Error-corrected phase estimation averaged over variable grids on a trapped-ion quantum computer: hyperacuity spectra of a CO molecule adsorbed onto $χ$-Fe$_5$C$_2$
Quantum phase estimation (QPE) is an underlying technology for extracting the excitation spectra of many-electron systems, yet its practical use on current hardware is hindered by low grid resolution and environmental noises. Here we propose QPE averaged over variable grids (QAVG), a vernier-type approach that combines low-resolution QPE with multiple origin shifts and physically motivated continuous parametrization to reconstruct the spectra accurately. We introduce this approach into an end-to-end workflow for the {\it ab initio}-based model system for a CO molecule adsorbed onto the $χ$-Fe$_5$C$_2$ surface. We perform experiments on Quantinuum H2-2 using both physical QPE circuits and logical QPE circuits encoded in the Steane code with offline bit-flip correction. We demonstrate that QAVG accurately reconstructs the spectra with deviations much smaller than the nominal QPE resolution, even when the noisy histograms are used. The cost landscapes averaged over the shifted grids substantially suppress the local minima arising from the spectral leakage, thereby stabilizing the optimization of trial parameters. These results indicate that QAVG provides a robust route to quantum simulations of correlated spectra toward the era of early-fault-tolerant quantum computers.
Error-corrected phase estimation averaged over variable grids on a trapped-ion quantum computer: hyperacuity spectra of a CO molecule adsorbed onto χ-Fe5C2
Quantum phase estimation (QPE) is an underlying technology for extracting the excitation spectra of many-electron systems, yet its practical use on current hardware is hindered by low grid resolution and environmental noises. Here we propose QPE averaged over variable grids (QAVG), a vernier-type approach that combines low-resolution QPE with multiple origin shifts and physically motivated continuous parametrization to reconstruct the spectra accurately. We introduce this approach into an end-to-end workflow for the {\it ab initio}-based model system for a CO molecule adsorbed onto the χ-Fe5C2 surface. We perform experiments on Quantinuum H2-2 using both physical QPE circuits and logical QPE circuits encoded in the Steane code with offline bit-flip correction. We demonstrate that QAVG accurately reconstructs the spectra with deviations much smaller than the nominal QPE resolution, even when the noisy histograms are used. The cost landscapes averaged over the shifted grids substantially suppress the local minima arising from the spectral leakage, thereby stabilizing the optimization of trial parameters. These results indicate that QAVG provides a robust route to quantum simulations of correlated spectra toward the era of early-fault-tolerant quantum computers.
For treating correlated electronic systems on quantum computers, we propose a quantum-classical hybrid scheme for dynamical mean-field theory (DMFT). In the quantum part of the scheme, we use modified quantum phase estimation (QPE) circuits suitable for the one-particle Green's function (GF) at a finite temperature so that we can extract spectral amplitudes and the excitation energies without knowing the excitation channel invoked at each measurement. In the classical part of the scheme, we adopt an approach that estimates reasonably the GF based on the data collected from the QPE sampling. We dub the approach the QPE averaged over variable grids (QAVG), that may help one to reconstruct the GF via optimization of trial parameters and modeling the probability distributions for various settings of the QPE circuits. We apply the QAVG-DMFT scheme to SrVO3 to demonstrate its validity via numerical simulations.
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-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.
Quantum phase estimation (QPE) serves as a building block of many different quantum algorithms and finds important applications in computational chemistry problems. Despite the rapid development of quantum hardware, experimental demonstration of QPE for chemistry problems remains challenging due to its large circuit depth and the lack of quantum resources to protect the hardware from noise with fully fault-tolerant protocols. In the present work, we take a step towards fault-tolerant quantum computing by demonstrating a QPE algorithm on a Quantinuum trapped-ion computer. We employ a Bayesian approach to QPE and introduce a routine for optimal parameter selection, which we combine with a [[ n + 2 , n , 2 ]] quantum error detection code carefully tailored to the hardware capabilities. As a simple quantum chemistry example, we take a hydrogen molecule represented by a two-qubit Hamiltonian and estimate its ground state energy using our QPE protocol. In the experiment, we use the quantum circuits containing as many as 920 physical two-qubit gates to estimate the ground state energy within 6 × 10 − 3 hartree of the exact value. Published by the American Physical Society 2024
In this study, we employed Fourier-based quantum phase estimation (QPE) to calculate X-ray absorption spectroscopy (XAS) spectra. The primary focus of this study is the calculation of the XAS spectra of transition metal L2,3-edges, which are dominated by strong correlation effects. First, the Fe L2,3-edge X-ray absorption near-edge structure of FePO4 is calculated using a noiseless simulator. The present computation involves a comparison of three types of input states: a uniform superposition state, optimal entangled input state, and Slater function state. Subsequently, we investigated the resolution error of the QPE and statistical error attributed to the measurements. It was revealed that post-processing to introduce Lorentzian broadening reduces the statistical error, which becomes a significant problem for a large number of qubits. Subsequently, we implemented QPE on a trapped-ion quantum computer, encompassing three orbitals within the active space. To this end, we implemented QPE using dynamic circuits to reduce ancilla qubits and [[k+2, k, 2]] quantum error detection code to mitigate the quantum noise inherent in current quantum computers. As a result, it was demonstrated that hardware noise was reduced, and spectra close to the noiseless ones were obtained.
Quantum phase estimation (QPE) is a key quantum algorithm, which has been widely studied as a method to perform chemistry and solid-state calculations on future fault-tolerant quantum computers.Recently, several authors have proposed statistical alternatives to QPE that have benefits on early faulttolerant devices, including shorter circuits and better suitability for error-mitigation techniques.However, experimental investigations of the algorithm on real quantum processors are lacking.Here, we implement statistical phase estimation on Rigetti's superconducting processors.Specifically, we use a modification of the Lin and Tong [PRX Quantum 3, 010318 (2022)] algorithm with the improved Fourier approximation of Wan et al. [Phys.Rev. Lett.129, 030503 (2022)] and apply a variational-compilation technique to reduce the circuit depth.We then incorporate error-mitigation strategies including zero-noise extrapolation and readout-error mitigation with bit-flip averaging.We propose a new method to estimate energies from the statistical phase estimation data, which is found to improve the accuracy in the final energy estimates by 1-2 orders of magnitude with respect to prior theoretical bounds, reducing the cost of performing accurate phase-estimation calculations.We apply these methods to chemistry problems for active spaces up to four electrons in four orbitals, including the application of a quantum embedding method, and use them to correctly estimate energies within chemical precision.Our work demonstrates that statistical phase estimation has a natural resilience to noise, particularly after mitigating coherent errors, and can achieve far higher accuracy than suggested by previous analysis, demonstrating its potential as a valuable quantum algorithm for early fault-tolerant devices.
We introduce a classical estimator for the post-processing of quantum phase estimation (QPE) data when a single target phase is isolated within a known interval, as is typical of ground state energy estimation of gapped systems. Our estimator filters the QPE signal within this promise region and recovers the phase through a moment-projection routine, which is robust to both external spurious phases and experimental noise. In the noiseless case this achieves an exponential suppression of bias with respect to a naive mean estimator. In the presence of global depolarizing noise the bias is exponentially small in the circuit depth t, and the variance is O(t−2F−2) for circuit fidelity F. This improves by a factor of t2 over a naive shifted-and-rescaled-mean approach. To mitigate realistic circuit-level noise, we combine our method with the explicit unbiasing scheme described in [Dutkiewicz et al., 2025]. This yields an overhead interpolating between the F−4 scaling typical of explicitly unbiased error mitigation and a reduced F−2 scaling when the noise samples fall outside the promise interval. We validate our estimators on a small-scale simulation of the Ising model, observing better-than-expected performance for a global depolarizing noise approximation. This robustness to both multiple eigenvalues and realistic noise makes limited-depth phase estimation practical for early fault tolerant quantum experiments.