Finite-Field Second Hyperpolarizability via Adaptive Bayesian Quantum Phase Estimation for Early Fault-Tolerant Quantum Computers: Ground-State Energies Beyond Chemical Accuracy
- 1Research Center for Computational Science and Informatics, Resonac Corporation, 8, Ebisu-cho, Yokohama, Kanagawa, Japan, 221-8517
Abstract
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.
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BibTeX
@misc{okuno2026finitefield,
title = {{Finite-Field Second Hyperpolarizability via Adaptive Bayesian Quantum Phase Estimation for Early Fault-Tolerant Quantum Computers: Ground-State Energies Beyond Chemical Accuracy}},
author = {Katsuki Okuno},
howpublished = {ChemRxiv},
year = {2026},
month = sep,
doi = {10.26434/chemrxiv.15008602/v1},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15008602/v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - Finite-Field Second Hyperpolarizability via Adaptive Bayesian Quantum Phase Estimation for Early Fault-Tolerant Quantum Computers: Ground-State Energies Beyond Chemical Accuracy
AU - Katsuki Okuno
PY - 2026
DA - 2026/09/09
PB - ChemRxiv
DO - 10.26434/chemrxiv.15008602/v1
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15008602/v1
AB - 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.
ER -