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Unbiasing Fermionic Quantum Monte Carlo with a Quantum Computer

William J. Huggins, Bryan A. O'Gorman, Nicholas C. Rubin, David R. Reichman, Ryan Babbush, Joonho Lee

arXiv:2106.16235v2quant-phphysics.chem-ph

TL;DR

Scalable constrained QMC controls the fermionic sign problem but can bias energies, while useful correlated trial wavefunctions are difficult to evaluate classically. The paper combines quantum overlap and local-energy evaluation with classical QMC, experimentally demonstrating this approach on chemical systems and reporting competitive accuracy.

  • Problem

    Constrained QMC is needed for scalable simulations but introduces bias, while correlated trial-wavefunction overlaps can be classically intractable.

  • Method

    QC-QMC uses quantum computing to evaluate trial-wavefunction overlaps and local energies, while a classical coprocessor performs most QMC evolution; shadow tomography reduces repeated experiments.

  • Results

    QC-AFQMC produced H4 energies of -1.96655(4) in STO-3G and -2.10910(8) in cc-pVQZ, compared with CCSD(T) values of -1.961308 and -2.114275, respectively.

  • Takeaways & Limitations

    The experiments demonstrate a hybrid quantum-classical route for expanding QMC trial-wavefunction choices beyond classically tractable forms.

Abstract

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Many-electron problems pose some of the greatest challenges in computational science, with important applications across many fields of modern science. Fermionic quantum Monte Carlo (QMC) methods are among the most powerful approaches to these problems. However, they can be severely biased when controlling the fermionic sign problem using constraints, as is necessary for scalability. Here we propose an approach that combines constrained QMC with quantum computing tools to reduce such biases. We experimentally implement our scheme using up to 16 qubits in order to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed on quantum computers (more than doubling the size of prior electron correlation calculations), while obtaining accuracy competitive with state-of-the-art classical methods. Our results demonstrate a new paradigm of hybrid quantum-classical algorithm, surpassing the popular variational quantum eigensolver in terms of potential towards the first practical quantum advantage in ground state many-electron calculations.

Appendix A: Technical Introduction

Projector QMC targets exact many-electron ground states through imaginary-time evolution, but scalable stochastic implementations face the fermionic sign problem. Constraints restore statistical efficiency while introducing bias determined by the trial wavefunction, motivating quantum-assisted trial-state projections.

  • Accurate many-electron ground-state solutions are important across science, but the Schrödinger equation becomes increasingly difficult as electron number grows.
  • QMC accuracy depends on the trial wavefunction, yet flexible correlated trial states can make basis projections exponentially expensive or otherwise classically intractable.
  • Projector QMC uses stochastic imaginary-time evolution to estimate exact ground-state energies without explicitly storing exponentially large Hamiltonians or wavefunctions.
  • The fermionic sign problem causes exponentially growing estimator variance, making exact unbiased QMC practical only for small or sign-problem-free systems.
  • Constraints such as phaseless AFQMC control the sign problem and improve statistical efficiency, but they introduce bias into final estimates.

2. Auxiliary-field quantum Monte Carlo

AFQMC performs constrained projector QMC in second-quantized space using walker determinants and trial-wavefunction overlaps. Its phaseless constraint keeps weights suitable for low-variance estimation, while trial-state evaluation limits accuracy and scalability.

  • AFQMC is a second-quantized projector QMC method whose walkers are generally single Slater determinants.
  • AFQMC represents the propagated state with weighted walker wavefunctions, using walker–trial overlaps for importance sampling.
  • The Hubbard–Stratonovich transformation preserves the single-determinant walker manifold, enabling polynomial cost growth with system size.
  • The phaseless constraint updates walker weights so they remain real and positive, reducing variance in the mixed energy estimator.
  • Classical single-determinant trials scale to roughly 500 electrons but may be inaccurate, whereas determinant expansions can be accurate yet are limited to roughly 16 electrons.

Appendix C: Quantum-Classical Hybrid Auxiliary-Field QMC (QC-AFQMC) Algorithms

QC-AFQMC separates quantum trial-wavefunction projections from classical imaginary-time evolution. The quantum computer evaluates difficult overlaps and local-energy quantities, while the classical processor performs most of the QMC simulation.

  • QC-AFQMC uses a quantum computer to evaluate trial-wavefunction projections while retaining most imaginary-time evolution on a classical processor.
  • The approach targets trial wavefunctions whose overlaps with AFQMC walkers lack known efficient exact classical evaluation.
  • Shadow tomography avoids separate quantum experiments for every walker and timestep by characterizing the trial wavefunction offline.

1. Quantum trial wavefunctions

The paper uses quantum trial wavefunctions based on coupled-cluster ideas, including a generalized valence-bond pair-product ansatz. Hardware-efficient layers extend this ansatz, while quantum evaluation addresses otherwise difficult walker overlaps.

  • 1. Quantum trial wavefunctions: Coupled-cluster wavefunctions use exponential parametrization and can be systematically improved by adding singles, doubles, triples, and higher excitations.
  • 1. Quantum trial wavefunctions: Exact CCSD–Slater-determinant overlaps lack an efficient classical evaluation, complicating coupled-cluster use within AFQMC.
  • 1. Quantum trial wavefunctions: The experiments use a generalized valence-bond pair-product ansatz as a simplified coupled-cluster trial wavefunction.
  • 1. Quantum trial wavefunctions: The pair-product state maps spin-orbitals to qubits through the Jordan–Wigner transformation, with paired virtual orbitals associated with occupied orbitals.
  • 1. Quantum trial wavefunctions: The pair-product wavefunction can be insufficient for systems where inter-pair correlation, such as multiple-bond breaking effects, is important.
  • 1. Quantum trial wavefunctions: Hardware-efficient density-density and same-spin nearest-neighbor hopping layers are alternated around the pair-product ansatz to improve accuracy.

2. Overlap and Local energy evaluation

The method uses quantum trial-wavefunction overlaps as the quantum subroutine in a largely classical QMC calculation, enabling local-energy estimation and active-space treatments of virtual correlation.

  • Overlap and local energy evaluation: Shadow tomography replaces repeated Hadamard-test evaluations by estimating overlaps between quantum trial states and arbitrary Slater determinants.The overlap and local-energy evaluations are the key quantum-dependent subroutines.
  • Overlap and local energy evaluation: Estimating overlaps with single and double excitations is sufficient to evaluate the full local energy for a two-body Hamiltonian.The remaining terms follow from the Slater-Condon rule.
  • Overlap and local energy evaluation: O(N^4) quantum overlap queries are required, where N is the system size.The same approach can compute other mixed local observables.
  • Active-space treatment: The active-space construction incorporates virtual correlation energy by combining a quantum active-space trial state with classically represented frozen core and virtual sectors.The trial state factors into an active-space state, occupied core orbitals, and an unoccupied virtual vacuum.
  • Active-space treatment: Contracting the non-active sectors with the Slater determinant produces a normalized matchgate tensor in the active-space Hilbert space.The resulting tensor can be handled efficiently classically.
  • Active-space treatment: Local-energy evaluation for the active-space formulation uses only Na quantum qubits, with the associated matchgate contractions computed classically.This applies to the relevant numerator terms and their single- and double-excitation counterparts.

Appendix D: Experimental Implementation via Shadow Tomography

Shadow tomography estimates the overlaps needed by QC-QMC from randomized measurements of a reference state related to the quantum trial wavefunction. The protocol avoids iterative quantum-processor queries and can estimate many overlaps from shared measurement data.

  • Experimental implementation: Shadow tomography estimates quantum-state properties without performing full state tomography.Here it approximates the overlap quantities required for AFQMC.
  • Shadow construction: An invertible measurement channel enables classical-shadow reconstruction from randomized unitary measurements.Tomographic completeness is the condition required for applying the inverse channel.
  • Measurement ensembles: Random N-qubit Clifford circuits are used, with tensor products of smaller Clifford circuits also considered for implementation.The protocol requires measurement ensembles whose classical post-processing is tractable enough for the intended use.
  • Overlap estimation: The overlap between a walker and the trial state is rewritten as an expectation value of observables on the reference state.The construction uses |τ⟩=(|0⟩+|ΨT⟩)/√2 and the vacuum-overlap relation.
  • Scaling: Shadow tomography estimates overlaps for many walker wavefunctions using a measurement cost that scales logarithmically with the number of overlaps.The target accuracy is specified by additive error ϵ and failure probability δ.

3. Classical Post-processing for Wavefunction Overlaps

The implementation reduces Clifford-measurement circuits and evaluates wavefunction overlaps through classical post-processing of shadow data. For QC-AFQMC, this post-processing can become exponentially scaling, although special walker structures offer efficient alternatives.

  • Overlap post-processing: The experimental post-processing computes overlaps using overlaps between stabilizer states and computational-basis states.The method exploits the efficient evaluability of stabilizer-state overlaps.
  • Efficient special cases: If a walker is a polynomial-size linear combination of stabilizer states, the required overlap calculation becomes efficient.Green’s function Monte Carlo with computational-basis walkers is a special case.
  • Efficient special cases: For Slater-determinant walkers, no known efficient classical method computes the desired overlap exactly, motivating the quantum-assisted procedure.Approximate additive-error strategies can avoid exponential scaling when suitable sampling is available.
  • Overlap post-processing: QC-AFQMC overlap evaluation uses an exponentially scaling enumeration of computational-basis states.This choice is made because the prefactor in the alternative estimator precludes benefiting from additive-error overlap estimation.
  • Circuit reduction: Because final H-free operations only affect computational-basis labels before measurement, their action can be implemented in classical post-processing.This removes the need to physically apply the final H-free operator.
  • Circuit reduction: A Clifford circuit can be decomposed as F·H·F′, with H-free operators represented using X, CNOT, and CZ gates.The H-free action permutes computational-basis states and adds phases.
  • Partitioned measurements: Global Clifford ensembles can be replaced by partitioned random Clifford circuits to reduce NISQ circuit-depth demands.The partitioned protocol applies independent random Clifford circuits to qubit subsets.

6. Noise Resilience

The overlap-ratio formulation provides resilience to certain measurement noise in both global and partitioned shadow tomography. Under the stated noise assumptions, common rescaling factors cancel from the ratios.

  • Noise resilience: Noise can have negligible impact on overlap ratios used by the method.The result concerns the specific ratio observables evaluated in the QC-QMC procedure.
  • Depolarizing noise: A global depolarizing channel applied immediately before measurement has no effect on the overlap estimate under the stated assumptions.The conclusion neglects measurement-induced error in estimating the overlaps.
  • Robust reconstruction: Robust shadow tomography estimates a noise parameter and uses it to construct a corrected inverse measurement channel.In the noiseless case, the inverse-channel parameter reduces to f^-1=2^N+1.
  • Robust reconstruction: For global Clifford measurements, the noise-dependent factor also drops out when calculating overlap ratios, providing robustness without explicitly estimating the noise parameter.This is the paper’s “robustness for free” observation for ratios.
  • Partitioned measurements: For partitioned shadow tomography, noisy inverse channels rescale all estimated overlaps by the same parameter, so the factor cancels in overlap ratios.The same conclusion holds under gate-independent, time-stationary, Markovian noise.

Appendix E: Computational and Experimental Details and Supportive Numerical Results

The appendix documents the software, sampling, and numerical settings used for the AFQMC and shadow-tomography calculations.

  • Shadow-tomography measurements used 1,000 repetitions for each Clifford circuit.
  • AFQMC calculations used PAUXY and QMCPACK, with integrals generated by PySCF and selected calculations verified using Q-Chem.
  • Exact basis-set energies were obtained using heat-bath configuration interaction, while AFQMC used more than 1,000 walkers.
  • The numerical AFQMC time step was Δt = 0.005.

1. H4, 8-qubit experiment

The 8-qubit H4 study tests QC-QMC on a square geometry and dissociation problem where single-reference classical methods are unreliable, using two basis sets and repeated shadow-tomography experiments.

  • H4 was studied in a square geometry with side length 1.23 Å and in its dissociation into four hydrogen atoms.
  • Classical AFQMC produced -1.96655(4) for STO-3G and -2.10910(8) for cc-pVQZ, while CCSD(T) produced -1.961308 and -2.114275, respectively.
  • Shadow tomography was repeated four times for STO-3G and twice for cc-pVQZ, using partitioned and unpartitioned variants.
  • Although variational energies varied with Clifford count and tomography partitioning, QC-AFQMC energies were nearly converged at 15,000 Cliffords with negligible run-to-run variation.

2. N2, 12-qubit experiment

The N2 experiment used partitioned shadow tomography with 15,000 Cliffords and compared correlated calculations using a cc-pVTZ basis.

  • For N2, the study performed one partitioned shadow-tomography experiment using 15,000 Cliffords and a cc-pVTZ basis.
  • Classical AFQMC used UHF trial wavefunctions, while CCSD(T) used UHF reference states.
  • Spin projection did not change the N2 results discussed in the study.

3. Diamond, 16-qubit experiment

The 16-qubit diamond experiment used a small periodic cell, specified pseudopotential and basis, restricted-Hartree–Fock references, and HCI benchmarks.

  • The diamond calculation used the GTH-PADE pseudopotential and DZVP-GTH basis, sampling only the Γ point for a two-carbon unit cell.
  • Classical AFQMC and CCSD(T) used spin-restricted Hartree–Fock references.
  • HCI supplied the exact results, with the second-order perturbation correction below 0.0001 a.u.

4. Quantum Circuit Details

The experiments used quantum circuits to prepare trial wavefunctions and perform shadow-tomography measurements for AFQMC energy calculations. Resource tables and repeated measurements document the experimental setups and resulting energies.

  • Circuit structure: The circuits combine trial-wavefunction preparation with a shadow-tomography measurement stage.The preparation creates a superposition of the trial wavefunction and the zero state, while shadow tomography implements the measurement operator.
  • Trial wavefunctions: The experiments used perfect pairing states, with additional number-preserving fermionic gates in the eight-qubit experiment.These states and gates provide the quantum trial wavefunctions used in the experiments.
  • Data and resources: The reported data include repeated H4 measurements, resource counts for the QC-AFQMC experiments, and comparisons with prior fermionic simulations.The N2 and diamond datasets note that quantum-trial energies came from single experimental runs and may vary significantly between runs.
  • Trial-wavefunction preparation: A Hadamard gate followed by CNOT and SWAP ladders prepares the required state structure across the qubits.For each set of four qubits representing a pair of spatial orbitals, CNOT and iSWAP gates preserve the zero component while preparing the paired state.
  • Experimental data: Four independent partitioned shadow-tomography experiments produced AFQMC energies for H4 in a cc-pVQZ basis, with the exact ground-state energy equal to -2.11216599.The unpartitioned measurements are reported separately in Table IX.
  • Measurement circuits: Shadow-tomography measurement operators use a CZ layer between single-qubit gate layers, with qubit reversal handled during post-processing.The implementation exploits the fact that the CZ layer is followed by single-qubit gates and computational-basis measurement.

Appendix F: Outlook on Potential Quantum Advantage

The outlook examines whether QC-AFQMC can scale toward quantum advantage in overlap estimation and ground-state energy computation. It identifies exponentially shrinking overlaps and classical post-processing as key constraints, while proposing active-space and improved-wavefunction strategies.

  • System size scaling: Walker–trial-wavefunction overlaps can decay exponentially with system size, reaching 10^-5 for 16 atoms, 10^-16 for 54 atoms, and 10^-38 for 128 atoms.Maintaining fixed relative precision would therefore formally require exponentially more measurements as system size approaches the thermodynamic limit.
  • System size scaling: QC-AFQMC may need formulations beyond the experiment’s setup to address overlap-scaling challenges.More sophisticated efficiently preparable wavefunctions could maintain better overlap with the walker state.
  • Scaling strategies: Virtual correlation can use a small quantum active space while computing correlation energy in a larger space, potentially maintaining larger overlaps for finite-correlation-length systems.This approach avoids requiring the quantum processor to represent the full physical space directly.
  • Overlap estimation: Shadow tomography measurements are efficient in number but incurred exponential classical post-processing overhead because all Hilbert-space determinants were enumerated.Removing this overhead remains an open question; fermionic shadow tomography is identified as a promising direction.
  • Overlap estimation: The perfect-pairing trial wavefunction does not provide quantum advantage for overlap evaluation because its overlap with an arbitrary Slater determinant can be approximated efficiently classically.This limitation concerns the specific workhorse trial state used in the experiments.
  • Potential quantum advantage: The authors identify overlap estimation and ground-state energy computation as possible routes toward practical quantum advantage in NISQ fermionic simulations.The outlook treats both the general ground-state-energy objective and the specific overlap-estimation subroutine as potential sources of advantage.
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