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arXiv:2506.22408 · Physical Review Research

Quantum-classical auxiliary field quantum Monte Carlo with matchgate shadows on trapped ion quantum computers

Luning Zhao, Joshua J. Goings, Willie Aboumrad, Andrew Arrasmith, Lázaro Calderín, Spencer Churchill, Dor Gabay, Thea Harvey-Brown, M.L. Hiles, Magdalena Kaja, Matthew Keesan, Karolina Kulesz, Andrii Maksymov, Mei Maruo, Mauricio Muñoz, Bas Nijholt, Rebekah Schiller, Yvette de Sereville, Amy Smidutz, Felix Tripier, Grace Yao, Trishal Zaveri, Coleman Collins, Martin Roetteler, Evgeny Epifanovsky, Arseny Kovyrshin, Lars Tornberg, Anders Broo, Jeff R. Hammond, Zohim Chandani, Pradnya Khalate, Elica S. Kyoseva, Yi-Ting Chen, E. M. Kessler, Cedric Yen-Yu Lin, G. Ramu, Ryan Shaffer, Michael Brett, Benchen Huang, Maxime R. Hugues, Tyler Y. Takeshita

Published Jun 17, 2026• 2 citations
arXivHTMLPDF

Abstract

We demonstrate an end-to-end workflow to model chemical reaction barriers with the quantum-classical auxiliary field quantum Monte Carlo (QC-AFQMC) algorithm with quantum tomography using matchgate shadows. The workflow operates within an accelerated quantum supercomputing environment with the IonQ Forte quantum computer and NVIDIA graphic processing units (GPUs) on Amazon Web Services. We present several algorithmic innovations and an efficient GPU-accelerated execution that achieves a several orders of magnitude speedup over the state-of-the-art implementation of QC-AFQMC. We apply the algorithm to simulate the oxidative addition step of the nickel-catalyzed Suzuki-Miyaura reaction using 24 qubits of IonQ Forte with 16 qubits used to represent the trial state, plus 8 additional ancilla qubits for error mitigation, resulting in the largest QC-AFQMC with matchgate shadow experiments ever performed on quantum hardware. We achieve a 9 × speedup in collecting matchgate circuit measurements, and our distributed-parallel postprocessing implementation attains a 656 × time-to-solution improvement over the prior state of the art. Chemical reaction barriers for the model reaction evaluated with active-space QC-AFQMC are within the uncertainty interval of ± 4 kcal/mol from the reference CCSD(T) (coupled cluster singles and doubles with perturbative triples) result when matchgates are sampled on the ideal simulator and within 10 kcal/mol from reference when measured on quantum processing unit. This work marks a step toward practical quantum chemistry simulations on quantum devices while identifying several opportunities for further development.

DOI
10.1103/n1tf-8kr7

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We demonstrate an end-to-end workflow to model chemical reaction barriers with the quantum-classical auxiliary field quantum Monte Carlo (QC-AFQMC) algorithm with quantum tomography using matchgate shadows. The workflow operates within an accelerated quantum supercomputing environment with the IonQ Forte quantum computer and NVIDIA graphic processing units (GPUs) on Amazon Web Services. We present several algorithmic innovations and an efficient GPU-accelerated execution that achieves a several orders of magnitude speedup over the state-of-the-art implementation of QC-AFQMC. We apply the algorithm to simulate the oxidative addition step of the nickel-catalyzed Suzuki-Miyaura reaction using 24 qubits of IonQ Forte with 16 qubits used to represent the trial state, plus 8 additional ancilla qubits for error mitigation, resulting in the largest QC-AFQMC with matchgate shadow experiments ever performed on quantum hardware. We achieve a 9 × speedup in collecting matchgate circuit measurements, and our distributed-parallel postprocessing implementation attains a 656 × time-to-solution improvement over the prior state of the art. Chemical reaction barriers for the model reaction evaluated with active-space QC-AFQMC are within the uncertainty interval of ± 4 kcal/mol from the reference CCSD(T) (coupled cluster singles and doubles with perturbative triples) result when matchgates are sampled on the ideal simulator and within 10 kcal/mol from reference when measured on quantum processing unit. This work marks a step toward practical quantum chemistry simulations on quantum devices while identifying several opportunities for further development.
We introduce QC-CBT-AFQMC, a hybrid algorithm that incorporates computational basis tomography (CBT) into the quantum-classical auxiliary field quantum Monte Carlo (QC-AFQMC) method proposed by Huggins [], replacing the use of classical shadows. While the original QC-AFQMC showed high accuracy for quantum chemistry calculations, it required exponentially costly postprocessing. Subsequent work using Matchgate shadows [Wan ] improved scalability, but still suffers from prohibitive computational requirements that limit practical applications. Our QC-CBT-AFQMC approach uses shallow Clifford circuits with a quadratic reduction of two-qubit gates over the original algorithm, significantly reducing computational requirements and enabling accurate calculations under limited measurement budgets. We demonstrate its effectiveness on the hydroxyl radical, ethylene, and nitrogen molecule, producing potential energy curves that closely match established benchmarks. We also examine the influence of CBT measurement counts on accuracy, showing that subtracting the active space AFQMC energy mitigates measurement-induced errors. Furthermore, we apply QC-CBT-AFQMC to estimate reaction barriers in [ 3 + 2 ] cycloaddition reactions, achieving agreement with high-level references and successfully incorporating complete basis set extrapolation techniques. These results highlight QC-CBT-AFQMC as a practical quantum-classical hybrid method that bridges the capabilities of quantum devices and accurate chemical simulations.
Quantum-classical auxiliary-field quantum Monte Carlo (QC-AFQMC) is a promising route to using quantum computers for strongly correlated electronic structure problems, but its scalability is limited by a classical post-processing bottleneck: the measured quantum trial state must be used to evaluate overlaps, force biases, and local energies for many walkers throughout the AFQMC propagation. Existing matchgate-shadow approaches avoid the exponential cost of generic classical shadows, but their post-processing remains expensive for large systems. In this work, we show that particle-number-preserving fermionic shadows, implemented as orbital-rotation shadows, are able to drastically reduce QC-AFQMC post-processing costs. We derive efficient estimators for overlaps and their derivatives, giving O(N3)O(N^3)O(N3) scaling for overlaps and force biases and O(N4)O(N^4)O(N4) scaling for local energies per shadow and walker and with O(N)O(N)O(N) Cholesky vector. For fixed normalized target determinants, we also prove that the overlap estimator is unbiased with variance upper bounded by O(η)O(η)O(η), where ηηη is the particle number, yielding a sufficient budget of O(η/ε2)O(η/ε^2)O(η/ε2) ideal independent measurements for additive overlap error εεε. We combine these estimators with a novel virtual-correlation-energy treatment and a GPU implementation and demonstrate QC-AFQMC using orbital-rotation shadows. Performance projections for πππ-conjugated molecules up to pentacene with a (22e, 22o) active space and 356 total orbitals show that the full classical post-processing workflow can be completed in slightly more than two days on 8×\times×A100 graphics processing units (GPUs). These results significantly reduce classical post-processing costs for QC-AFQMC and motivate further improvements in preparation and measurement of the trial state on quantum hardware.
We introduce algorithmic improvements to quantum-classical auxiliary-field quantum Monte Carlo (QC-AFQMC) that reduce the dominant per-step classical scaling from O~(N5.5)\tilde{\mathcal{O}}(N^{5.5})O~(N5.5) to O~(N4.5)\tilde{\mathcal{O}}(N^{4.5})O~(N4.5) as a function of the number of molecular spin-orbitals NNN. Central to this improvement is the application of Aitken's block transformation to handle singular Pfaffians arising in the estimation of overlaps between a quantum trial state and classical Slater-determinant walkers. Together with the use of algorithmic differentiation for the computation of the force bias, this yields a 248×248\times248× estimated runtime improvement for a system of 100 molecular orbitals. Using our workflow, we demonstrate a ground-state energy calculation for H8H_8H8​ from quantum data collected on IQM Emerald and post-processed with a tensor-network-based error-mitigation technique. We further validate the method's scalability through noiseless simulation of hydrogen chains up to H12H_{12}H12​, and on the lithium-air battery related rearrangement pathway of the Li2O4Li_2O_4Li2​O4​ lithium superoxide dimer in a (26e, 20o) active space. We estimate both quantum and classical runtimes for a potential fault-tolerant implementation of QC-AFQMC, showing that the method holds promise for the early fault-tolerant era. These results move QC-AFQMC a step closer to treating chemically relevant systems.
Using trial wave functions prepared on quantum devices to reduce the bias of auxiliary-field quantum Monte Carlo (QC-AFQMC) has established itself as a promising hybrid approach to the simulation of strongly correlated many body systems. Here, we further reduce the required quantum resources by decomposing the trial wave function into classical and quantum parts, respectively treated by classical and quantum devices, within the contextual subspace projection formalism. Importantly, we show that our algorithm is compatible with the recently developed matchgate shadow protocol for efficient overlap calculation in QC-AFQMC. Investigating the nitrogen dimer and the reductive decomposition of ethylene carbonate in lithium-based batteries, we observe that our method outperforms a number of established algorithm for ground state energy computations, while reaching chemical precision with less than half of the original number of qubits.
We extend correlated sampling to quantum-classical auxiliary-field quantum Monte Carlo (QC-AFQMC), enabling accurate nuclear force evaluation in strongly correlated systems. Computing forces via finite differences typically incurs prohibitive statistical noise in stochastic methods. We suppress this noise by maximizing correlation between geometries through synchronized random streams, orbital alignment, deterministic integral decomposition, and consistent classical shadow measurements. Crucially, a single shadow ensemble defined at the reference geometry suffices for all displaced structures, eliminating additional quantum measurements. This approach substantially reduces force variance while preserving accuracy. We validate the method on hydrogen chains across varying correlation regimes and demonstrate accurate forces for N 2 dissociation and stretched H 4 in strongly correlated regions where restricted coupled cluster methods fail qualitatively. Application to the MEA-CO 2 carbon capture reaction, integrating quantum information metrics for active space selection and matchgate shadows for overlap estimation, demonstrates that QC-AFQMC delivers accurate forces for complex reaction pathways in strongly correlated systems where conventional methods are unreliable or prohibitively expensive.
Abstract Quantum-enhanced auxiliary field quantum Monte Carlo (QC-AFQMC) uses output from a quantum computer to increase the accuracy of its classical counterpart. The algorithm requires the estimation of overlaps between walker states and a trial wavefunction prepared on the quantum computer. We study the applicability of this algorithm in terms of the number of measurements required from the quantum computer and the classical costs of post-processing those measurements. We compare the classical post-processing costs of state-of-the-art measurement schemes using classical shadows to determine the overlaps and argue that the overall post-processing cost stemming from overlap estimations scales like O ( N 9 ) per walker throughout the algorithm. With further numerical simulations, we compare the variance behavior of the classical shadows when randomizing over different ensembles, e.g. Cliffords and (particle-number restricted) matchgates beyond their respective bounds, and uncover the existence of covariances between overlap estimations of the AFQMC walkers at different imaginary time steps. Moreover, we include analyses of how the error in the overlap estimation propagates into the AFQMC energy and discuss its scaling when increasing the system size.