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Cited by 1,560Quantum Computing Algorithms and ArchitectureAdvanced Chemical Physics StudiesQuantum many-body systems

The DOI matches its Crossref record; 396 of the 396 references it cites were found in a registry or an index; none of the 446 sources its claims cite is checked for support yet.

Quantum Chemistry in the Age of Quantum Computing

Open access
Cao, Yudong, Romero, Jonathan, Olson, Jonathan P., Degroote, Matthias, Johnson, Peter D., Kieferová, Mária, Kivlichan, Ian D., Menke, Tim, Peropadre, Borja, Sawaya, Nicolas P. D., Sim, Sukin, Veis, Libor, Aspuru-Guzik, Alán·2019·Chemical Reviews
DOI 10.1021/acs.chemrev.8b00803
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Abstract

Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry, such as the electronic structure of molecules. In the past two decades, significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This Review provides an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.

Overview

AI snapshot

Written from the paper’s full text.

TL;DR

This review surveys how quantum computing can address quantum chemistry, covering classical limitations, computational complexity, fault-tolerant simulation algorithms, and NISQ-era variational methods. It argues quantum computers may efficiently handle wave functions and dynamics that are classically intractable, while near-term VQE remains heuristic and fault-tolerant QPE is longer-term.

Key findings

  • Classical quantum chemistry is limited by exponential wave-function scaling; quantum computers can natively represent and evolve entangled states, and QPE with Hamiltonian simulation can estimate spectra, but requires fault tolerance. The review states that even 30–100 error-corrected qubits could compute H₂O and LiH ground-state energies beyond classical accuracy.
  • For NISQ devices, VQE and hybrid quantum-classical algorithms are the leading approach. Hamiltonian averaging measurement cost scales roughly as \(M^2/\epsilon^2\); for electronic structure in local/Gaussian bases the expected scaling is \(O(N^6/\epsilon^2)\) versus the formal \(O(N^8/\epsilon^2)\), while a plane-wave dual basis reduces Hamiltonian terms to \(O(N^2)\) versus \(O(N^4)\) for Gaussians and gives a measurement bound of \(O(N^4/\epsilon^2)\).
  • Fault-tolerant algorithm resource estimates improved substantially: Trotter steps can be implemented with \(O(N)\) depth using a fermionic swap network; LCU costs are \(\tilde{O}(N^5)\) in second quantization and \(\tilde{O}(\eta^2 N^3)\) in first quantization; interaction-picture simulations achieve \(O(N^{11/3})\) and \(O(\eta \log N)\); Babbush et al. report T complexity \(O(N+\log 1/\epsilon)\) for Hamiltonian simulation and \(O(N^3/\epsilon + N^2 \log 1/\epsilon)\) for energy estimation.
  • Experimental VQE demonstrations include photonic HeH⁺ with 2 qubits, superconducting H₂ with 2 qubits, LiH with 4 qubits, BeH₂ with 6 qubits, and ion-trap H₂/LiH with 2/3 qubits; the review’s H₂ minimal-basis VQE simulation was numerically equal to FCI.

Methods

  • Design: a review article, not primary research, synthesizing algorithms and results relevant to quantum chemistry on quantum computers for both quantum chemists and quantum computing researchers.
  • Data or sample: no new experimental sample; it surveys cited literature and includes a pedagogical H₂ minimal-basis (STO-6G) VQE example, mapped via Bravyi-Kitaev to a four-qubit then two-qubit Hamiltonian.
  • Main analysis: narrative and taxonomic review across classical quantum chemistry limitations, computational complexity, fault-tolerant algorithms (Hamiltonian simulation, QPE, QFT), NISQ hybrid algorithms (VQE, VQS), and non-gate-based methods (adiabatic, linear optics).

Limitations

  • Stated: NISQ/VQE and related hybrid algorithms are heuristic with no performance guarantees; proper functioning must be established case-by-case and needs more hardware testing. Fault-tolerant algorithms require scalable error-corrected quantum computers, widely believed to be decades away. Complexity results are worst-case and may not reflect practical difficulty.
  • Evident: as a review, it reports no original data or new benchmarks; resource estimates and experimental results are drawn from cited works, and many timeline claims are forecasts.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1021/acs.chemrev.8b00803

Cites

396 of 396 references verified0 citation errors · 0 not checkable

Statistics

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Verification

Claims the paper makes

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Show all 234 claims

  • Probably not a substantial claim. 232
  • It may state two separate claims as one. 135
  • The quote only partly states this claim. 1
320 of 320

  • The authors state this as an assumption.

    There is always a gap between the ground state of H(s) and the first excited state during the adiabatic evolution.

    “Assuming there is always a gap between the ground state of H ( s ) and the first excited state, the end of the adiabatic evolution of the state of the system should be close to the ground state of H 1 .”

  • The authors state this as an assumption.

    The Born-Oppenheimer approximation treats nuclei as stationary point charges because nuclear mass is three orders of magnitude greater than electron mass.

    “Because the mass of a nucleus is three orders of magnitude greater than that of an electron, a common simplifying assumption made in quantum chemistry calculations, called the Born-Oppenheimer approximation (BOA), treats the nuclei as stationary point charges.”

  • The authors state this as an assumption.

    In most applications of time-dependent DFT, the time-dependent part of the Hamiltonian is assumed to be small.

    “Even though the original formulation of Time-Dependent DFT 158 does not require the time-dependent part of the Hamiltonian to be small, it has mostly been applied using this assumption 159-161 .”

  • The authors state this as an assumption.

    In early Hamiltonian simulation works, the Hamiltonian H can be written as a sum of local terms H = ∑_j H_j where each H_j acts on a subsystem.

    “Early works on Hamiltonian simulation 37,45,186 assume that H can be written as a sum of local terms H = ∑ j H j where each H j acts on a subsystem.”

  • The authors state this as an assumption.

    In the oracle model for Hamiltonian simulation, the Hamiltonian is a sparse matrix and its elements are provided by a black box or oracle.

    “An alternative setting of Hamiltonian simulation 187,188 considers an 'oracle model' where the Hamiltonian is assumed to be a sparse matrix and its elements are provided by a black box (or oracle )”

  • The authors state this as an assumption.

    In the oracle model, one can make queries to the oracle in superposition and obtain answers in superposition.

    “It is also assumed that one can make queries to the oracle in superposition and obtain answers in superposition.”

  • The authors state this as an assumption.

    In algorithms for simulating open quantum systems, the Hamiltonian and Lindblad operators are sparse and given by oracles or black boxes.

    “These algorithms focus on the Hamiltonian and Lindblad operators being sparse and given by oracles (black-boxes).”

  • The authors state this as an assumption.

    The number of terms ℓ in the Hamiltonian scales polynomially with the system size N.

    “Assuming that the number of terms ℓ in the Hamiltonian scales polynomially with the system size N (such as the number of particles or the number of qubits)”

  • The authors state this as an assumption.

    The underlying quantum computer is fault tolerant for the quantum algorithms discussed in Section 4.

    “here we focus on quantum algorithms which assume the underlying quantum computer is fault tolerant.”

  • The paper relies on this without support.

    The amplitudes of a wave function stored in quantum bits cannot be efficiently accessed.

    “The caveat is that the amplitudes of a wave function stored in quantum bits cannot be efficiently accessed.”

  • The paper relies on this without support.

    Creating a thermal state is computationally difficult, but an approximation may suffice.

    “While creating a thermal state is a computationally difficult problem by itself, an approximation may suffice.”

  • The paper relies on this without support.

    The representation of the Hamiltonian in the basis of Slater determinants (the configuration interaction matrix) is sparse.

    “The representation of the Hamiltonian in the basis of Slater determinants (the configuration interaction matrix) is sparse and therefore techniques from Section 4.1 can be used for simulation.”

  • The authors state this as an assumption.

    In adiabatic state preparation, the transition between the initial Hamiltonian H0 and the target Hamiltonian H1 is continuous.

    “This process assumes that the transition between H 0 and H 1 is continuous and that the ground state energy of λH 0 +(1 -λ ) H 1 remains sufficiently separated from the first excited state energy during the preparation”

  • The authors state this as an assumption.

    In adiabatic state preparation, the ground state energy of λH0+(1-λ)H1 remains sufficiently separated from the first excited state energy during the preparation.

    “This process assumes that the transition between H 0 and H 1 is continuous and that the ground state energy of λH 0 +(1 -λ ) H 1 remains sufficiently separated from the first excited state energy during the preparation”

  • The paper relies on this without support.

    Reaction rates are exponentially sensitive to changes in energy; a change of only 1.4 kcal/mol in the free energy generates a change of one order of magnitude in the estimate of a reaction rate.

    “Reaction rates, for example, are exponentially sensitive to changes in energy: a change of only 1.4 kcal/mol in the free fenergy (which is proportional to the energy) generates a change of one order of magnitude in the estimate of a reaction rate.”

  • The authors state this as an assumption.

    All Hamiltonian terms are measured independently, i.e. Cov(〈O_i〉, 〈O_j〉) = 0 for all i ≠ j.

    “Assuming that all the terms are measured independently i.e Cov( 〈 O i 〉 , 〈 O j 〉 ) = 0 ∀ i = j ), the total variance of the estimator of 〈 H 〉 , will be given by the sum of the variances of the individual Hamiltonian terms”

  • The authors state this as an assumption.

    Parallel rotation and orbital readout is available for Hamiltonian averaging.

    “Hamiltonian averaging only adds a constant factor to the state preparation procedure, assuming parallel”

  • The authors state this as an assumption.

    The bound on the number of measurements for the plane wave dual basis assumes a neutral system and a fixed density.

    “This bound is obtained by bounding the norms of the different terms in the Hamiltonian assuming a neutral system and a fixed density.”

  • The paper relies on this without support.

    In finite molecules and bulk materials, the number of electrons η is in O(N).

    “Since in finite molecules and bulk materials, η ∈ O ( N ), this shows that the number of measurements does not scale worse than O ( N 4 /ϵ 2 ) for this representation of the Hamiltonian.”

  • The authors state this as an assumption.

    In AAVQE, the gap of H(s) does not become too small.

    “As long as the gap of H ( s ) does not become too small (see Section 5.3.1), the ground state of H ( s ) will lie close to the ground state of the Hamiltonian H ( s +∆ s ) in the subsequent step.”

  • The paper relies on this without support.

    Measurement noise has a significant impact on VQE-based calculations, especially when iterative procedures are involved.

    “In all these calculations, an important factor to consider is the impact of measurement noise, especially when iterative procedures are involved.”

  • The authors state this as an assumption.

    The Born-Oppenheimer approximation, treating the nuclei as stationary classical particles, is valid for the molecular hydrogen problem.

    “To simplify the problem, we apply the Born-Oppenheimer approximation, in which we treat the nuclei as stationary classical particles.”

  • The paper relies on this without support.

    In periodic systems, the periodic density does not vanish exponentially far away from the nuclei.

    “In periodic systems, the periodic density does not vanish exponentially far away from the nuclei, unlike the case of isolated molecules.”

  • The paper relies on this without support.

    In isolated molecules, the electronic density is highly peaked around the nuclei and vanishes exponentially away from them.

    “In isolated molecules, the electronic density is highly peaked around the nuclei and vanishes exponentially away from them”

  • The paper relies on this without support.

    Compared to GTOs, PW basis sets generally require an order of magnitude more functions to achieve a similar accuracy.

    “Compared to GTOs, PW basis sets generally require an order of magnitude more functions to achieve a similar accuracy.”

  • The paper relies on this without support.

    So far, the parity mapping has not been considered as a useful contender for use in quantum simulation.

    “While the parity mapping is a valid transformation from fermionic operators to qubits, so far, it has not been considered as a useful contender for use in quantum simulation.”
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Each contradiction between two passages listed here was confirmed by an independent check.

  • The stated creation action uses a matrix that annihilates the unoccupied state

    Minor

    The paper says σ+ performs the occupation-changing action of a creation operator, but its displayed matrix sends |0〉 to zero, not to |1〉. This conflicts with its stated convention that occupation is represented by 1.

  • Pauli-operator indices are defined in opposite orders

    Minor

    The text defines σ i,j as the j-th operator in the i-th Hamiltonian term, but the glossary calls it the i-th operator acting on the j-th qubit. These assign different meanings to the indices.

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Citations

Cites

Items 201–396 of 396

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Research works printed but not linked: 34

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Metadata

Authors
Cao, Y., Romero, J., Olson, J. P., Degroote, M., Johnson, P. D., Kieferová, M., Kivlichan, I. D., Menke, T., Peropadre, B., Sawaya, N. P. D., Sim, S., Veis, L., & Aspuru-Guzik, A.
Affiliations
  • Harvard University (US)
  • University of Waterloo (CA)
  • Zapata (United States) (US)
  • Intel (United States) (US)
  • Czech Academy of Sciences (CZ)
  • Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States
ORCID
Venue
Chemical Reviews
Publisher
American Chemical Society (ACS)
Published
2019-08-30
Volume
119
Issue
19
Pages
10856-10915
ISSN
  • 0009-2665 (ISSN-L)
  • 1520-6890
DOI
10.1021/acs.chemrev.8b00803
OpenAlex
W2906538035
Semantic Scholar
1eaab9b33f1261744567455a14830e8a92796cf5
Cited by
1,560
References
368
Citations by year
2026: 233 · 2025: 316 · 2024: 275 · 2023: 272 · 2022: 213 · 2021: 164 · 2020: 77 · 2019: 9
FWCI
7.53
Type
journal article
Language
en
Open access
Licence
Locations
  • Chemical Reviews (publishedVersion)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • Government of Canada
  • Army Research Office (W911NF-15-1-0256)
  • Grantov? Agentura Cesk? Republiky (18-18940Y)
  • Intelligence Advanced Research Projects Activity (W911NF-17-C-0050)
  • Office of Naval Research (ONR 00014-16-1-2008)
  • U.S. Department of Energy (DE-FG02-97ER25308)
  • Natural Sciences and Engineering Research Council of Canada
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Advanced Chemical Physics Studies
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
Keywords
quantum computing, quantum chemistry, quantum simulation, quantum algorithms, molecular electronic structure, quantum computing hardware, superposition, entanglement
MeSH terms
Algorithms, Computing Methodologies, Models, Chemical, Quantum Theory, Molecular Dynamics Simulation

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