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Finite-Field Second Hyperpolarizability via Adaptive Bayesian Quantum Phase Estimation for Early Fault-Tolerant Quantum Computers: Ground-State Energies Beyond Chemical Accuracy

Katsuki Okuno

Published Sep 9, 2026• 0 citations
Publisher

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.

DOI
10.26434/chemrxiv.15008602/v1

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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.
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 CO2_22​-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\sim 10^5∼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.
Under suitable assumptions, the quantum-phase-estimation (QPE) algorithm is able to achieve Heisenberg-limited precision scaling in estimating the ground-state energy.However, QPE requires a large number of ancilla qubits and a large circuit depth, as well as the ability to perform inverse quantum Fourier transform, making it expensive to implement on an early fault-tolerant quantum computer.We propose an alternative method to estimate the ground-state energy of a Hamiltonian with Heisenberglimited precision scaling, which employs a simple quantum circuit with one ancilla qubit, and a classical postprocessing procedure.Besides the ground-state energy, our algorithm also produces an approximate cumulative distribution function of the spectral measure, which can be used to compute other spectral properties of the Hamiltonian.
We compare several quantum phase estimation (QPE) protocols intended for early fault-tolerant quantum computers (EFTQCs) in the context of models of their implementations on a surface code architecture. We estimate the logical and physical resources required to use these protocols to calculate the ground state energy of molecular hydrogen in a minimal basis with error below 10−310^{-3}10−3 atomic units in the presence of depolarizing logical errors. Accounting for the overhead of rotation synthesis and magic state distillation, we find that the total TTT-gate counts do not vary significantly among the EFT QPE protocols at fixed state overlap. In addition to reducing the number of ancilla qubits and circuit depth, the noise robustness of the EFT protocols can be leveraged to reduce resource requirements below those of textbook QPE, realizing approximately a 300-fold reduction in computational volume in some cases. Even so, our estimates are well beyond the scale of existing early fault-tolerance demonstrations.
We present several refinements and extensions of the statistical quantum phase estimation (SQPE) framework to address some of its key practical limitations, improving its applicability to realistic cases. Recently, a family of statistical approaches for QPE have been proposed where each run uses only a few ancillae and shorter circuits than standard QPE and thus is better suited for early fault-tolerant quantum computers that are qubit-and depth-limited. SQPE method within that family estimates the cumulative distribution function (CDF) associated with spectral density of the Hamiltonian for a given trial state by using its Fourier approximation and then identifies the first jump discontinuity of the CDF to determine the ground state energy (GSE) of the Hamiltonian. It relies on random compilation procedure based on linear combination of unitaries (LCU) decomposition of the Hamiltonian assuming positive Pauli weights and requires a good estimate of lower bound on the overlap between the trial and true ground state, both of which may be difficult to achieve in practice. We address these limitations by generalizing the random compilation procedure for negative Pauli weights and employing a changepoint detection method for determining GSE which does not rely on an estimate of this overlap. We also show that by exploiting symmetry of the Fourier series one can reduce number of circuit runs/samples by a factor of 2x while keeping the GSE estimation accuracy the same. We illustrate these new developments numerically via a quantum simulator in Qiskit.