Exponential speed-up in VQE molecular energy ranking with Sridhara-compressed Hamiltonians

TL;DR

This work extends Sridhara Block Diagonalization to matrix secular equations and applies it to Hartree-Fock Hamiltonians mapped to qubits for six tetracyclic PAHs, aiming to accelerate ground-state energy ranking with VQE on NISQ hardware. By flexibly optimizing the SBD parameters and leveraging a truncated Newton-Schulz expansion, the method achieves substantial compression with controlled error, yielding a $77.8\%$ probability of reproducing the uncompressed VQE energy ranking and a median speed-up of $164.16\%$, while maintaining a small average active-space-reduction error of $0.09\%$. The study demonstrates that SBD-VQE outperforms SBD-Arnoldi for ranking tasks and provides insights into the relationship between block commutativity and estimation error, suggesting broader applications in PCA, vector compression, and Ansatz optimization. Overall, SBD offers a fast, flexible block-diagonalization tool to enable faster quantum chemistry simulations and energy-based molecular ranking on near-term quantum devices.

Abstract

Polycyclic aromatic hydrocarbons (PAHs) are residual and intermediary molecules in the Chemical Vapor Deposition (CVD) to produce graphene from methane. Ranking a combinatorial space of variants of PAHs by energy allows the CVD to be optimized, while simulations of PAHs are strong candidates for quantum advantage in quantum computers. We extend on Sridhara's root formula to perform block diagonalization (SBD) of six PAHs using Hartree-Fock Hamiltonians with STO-3G basis set and $(2,2)$, $(4,4)$, $(6,6)$ settings of active orbitals and active electrons. We show that the proposed SBD algorithm followed by Variational Quantum Eigensolver (VQE) allows ranking molecules by ground state energy with $77.8\%$ of success in comparison with the uncompressed VQE, while speeding up the VQE simulation in $164.16\%$ (median) keeping its average error of active space reduction down to $0.09\%$. We conclude that the flexibilization of constraints of the SBD algorithm makes it a fast and reliable estimator for active space reduction in molecular simulation.

Figures (8)

• Figure 1: The block diagonalization procedure rotates the molecular Hamiltonian into a high-energy and a low-energy block at each step of the iteration.
• Figure 2: Tetracene, benz(a)anthracene, triphenylene, benzo[c]phenanthracene, pyrene and chrysene, respectively, from a to f. TAHs are common residuals and precursors in graphene synthesis.
• Figure 3: Relative error versus adapted Frobenius norm for ground state (circles) and top state (diamonds) estimation after $1$, $2$, $3$ and $4$ steps of SBD compression, for SD (a), PSD (b) and G (c) populations.
• Figure 4: Relative error profiles for the full spectra after four steps of SBD compression for 10000 random Hermitian matrices of size $(2^6,2^6)$, from (a) simultaneously diagonalizable origin, (b) simultaneously diagonalizable origin with small perturbation and (c) general, non-commutative origin.
• Figure 5: Ground state energies as functions of the Hamiltonian size for three sets of electronic active space, using Arnoldi eigensolver and VQE eigensolver. Compression of up to half the number of qubits keeps the VQE ground state slightly stable for any active space.