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Chemical Reviews

Walking through Hilbert Space with Quantum Computers

Tong Jiang, Jinghong Zhang, Moritz K. A. Baumgarten, Meng‐Fu Chen, Hieu Q. Dinh, Aadithya Ganeshram, Nishad Maskara, Anton Z. Ni, Joonho Lee

Published May 2, 2025• 12 citations
Publisher

Abstract

Computations of chemical systems’ equilibrium properties and nonequilibrium dynamics have been suspected of being a “killer app” for quantum computers. This review highlights the recent advancements of quantum algorithms tackling complex sampling tasks in the key areas of computational chemistry: ground state, thermal state properties, and quantum dynamics calculations. We review a broad range of quantum algorithms, from hybrid quantum–classical to fully quantum, focusing on the traditional Monte Carlo family, including Markov chain Monte Carlo, variational Monte Carlo, projector Monte Carlo, path integral Monte Carlo, etc. We also cover other relevant techniques involving complex sampling tasks, such as quantum-selected configuration interaction, minimally entangled typical thermal states, entanglement forging, and Monte Carlo-flavored Lindbladian dynamics. We provide a comprehensive overview of these algorithms’ classical and quantum counterparts, detailing their theoretical frameworks and discussing the potentials and challenges in achieving quantum computational advantages.

DOI
10.1021/acs.chemrev.4c00508

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Computations of chemical systems’ equilibrium properties and nonequilibrium dynamics have been suspected of being a “killer app” for quantum computers. This review highlights the recent advancements of quantum algorithms tackling complex sampling tasks in the key areas of computational chemistry: ground state, thermal state properties, and quantum dynamics calculations. We review a broad range of quantum algorithms, from hybrid quantum–classical to fully quantum, focusing on the traditional Monte Carlo family, including Markov chain Monte Carlo, variational Monte Carlo, projector Monte Carlo, path integral Monte Carlo, etc. We also cover other relevant techniques involving complex sampling tasks, such as quantum-selected configuration interaction, minimally entangled typical thermal states, entanglement forging, and Monte Carlo-flavored Lindbladian dynamics. We provide a comprehensive overview of these algorithms’ classical and quantum counterparts, detailing their theoretical frameworks and discussing the potentials and challenges in achieving quantum computational advantages.
Computations of chemical systems' equilibrium properties and non-equilibrium dynamics have been suspected of being a "killer app" for quantum computers. This review highlights the recent advancements of quantum algorithms tackling complex sampling tasks in the key areas of computational chemistry: ground state, thermal state properties, and quantum dynamics calculations. We review a broad range of quantum algorithms, from hybrid quantum-classical to fully quantum, focusing on the traditional Monte Carlo family, including Markov chain Monte Carlo, variational Monte Carlo, projector Monte Carlo, path integral Monte Carlo, etc. We also cover other relevant techniques involving complex sampling tasks, such as quantum-selected configuration interaction, minimally entangled typical thermal states, entanglement forging, and Monte Carlo-flavored Lindbladian dynamics. We provide a comprehensive overview of these algorithms' classical and quantum counterparts, detailing their theoretical frameworks and discussing the potentials and challenges in achieving quantum computational advantages.
This Perspective focuses on the several overlaps between quantum algorithms and Monte Carlo methods in the domains of physics and chemistry. We will analyze the challenges and possibilities of integrating established quantum Monte Carlo solutions into quantum algorithms. These include refined energy estimators, parameter optimization, real and imaginary-time dynamics, and variational circuits. Conversely, we will review new ideas for utilizing quantum hardware to accelerate the sampling in statistical classical models, with applications in physics, chemistry, optimization, and machine learning. This review aims to be accessible to both communities and intends to foster further algorithmic developments at the intersection of quantum computing and Monte Carlo methods. Most of the works discussed in this Perspective have emerged within the last two years, indicating a rapidly growing interest in this promising area of research.
Over the past several decades, computational approaches to studying strongly-interacting systems have become increasingly varied and sophisticated. This book provides a comprehensive introduction to state-of-the-art quantum Monte Carlo techniques relevant for applications in correlated systems. Providing a clear overview of variational wave functions, and featuring a detailed presentation of stochastic samplings including Markov chains and Langevin dynamics, which are developed into a discussion of Monte Carlo methods. The variational technique is described, from foundations to a detailed description of its algorithms. Further topics discussed include optimisation techniques, real-time dynamics and projection methods, including Green's function, reptation and auxiliary-field Monte Carlo, from basic definitions to advanced algorithms for efficient codes, and the book concludes with recent developments on the continuum space. Quantum Monte Carlo Approaches for Correlated Systems provides an extensive reference for students and researchers working in condensed matter theory or those interested in advanced numerical methods for electronic simulation.
Quantum chemistry is among the most promising applications of quantum computing, offering the potential to solve complex electronic structure problems more efficiently than classical approaches. A critical component of any quantum algorithm is the measurement step, where the desired properties are extracted from a quantum computer. This review focuses on recent advancements in quantum measurement techniques tailored for quantum chemistry, particularly within the second quantized framework suitable for current and near-term quantum hardware. We provide a comprehensive overview of measurement strategies developed primarily for the Variational Quantum Eigensolver (VQE) and its derivatives. These strategies address the inherent challenges posed by complexity of the electronic Hamiltonian operator. Additionally, we examine methods for estimating excited states and one- and two-electron properties, extending the applicability of quantum algorithms to broader chemical phenomena. Key aspects of the review include approaches for constructing measurement operators with reduced classical preprocessing and quantum implementation costs, techniques to minimize the number of measurements required for a given accuracy, and error mitigation strategies that leverage symmetries and other properties of the measurement operators. Furthermore, we explore measurement schemes rooted in Quantum Phase Estimation (QPE), which are expected to become viable with the advent of fault-tolerant quantum computing. This review emphasizes foundational concepts and methodologies rather than numerical benchmarks, serving as a resource for researchers aiming to enhance the efficiency and accuracy of quantum measurements in quantum chemistry.
Quantum chemistry is among the most promising applications of quantum computing, offering the potential to solve complex electronic structure problems more efficiently than classical approaches. A critical component of any quantum algorithm is the measurement step, where the desired properties are extracted from a quantum computer. This review focuses on recent advancements in quantum measurement techniques tailored for quantum chemistry, particularly within the second quantized framework suitable for current and near-term quantum hardware. We provide a comprehensive overview of measurement strategies developed primarily for the Variational Quantum Eigensolver (VQE) and its derivatives. These strategies address the inherent challenges posed by the complexity of the electronic Hamiltonian operator. Additionally, we examine methods for estimating excited states and one- and two-electron properties, extending the applicability of quantum algorithms to broader chemical phenomena. Key aspects of the review include approaches for constructing measurement operators with reduced classical preprocessing and quantum implementation costs, techniques to minimize the number of measurements required for a given accuracy, and error mitigation strategies that leverage symmetries and other properties of the measurement operators. Furthermore, we explore measurement schemes rooted in Quantum Phase Estimation (QPE), which are expected to become viable with the advent of fault-tolerant quantum computing. This review emphasizes foundational concepts and methodologies rather than numerical benchmarks, serving as a resource for researchers aiming to enhance the efficiency and accuracy of quantum measurements in quantum chemistry.
Solving the ground state of quantum many-body systems remains a fundamental challenge in physics and chemistry. Recent advancements in quantum hardware have opened new avenues for addressing this challenge. Inspired by the quantum-enhanced Markov chain Monte Carlo (QeMCMC) algorithm, which was originally designed for sampling the Boltzmann distribution of classical spin models using quantum computers, we introduce a quantum-assisted variational Monte Carlo (QA-VMC) algorithm for solving the ground state of quantum many-body systems by adapting QeMCMC to sample the distribution of a (neural-network) wave function in VMC. The central question is whether such a quantum-assisted proposal can potentially offer a computational advantage over classical methods. Through numerical investigations for the Fermi-Hubbard model and molecular systems, we demonstrate that the quantum-assisted proposal exhibits larger absolute spectral gaps and reduced autocorrelation times compared to conventional classical proposals, leading to more efficient sampling and faster convergence to the ground state in VMC as well as a more accurate and precise estimation of physical observables. This advantage is especially pronounced for specific parameter ranges, where the ground-state configurations are more concentrated in some configurations separated by large Hamming distances. Our results underscore the potential of quantum-assisted algorithms to enhance classical variational methods for solving the ground state of quantum many-body systems.