Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2410.18901 · Journal of Chemical Theory and Computation

Quantum Computing Approach to Fixed-Node Monte Carlo Using Classical Shadows

Nick S. Blunt, Laura Caune, Javiera Quiroz-Fernandez

Published Feb 5, 2025• 3 citations
arXivHTMLPDF

Abstract

High Resolution Image Download MS PowerPoint Slide Quantum Monte Carlo (QMC) methods are powerful approaches for solving electronic structure problems. Although they often provide high-accuracy solutions, the precision of most QMC methods is ultimately limited by the trial wave function that must be used. Recently, an approach has been demonstrated to allow the use of trial wave functions prepared on a quantum computer [Huggins et al., Unbiasing fermionic quantum Monte Carlo with a quantum computer. Nature 2022, 603, 416] in the auxiliary-field QMC (AFQMC) method using classical shadows to estimate the required overlaps. However, this approach has an exponential post-processing step to construct these overlaps when performing classical shadows obtained using random Clifford circuits. Here, we study an approach to avoid this exponential scaling step by using a fixed-node Monte Carlo method based on full configuration interaction quantum Monte Carlo. This method is applied to the local unitary cluster Jastrow ansatz. We consider H 4, ferrocene, and benzene molecules using up to 12 qubits as examples. Circuits are compiled to native gates for typical near-term architectures, and we assess the impact of circuit-level depolarizing noise on the method. We also provide a comparison of AFQMC and fixed-node approaches, demonstrating that AFQMC is more robust to errors, although extrapolations of the fixed-node energy reduce this discrepancy. Although the method can be used to reach chemical accuracy, the sampling cost to achieve this is high even for small active spaces, suggesting caution about the prospect of outperforming conventional QMC approaches.

DOI
10.1021/acs.jctc.4c01468

Related papers

Evaluating a quantum-classical quantum Monte Carlo algorithm with Matchgate shadows

Benchen Huang, Yi-Ting Chen, Brajesh Gupt +4 more

Solving the electronic structure problem of molecules and solids to high accuracy is a major challenge in quantum chemistry and condensed matter physics. The rapid emergence and development of quantum computers offer a promising route to systematically tackle this problem. Recent work by [Huggins , ] proposed a hybrid quantum-classical quantum Monte Carlo (QC-QMC) algorithm using Clifford shadows to determine the ground state of a Fermionic Hamiltonian. This approach displayed inherent noise resilience and the potential for improved accuracy compared to its purely classical counterpart. Nevertheless, the use of Clifford shadows introduces an exponentially scaling postprocessing cost. In this work, we investigate an improved QC-QMC scheme utilizing the recently developed Matchgate shadows technique [], which removes the aforementioned exponential bottleneck. We observe from experiments on quantum hardware that the use of Matchgate shadows in QC-QMC is inherently noise robust. We show that this noise resilience has a more subtle origin than in the case of Clifford shadows. Nevertheless, we find that classical postprocessing, while asymptotically efficient, requires hours of runtime on thousands of classical CPUs for even the smallest chemical systems, presenting a major challenge to the scalability of the algorithm. Published by the American Physical Society 2024

arXiv:2404.18303•Physical Review Research•Oct 24, 2024•19 citations

Contextual Subspace Auxiliary-Field Quantum Monte Carlo: Improved Bias with Reduced Quantum Resources

Matthew Kiser, Matthias Beuerle, Fedor Šimkovic

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.

arXiv:2408.06160•Journal of Chemical Theory and Computation•Feb 20, 2025•8 citations

Orbital-Rotation Shadow Tomography Reduces the Classical Cost of Quantum-Classical Auxiliary-Field Quantum Monte Carlo

Luning Zhao, Joshua Goings, Evgeny Epifanovsky +1 more

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 $O(N^3)$ scaling for overlaps and force biases and $O(N^4)$ scaling for local energies per shadow and walker and with $O(N)$ Cholesky vector. For fixed normalized target determinants, we also prove that the overlap estimator is unbiased with variance upper bounded by $O(η)$, where $η$ is the particle number, yielding a sufficient budget of $O(η/ε^2)$ ideal independent measurements for additive overlap error $ε$. 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 $π$-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.

quant-pharXiv:2610.03103•Oct 5, 2026•0 citations

Classical and quantum cost of measurement strategies for quantum-enhanced auxiliary field quantum Monte Carlo

Matthew Kiser, Anna K. Schroeder, Gian-Luca Anselmetti +4 more

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.

arXiv:2312.09872•New Journal of Physics•Mar 1, 2024•14 citations

A brief introduction to the diffusion Monte Carlo method and the fixed-node approximation

Alfonso Annarelli, Dario Alfè, Andrea Zen

Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.

arXiv:2412.06006•The Journal of Chemical Physics•Dec 28, 2024•9 citations

Benchmark Phaseless Auxiliary-Field Quantum Monte Carlo Method for Small Molecules

Zoran Sukurma, Martin Schlipf, Moritz Humer +2 more

High Resolution Image Download MS PowerPoint Slide We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO 2, and O 2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.

arXiv:2303.04256•Journal of Chemical Theory and Computation•Jul 20, 2023•17 citations
High Resolution Image Download MS PowerPoint Slide Quantum Monte Carlo (QMC) methods are powerful approaches for solving electronic structure problems. Although they often provide high-accuracy solutions, the precision of most QMC methods is ultimately limited by the trial wave function that must be used. Recently, an approach has been demonstrated to allow the use of trial wave functions prepared on a quantum computer [Huggins et al., Unbiasing fermionic quantum Monte Carlo with a quantum computer. Nature 2022, 603, 416] in the auxiliary-field QMC (AFQMC) method using classical shadows to estimate the required overlaps. However, this approach has an exponential post-processing step to construct these overlaps when performing classical shadows obtained using random Clifford circuits. Here, we study an approach to avoid this exponential scaling step by using a fixed-node Monte Carlo method based on full configuration interaction quantum Monte Carlo. This method is applied to the local unitary cluster Jastrow ansatz. We consider H 4, ferrocene, and benzene molecules using up to 12 qubits as examples. Circuits are compiled to native gates for typical near-term architectures, and we assess the impact of circuit-level depolarizing noise on the method. We also provide a comparison of AFQMC and fixed-node approaches, demonstrating that AFQMC is more robust to errors, although extrapolations of the fixed-node energy reduce this discrepancy. Although the method can be used to reach chemical accuracy, the sampling cost to achieve this is high even for small active spaces, suggesting caution about the prospect of outperforming conventional QMC approaches.
Solving the electronic structure problem of molecules and solids to high accuracy is a major challenge in quantum chemistry and condensed matter physics. The rapid emergence and development of quantum computers offer a promising route to systematically tackle this problem. Recent work by [Huggins , ] proposed a hybrid quantum-classical quantum Monte Carlo (QC-QMC) algorithm using Clifford shadows to determine the ground state of a Fermionic Hamiltonian. This approach displayed inherent noise resilience and the potential for improved accuracy compared to its purely classical counterpart. Nevertheless, the use of Clifford shadows introduces an exponentially scaling postprocessing cost. In this work, we investigate an improved QC-QMC scheme utilizing the recently developed Matchgate shadows technique [], which removes the aforementioned exponential bottleneck. We observe from experiments on quantum hardware that the use of Matchgate shadows in QC-QMC is inherently noise robust. We show that this noise resilience has a more subtle origin than in the case of Clifford shadows. Nevertheless, we find that classical postprocessing, while asymptotically efficient, requires hours of runtime on thousands of classical CPUs for even the smallest chemical systems, presenting a major challenge to the scalability of the algorithm. Published by the American Physical Society 2024
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.
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.
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.
Quantum Monte Carlo (QMC) methods represent a powerful family of computational techniques for tackling complex quantum many-body problems and performing calculations of stationary state properties. QMC is among the most accurate and powerful approaches to the study of electronic structure, but its application is often hindered by a steep learning curve; hence it is rarely addressed in undergraduate and postgraduate classes. This tutorial is a step toward filling this gap. We offer an introduction to the diffusion Monte Carlo (DMC) method, which aims to solve the imaginary time Schrödinger equation through stochastic sampling of the configuration space. Starting from the theoretical foundations, the discussion leads naturally to the formulation of a step-by-step algorithm. To illustrate how the method works in simplified scenarios, examples such as the harmonic oscillator and the hydrogen atom are provided. The discussion extends to the fixed-node approximation, a crucial approach for addressing the fermionic sign problem in multi-electron systems. In particular, we examine the influence of trial wave function nodal surfaces on the accuracy of DMC energy by evaluating results from a non-interacting two-fermion system. Extending the method to excited states is feasible in principle, but some additional considerations are needed, supported by practical insights. By addressing the fundamental concepts from a hands-on perspective, we hope this tutorial will serve as a valuable guide for researchers and students approaching DMC for the first time.
High Resolution Image Download MS PowerPoint Slide We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO 2, and O 2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.