⚛️ quantum physics

Quantum-Classical Auxiliary Field Quantum Monte Carlo with Matchgate Shadows on Trapped Ion Quantum Computers

This paper demonstrates a record-breaking, GPU-accelerated end-to-end workflow for simulating chemical reaction barriers using Quantum-Classical Auxiliary Field Quantum Monte Carlo with matchgate shadows on an IonQ Forte quantum computer, achieving significant speedups and results within 10 kcal/mol of reference values.

Original authors: Luning Zhao, Joshua J. Goings, Willie Aboumrad, Andrew Arrasmith, Lazaro Calderin, Spencer Churchill, Dor Gabay, Thea Harvey-Brown, Melanie Hiles, Magda Kaja, Matthew Keesan, Karolina Kulesz, Andrii M

Published 2026-07-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Luning Zhao, Joshua J. Goings, Willie Aboumrad, Andrew Arrasmith, Lazaro Calderin, Spencer Churchill, Dor Gabay, Thea Harvey-Brown, Melanie Hiles, Magda 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 Kyoseva, Yi-Ting Chen, Eric M. Kessler, Cedric Yen-Yu Lin, Gandhi Ramu, Ryan Shaffer, Michael Brett, Benchen Huang, Maxime R. Hugues, Tyler Y. Takeshita

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to predict how a complex machine will behave just by looking at its blueprints. In the world of chemistry, that machine is a molecule, and the blueprint is a set of equations describing how electrons dance around atoms. For decades, scientists have used powerful classical computers to solve these equations, but when the molecules get tricky—like those involving transition metals used in life-saving drugs or industrial catalysts—the math becomes so incredibly tangled that even the world's fastest supercomputers get stuck. It's like trying to untangle a knot of a billion strings while blindfolded; the more strings you add, the harder it gets, until the task becomes impossible. This is where quantum computing enters the story. Instead of trying to untangle the knot with a single pair of hands, quantum computers use the strange rules of the quantum world to explore many possibilities at once, offering a potential shortcut to understanding these complex chemical reactions. The big question is: can we actually use these fragile, noisy new machines to solve real-world chemistry problems today, or are we still waiting for a perfect, error-free future?

This paper tells the story of a team that decided to stop waiting and start building a bridge between the quantum and classical worlds to solve a specific, difficult chemistry puzzle. They focused on a reaction called the Suzuki–Miyaura cross-coupling, a vital process used to stitch together carbon atoms to create medicines. Specifically, they wanted to calculate the energy barrier for a step where a nickel atom grabs a piece of a molecule to get the reaction started. This is a notoriously hard problem because the nickel atom has electrons that are strongly "correlated," meaning they act in a highly coordinated, chaotic way that breaks standard computer models. The researchers didn't just simulate this on a computer; they ran a real experiment using a quantum computer from IonQ, which uses trapped ions (charged atoms) as its processing units, combined with a massive army of NVIDIA GPUs to do the heavy lifting.

The team developed a clever workflow called "Quantum-Classical Auxiliary Field Quantum Monte Carlo" (QC-AFQMC). Think of this as a relay race. First, the quantum computer prepares a "trial state," which is a rough guess of what the molecule's electrons look like. Instead of asking the quantum computer to do all the hard math, the team used a technique called "matchgate shadows." Imagine taking a photo of a 3D object from many different, random angles to create a "shadow" that captures its essence without needing to see every single detail. These shadows are then sent to the classical supercomputers, which use advanced math tricks (like algorithmic differentiation) to finish the calculation and predict the energy barrier.

The results were a mix of triumph and reality checks. When they used the quantum computer's "ideal simulator" (a perfect, noise-free version), their results were incredibly close to the gold-standard reference, landing within ±4 kcal/mol of the expected value. However, when they ran the experiment on the actual, noisy IonQ Forte quantum hardware, the results drifted a bit, ending up within 10 kcal/mol of the reference. While this isn't perfect yet, it's a massive leap forward. The paper explicitly rules out the idea that this method is too slow or expensive to be useful; in fact, they achieved a 656-fold speedup in the time it takes to solve the problem compared to previous state-of-the-art methods, thanks to their new algorithms and GPU acceleration. They also found that while the quantum hardware introduced some noise that shifted the energy values, the method remained robust enough to give useful chemical insights.

The team also discovered that tuning the quantum computer specifically for this job made a huge difference. By optimizing how the circuits were sent and executed, they sped up the data collection by 9 times. They used 24 qubits in total: 16 to represent the trial state of the molecule and 8 extra "ancilla" qubits to help catch and fix errors. The final takeaway is that while we aren't there yet for perfect, error-free chemistry on quantum devices, this end-to-end workflow proves that we can now tackle real, complex catalytic reactions that were previously out of reach. It's a significant step toward making quantum chemistry a practical tool for drug discovery, showing that with the right mix of quantum hardware, classical supercomputing power, and clever algorithms, we can start solving the tangled knots of molecular chemistry.

Technical Summary: Quantum-Classical Auxiliary Field Quantum Monte Carlo with Matchgate Shadows on Trapped Ion Quantum Computers

Problem Statement
Accurate modeling of chemical reaction barriers involving transition metal (TM) catalysts, such as the Suzuki–Miyaura cross-coupling reaction, remains a significant challenge for classical computational chemistry. These systems exhibit strong electron correlation and multi-reference character due to unpaired d- or f-electrons, causing standard single-reference methods like Density Functional Theory (DFT) and even the "gold standard" Coupled Cluster Singles and Doubles with perturbative Triples [CCSD(T)] to fail or break down. While Quantum Phase Estimation (QPE) offers exact solutions, it requires fault-tolerant hardware beyond current capabilities. Variational Quantum Eigensolvers (VQE) are noise-resilient but suffer from prohibitive measurement costs and sensitivity to device noise. Consequently, there is a need for hybrid quantum-classical algorithms that can leverage near-term quantum hardware to prepare correlated trial states while mitigating the computational bottlenecks of classical post-processing.

Methodology
The authors present an end-to-end workflow for the Quantum-Classical Auxiliary Field Quantum Monte Carlo (QC-AFQMC) algorithm, executed within an accelerated quantum supercomputing environment. The workflow integrates the IonQ Forte trapped-ion quantum computer with NVIDIA GPUs on Amazon Web Services (AWS).

Algorithmic Framework: The method utilizes the classical AFQMC projector Monte Carlo approach, which evolves a wavefunction in imaginary time to find the ground state. To address the fermionic phase problem, a trial wavefunction ( $|\Psi_T\rangle$ ) is required. In this implementation, the trial state is prepared on a quantum computer using the Variational Quantum Eigensolver (VQE) with a unitary pair coupled-cluster doubles (upCCD) ansatz.
Matchgate Shadows: Instead of using the standard Hadamard test or Clifford-based classical shadows (which scale exponentially for overlap calculations in AFQMC), the workflow employs matchgate shadows. This technique allows for the efficient evaluation of overlaps $\langle \Psi_T | \phi \rangle$ between the quantum trial state and classical walker states. The overlap computation scales polynomially ( $O(N^{4.5})$ ) with system size $N$ .
Virtual Correlation Energy (VCE): To handle large chemical systems where the full orbital space exceeds available qubits, the system is partitioned into an active space (handled by the quantum trial state) and a virtual space. The VCE technique captures electron correlations outside the active space using a single Slater determinant, allowing the quantum resources to focus on the strongly correlated active space.
Algorithmic Optimizations: The authors introduce significant algorithmic improvements to the classical post-processing step. By utilizing algorithmic differentiation (specifically derivatives of matrix Pfaffians) rather than finite differences, the computational complexity for calculating local energies and force bias terms is reduced from $O(N^{8.5})$ and $O(N^{7.5})$ to $O(N^{5.5})$ and $O(N^{4.5})$ , respectively.
Hardware Execution:
- Quantum: The trial state is optimized on an ideal simulator, and matchgate shadow circuits are executed on the IonQ Forte QPU. Performance tuning of the QPU control software and firmware was implemented to maximize throughput.
- Classical: Post-processing, including imaginary time propagation and energy evaluation, is performed on AWS ParallelCluster using NVIDIA H100 and H200 GPUs, leveraging cuBLAS, cuSOLVER, and cuTENSOR libraries.

Key Contributions

First Large-Scale Demonstration: This work reports the largest QC-AFQMC experiment with matchgate shadows performed on quantum hardware to date, utilizing 24 qubits (16 for the trial state, 8 ancilla for error mitigation) on the IonQ Forte processor.
Algorithmic Speedup: The implementation achieves a 656× improvement in time-to-solution for the classical post-processing step compared to the state-of-the-art implementation by Huang et al. [41], primarily driven by the reduction in asymptotic complexity via algorithmic differentiation and GPU acceleration.
Hardware Throughput Optimization: Specific performance tuning of the IonQ Forte control stack resulted in a 9× speedup in collecting matchgate circuit measurements.
End-to-End Workflow: The paper establishes a fully integrated workflow combining quantum state preparation, shadow tomography, and GPU-accelerated classical Monte Carlo propagation for complex chemical problems.

Results
The algorithm was applied to simulate the oxidative addition step of a nickel-catalyzed Suzuki–Miyaura reaction (involving species B, transition state [B-C]‡, and product C) using an (8,8) active space.

Accuracy: When matchgates were sampled on an ideal simulator, the calculated reaction barriers were within the uncertainty interval of ±4 kcal/mol of the reference CCSD(T) result.
Noise Impact: When measured on the actual QPU, the results deviated by approximately 10 kcal/mol from the reference. The noise caused a qualitative shift in the predicted energy ordering of the reactant and product, highlighting the current limitations of error mitigation for this specific application.
Particle Number: While the ideal simulator yielded particle numbers close to the ground truth (8 electrons), the QPU measurements showed a deviation to approximately 10.5 electrons, indicating significant hardware noise, though the energy results remained more resilient to this noise than the particle number estimates.

Significance
The paper claims this work represents a "step towards practical quantum chemistry simulations on quantum devices." It challenges the prior consensus that QC-AFQMC is prohibitively expensive for systems of real scientific interest by demonstrating that, with algorithmic refinements and hardware acceleration, the method can be applied to catalytic transition metal complexes. The study identifies that while the method is viable, further development is required in error mitigation techniques and the selection of robust trial states to fully realize its potential on noisy intermediate-scale quantum (NISQ) hardware. The work underscores the necessity of co-designing quantum hardware control software with specific algorithmic workloads to achieve meaningful speedups.

Technical Summary: Quantum-Classical Auxiliary Field Quantum Monte Carlo with Matchgate Shadows on Trapped Ion Quantum Computers

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