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arXiv:2610.03103

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

Luning Zhao, Joshua Goings, Evgeny Epifanovsky, Martin Roetteler

quant-phphysics.class-phSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

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.

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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 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.
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.
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 consider the problem of learning expectation values of particle-preserving operators with respect to an unknown ηηη-particle nnn-mode fermionic state via classical shadows. Our main application is to estimating overlaps with arbitrary Slater determinant states: While it is known that such overlaps can, in the average case, be learnt to a fixed additive precision with a constant number of samples, the best-known worst case bound is O(nlog⁡n)\mathcal{O}(\sqrt n \log n)O(n​logn); here we improve this to O(ηlog⁡η)\mathcal{O}(η\logη)O(ηlogη), achieving a mode-independent sample cost. Our procedure is also computationally efficient, requiring only classical post-processing which for a generic dense orbital runs in time O(nη2)\mathcal{O}(nη^2)O(nη2). For the task of estimating the expectation value of a general particle-preserving quadratic fermionic observable hhh, we prove a sample complexity bound of O(η∥h0∥22)\mathcal{O}(η\|h_0\|_2^2)O(η∥h0​∥22​), where h0h_0h0​ is the traceless component of hhh; the associated classical post-processing scales as O(n2η)\mathcal{O}(n^2η)O(n2η). Finally, we discuss implementation of the required randomization: in a first-quantized encoding, approximate unitary designs give circuit depths polylogarithmic in the number of modes, contrasting with linear-depth requirements for nearest-neighbor second-quantized matchgate implementations. On the technical side, our proof reduces the extremal shadow variance to harmonic analysis on the AIII symmetric space U(n)/(U(η)×U(n−η))U(n)/(U(η)\times U(n-η))U(n)/(U(η)×U(n−η)) and evaluates the resulting integral using techniques from the theories of Jacobi ensembles and orthogonal polynomials, in a calculation which may be of independent interest.
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.