Classically Driven Hybrid Quantum Algorithms with Sequential Givens Rotations for Reduced Measurement Cost

TL;DR

A diagonalization-driven framework is introduced that progressively drives the electronic Hamiltonian toward a (block-)diagonal form in the Slater-determinant basis using sequential Givens rotations, adopting a Heisenberg-picture viewpoint.

Abstract

Quantum algorithms for electronic-structure simulations are actively being developed, yet many hybrid quantum-classical approaches are bottlenecked by the measurement overhead associated with large molecular Hamiltonians. Here we introduce a diagonalization-driven framework that progressively drives the electronic Hamiltonian toward a (block-)diagonal form in the Slater-determinant basis using sequential Givens rotations. In contrast to Schrödinger-picture methods that variationally optimize a wave function, our approach adopts a Heisenberg-picture viewpoint: the Hamiltonian is iteratively transformed, and rotation angles are determined classically from low-dimensional effective blocks, reducing the quantum workload to a small, fixed set of matrix-element measurements per iteration. Candidate generators are estimated via approximate Baker-Campbell-Hausdorff updates with truncation and cumulant-based approximations that control Hamiltonian growth, complemented by stochastic selection to avoid stagnation. We further introduce an angle-merging procedure that reduces circuit depth by consolidating repeated small-angle rotations. We benchmark the framework on N$_2$ and strongly correlated hydrogen systems, assessing convergence behavior, residual-structure diagnostics, measurement-accuracy trade-offs, circuit costs, and robustness under finite sampling.

Figures (9)

• Figure 1: (a) General representation of the Hamiltonian transformation, where the unitary operator eliminates off-diagonal couplings with respect to the Hartree--Fock determinant $|\Phi_0\rangle$ (red blocks). (b) Specific Hamiltonian update under deterministic selection, with G highlighted in cyan boxes; white and gray entries denote zero and non-zero elements, respectively.
• Figure 2: Schematic outline of the algorithmic workflow. Blue blocks indicate classical processing steps, while pink block represents operations in quantum circuits.
• Figure 3: Energy error relative to FCI (top panels) and Hamiltonian term count (bottom panels) as a function of the number of expectation values for N$_2$ at the near-equilibrium bond length of 1.0977 Å. Results are shown for PQJ, FQJ, and CFQJ at truncation thresholds $\epsilon = 10^{-3}$ and $10^{-4}$, with comparisons to the exact BCH expansion and UCCSD. The red dashed lines in the top panels denote the chemical accuracy threshold.
• Figure 4: Energy error relative to FCI (top panels) and Hamiltonian term count (bottom panels) as a function of the number of expectation values for N$_2$ at the stretched bond length of 1.8 Å. Results are shown for PQJ, FQJ, and CFQJ at truncation thresholds $\epsilon = 10^{-3}$ and $10^{-4}$, with comparisons to the exact BCH expansion and UCCSD. The red dashed lines in the top panels denote the chemical accuracy threshold.
• Figure 5: Residual excitation amplitudes for CFQJ method with a truncation threshold of $10^{-3}$ for N$_2$ at a bond length of 1.8 Å. Each panel corresponds to a different iteration cycle ($k = 0,10,20$ and $k_c=27$), with insets providing zoomed views of selected regions for clarity.