The static second hyperpolarizability governs third-order nonlinear-optical responses but is difficult to compute by first-principles methods because finite-field extraction requires a fourth-order energy derivative. The resulting error amplification demands sub-µHa ground-state energies, far beyond conventional chemical accuracy. Standard multi-qubit QPE can in principle meet this precision but requires a wide ancilla register, while single-ancilla iterative QPE (IQPE) is limited by a binary-grid quantization floor. We develop an adaptive Bayesian QPE (BQPE) workflow that uses a single ancilla, terminal measurements, and real-time classical feedback to estimate the eigenphase through a continuous posterior rather than a fixed binary grid. For a linear H4 chain in the STO-3G basis, represented by a six-qubit parity Hamiltonian with one ancilla, BQPE reaches a ground-state energy error of ≈ 10−7 Ha at controlled-U depth mmax ≈ 215, about two orders of magnitude below the IQPE floor. Propagated through the finite-field stencil and Richardson extrapolation, these energies recover the second hyperpolarizability within ±10% of the FCI reference over the stable field-strength plateau, whereas IQPE at the same depth fails to do so. These results indicate that grid-free Bayesian phase estimation can make nonlinear-optical properties accessible on early fault-tolerant quantum computers.
Quantum simulation of molecular electronic structure is one of the most promising applications of quantum computing. However, achieving chemically accurate predictions for strongly correlated systems requires quantum phase estimation (QPE) on fault-tolerant quantum computing (FTQC) devices. Existing resource estimates for typical FTQC architectures suggest that such calculations demand millions of physical qubits, thereby placing them beyond the reach of near-term devices. Here, we investigate the feasibility of performing QPE for chemically relevant molecular systems in an early-FTQC regime, characterized by partial fault tolerance, constrained qubit budgets, and limited circuit depth. Our framework is based on single-ancilla, Trotter-based QPE implementations combined with partially randomized time evolution. Within this framework, we develop a novel Hamiltonian optimization strategy, termed unitary weight concentration, that reduces algorithmic cost by reshaping linear-combination-of-unitaries representations. Applying this framework to active-space models of iron-sulfur clusters, cytochrome P450 active sites, and CO
2-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using
∼105 physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.
The static second hyperpolarizability governs third-order nonlinear-optical responses but is difficult to compute by first-principles methods because finite-field extraction requires a fourth-order energy derivative. The resulting error amplification demands sub-µHa ground-state energies, far beyond conventional chemical accuracy. Standard multi-qubit QPE can in principle meet this precision but requires a wide ancilla register, while single-ancilla iterative QPE (IQPE) is limited by a binary-grid quantization floor. We develop an adaptive Bayesian QPE (BQPE) workflow that uses a single ancilla, terminal measurements, and real-time classical feedback to estimate the eigenphase through a continuous posterior rather than a fixed binary grid. For a linear H4 chain in the STO-3G basis, represented by a six-qubit parity Hamiltonian with one ancilla, BQPE reaches a ground-state energy error of ≈ 10−7 Ha at controlled-U depth mmax ≈ 215, about two orders of magnitude below the IQPE floor. Propagated through the finite-field stencil and Richardson extrapolation, these energies recover the second hyperpolarizability within ±10% of the FCI reference over the stable field-strength plateau, whereas IQPE at the same depth fails to do so. These results indicate that grid-free Bayesian phase estimation can make nonlinear-optical properties accessible on early fault-tolerant quantum computers.