bounds.

By unifying these hierarchies, the study provides a comprehensive protocol for deriving electronic structure Hamiltonian representations and connecting them to the v2RDM method, known for its applications in quantum chemistry. The weighted SOS ansatz, combined with the enforcement of symmetry constraints, represents a significant advancement in the field of quantum simulation.

Sum-of-squares decomposition and two-particle reduced density matrix duality for symmetry-constrained variational optimisation

A sum-of-squares (SOS) hierarchy forms the basis of this work, providing rigorous lower bounds on ground-state energies and aiding in the design of efficient classical and quantum simulations. Hermitian operators within a finite basis set are expressed as a Hermitian sum of squares plus a constant shift, mathematically defined as H − ESOS = Σα O†αOα, establishing a non-negative representation and certifying a lower bound on ground-state energy.

The research unifies this SOS decomposition with two-particle reduced density matrix (v2RDM) theory, demonstrating that a “weighted” SOS ansatz naturally recovers the dual of the v2RDM program. This connection enables the strict enforcement of symmetry constraints, including particle number and spin, within the SOS framework.

Explicit SOS constructions were developed for both the Hubbard model and electronic structure Hamiltonians, progressing from spin-free approximations to full rank-2 expansions. These constructions leverage the ability to improve lower bounds by increasing the degree of the SOS polynomials or expanding their spatial locality, unifying p-positivity and locality considerations.

The study highlights theoretical connections to block-invariant symmetry shifts, revealing how these emerge naturally from a weighted SOS positive ansatz. Numerical benchmarks were performed on molecular systems and Iron-Sulfur clusters to validate these near frustration-free representations, demonstrating their utility in improving spectral gap amplification and reducing block encoding costs. The work also details the construction of spin-adapted SOS programs and provides the mathematical programs and codes necessary to solve for the SOS Hamiltonian representation using different approximations to the quadratic generating algebra.

Symmetry-constrained sum-of-squares decompositions accurately model molecular potential energy surfaces

Researchers demonstrate a unified framework connecting sum-of-squares (SOS) decompositions with two-particle reduced density matrix (v2RDM) theory, enabling strict enforcement of symmetry constraints. Numerical benchmarks on molecular systems and Iron-Sulfur clusters validate these near frustration-free representations, improving spectral gap amplification and reducing block encoding costs.

For molecular nitrogen dissociation, the error in lower-bound estimates from an approximate rank-2 SOS algebra was more than two orders of magnitude larger than that from the full rank-2 algebra at the dissociation limit. Similarly, even larger errors were observed when using a spin-free algebra for the same system.

Analysis of potential energy curves for the dissociation of molecular nitrogen and the symmetric double dissociation of water revealed significant differences in shape when using approximate SOS representations compared to full configuration interaction. Specifically, the absolute energy error for the spin-free algebra was substantially higher than that of the full rank-2 algebra, indicating a diminished quality of the lower-bound estimate.

For the Fe2S2 complex, the spin-free SOS representation yielded a gap of 0.9764, while the spinful representation with the full level-2 spin-adapted algebra achieved a gap of 0.281. This corresponds to a 1.861-fold improvement in query complexity by including higher algebra components. Considering the Fe4S4 complex, the spin-free representation exhibited a gap of 1.8664, contrasted with a gap of 0.718 for the spinful representation, resulting in a 1.611-fold reduction in block encoding calls.

For the larger FeMoco-54 complex, the spin-free and spinful representations showed gaps of 3.4638 and 1.569 respectively, indicating a 1.485-fold improvement in query complexity. Scalings with two SDP solvers for Hydrogen rings of size ten to 30 Hydrogen atoms demonstrate computational feasibility, with extrapolated times suggesting the spin-free dual SOS Hamiltonian construction is feasible within a day of preprocessing for 100 orbital systems.

Sum-of-squares decompositions unify reduced density matrix theory for symmetry-constrained electronic structure calculations

Hamiltonian representations employing the sum-of-squares (SOS) hierarchy offer rigorous lower bounds on ground-state energies and aid in designing efficient classical and quantum simulations. This work unifies SOS decompositions with two-particle reduced density matrix (v2RDM) theory, demonstrating that a weighted SOS ansatz recovers the dual of the v2RDM program and rigorously enforces symmetry constraints like particle number and spin.

Explicit SOS constructions were developed for the Hubbard model and electronic structure Hamiltonians, ranging from spin-free approximations to full rank-2 expansions, and theoretical links to block-invariant symmetry shifts were highlighted. Numerical benchmarks, conducted on molecular systems and Iron-Sulfur clusters, validate these near frustration-free representations, confirming their potential to improve spectral gap amplification and reduce block encoding costs within quantum algorithms.

The research establishes a framework where both increasing the polynomial degree and expanding the locality of the SOS generators can be combined to achieve tighter lower bounds on ground-state energy. However, the authors acknowledge that the SOS ansatz alone is insufficient for constructing witness Hamiltonians when considering symmetries such as a fixed particle manifold. Future research may focus on further exploring the interplay between p-positivity and locality within the SOS framework to optimise Hamiltonian representations for both classical and quantum computation.