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.
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 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) to O~(N4.5) as a function of the number of molecular spin-orbitals N. 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× estimated runtime improvement for a system of 100 molecular orbitals. Using our workflow, we demonstrate a ground-state energy calculation for H8 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 H12, and on the lithium-air battery related rearrangement pathway of the Li2O4 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.
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.
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.
The auxiliary-field quantum Monte Carlo (AFQMC) method is a general numerical method for correlated many-electron systems, which is being increasingly applied in lattice models, atoms, molecules, and solids. Here, we introduce the theory and algorithm of the method specialized for real materials and present several recent developments. We give a systematic exposition of the key steps of AFQMC, closely tracking the framework of a modern software library we are developing. The building of a Monte Carlo Hamiltonian, projecting to the ground state, sampling two-body operators, phaseless approximation, and measuring ground state properties are discussed in detail. An advanced implementation for multi-determinant trial wave functions is described, which dramatically speeds up the algorithm and reduces the memory cost. We propose a self-consistent constraint for real materials, and discuss two flavors for its realization, either by coupling the AFQMC calculation to an effective independent-electron calculation or via the natural orbitals of the computed one-body density matrix.
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.
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.