Simulation of electronic structure Hamiltonians using quantum computers
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Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, obtains the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule of how a chemical Hamiltonian can be simulated using a quantum computer.
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The time the quantum computer must remain coherent in QPE is determined by the necessity of O(p−1) successive applications, each of which can require on the order of millions or billions of quantum gates for practical applications, as compared to the tens to hundreds of gates achievable in the short term.
“The time the quantum computer must remain coherent is determined by the necessity of O(p−1) successive applications of , each of which can require on the order of millions or billions of quantum gates for practical applications1719, as compared to the tens to hundreds of gates achievable in the short term.”
From A variational eigenvalue solver on a photonic quantum processor (2014) · cites this paper as [17]
Split into 2 atoms — 1 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.
Experimental — accuracy not yet measuredThere are already proposals for NISQ algorithms based on variational principles.
“Indeed, there are already proposals 200 for NISQ algorithms based on variational principles 201,202 .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [201]
The standard quantum-circuit treatment for implementing each non-local operator involves as many as 2N two-qubit gates for simulating N fermionic modes; in our protocol one needs only two Mølmer-Sørensen gates.
“It is remarkable that the standard quantum-circuit treatment (e.g. see Ref.47) for implementing each involves as many as 2N two-qubit gates for simulating N fermionic modes; in our protocol one needs only two Mølmer-Sørensen gates, which are straightforwardly implementable with current trapped-ion technology.”
From From transistor to trapped-ion computers for quantum chemistry (2014) · cites this paper as [47]
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BackgroundMethodSelf-citationQASMBench: A Low-Level Quantum Benchmark Suite for NISQ Evaluation and Simulation
Li, A., Stein, S., Krishnamoorthy, S., & Ang, J. (2023) · ACM Transactions on Quantum Computing · doi:10.1145/3550488 · Cited by 183 · see citing works
“We also include two Unitary Coupled Cluster Single-Double VQE ansatz [132] for estimating the ground energy of molecules due to their ultra deep circuits.”
MethodQuantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments
O’Brien, T. E., Tarasinski, B., & Terhal, B. M. (2019) · New Journal of Physics · doi:10.1088/1367-2630/aafb8e · Cited by 175 · see citing works
Emerging quantum computing algorithms for quantum chemistry
Motta, M., & Rice, J. E. (2021) · Wiley Interdisciplinary Reviews Computational Molecular Science · doi:10.1002/wcms.1580 · Cited by 180 · see citing works
“Techniques of circuit recompilation aimed at generating optimal circuit are also intensely investigated [132, 205–207].”
BackgroundTakeshita, T. Y., Rubin, N. C., Zhang, J., Lee, E., Babbush, R., & McClean, J. R. (2020) · Physical Review X · doi:10.1103/physrevx.10.011004 · Cited by 169 · see citing works
“Starting from the original proposal to use quantum phase estimation for chemical problems [2], the precise costs of these algorithms and methods to reduce these costs by orders of magnitude have now been developed in detail [3 – 15].”
MethodKivlichan, I. D., Gidney, C., Berry, D. W., Wiebe, N., McClean, J., Sun, W., Jiang, Z., Rubin, N., Fowler, A., Aspuru-Guzik, A., et al. (2020) · Quantum · doi:10.22331/q-2020-07-16-296 · Cited by 173 · see citing works
“The first quantum circuits for these simulations [8] had O ( N 10 ) scaling where N is the number of single-particle basis functions [9].”
BackgroundSelf-citationTomography and generative training with quantum Boltzmann machines
Kieferová, M., & Wiebe, N. (2017) · Physical Review A · doi:10.1103/physreva.96.062327 · Cited by 170 · see citing works
Nakazawa, S., Nishida, S., Ise, T., Yoshino, T., Mori, N., Rahimi, R. D., Sato, K., Morita, Y., Toyota, K., Shiomi, D., et al. (2012) · Angewandte Chemie International Edition · doi:10.1002/anie.201204489 · Cited by 164 · see citing works
Statistical assertions for validating patterns and finding bugs in quantum programs
Huang, Y., & Martonosi, M. (2019) · Proceedings - International Symposium on Computer Architecture/Proceedings · doi:10.1145/3307650.3322213 · Cited by 169 · see citing works
“The procedure for doing this was laid out in detail by Whitfield [54].”
MethodQuantum Algorithms to Simulate Many-Body Physics of Correlated Fermions
Jiang, Z., Sung, K. J., Kechedzhi, K., Smelyanskiy, V. N., & Boixo, S. (2018) · Physical Review Applied · doi:10.1103/physrevapplied.9.044036 · Cited by 175 · see citing works
“(C10) commute with each other, and the first part can be implemented with only single-qubit gates with the JWT. 20”
MethodInfluentialGreen, A. S., Lumsdaine, P. L., Ross, N. J., Selinger, P., & Valiron, B. (2013) · ACM SIGPLAN Notices · doi:10.1145/2499370.2462177 · Cited by 158 · see citing works
“ame of Hex. • Class Number (CL). To approximate the class group of a real quadratic number field [8]. • Ground State Estimation (GSE). To compute the ground state energy level of a particular molecule [23]. • Quantum Linear Systems (QLS). To solve a linear system of equations [9]. • Unique Shortest Vector (USV). To cho…”
BackgroundMethodInfluentialApplication of fermionic marginal constraints to hybrid quantum algorithms
Rubin, N. C., Babbush, R., & McClean, J. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aab919 · Cited by 152 · see citing works
A quantum–quantum Metropolis algorithm
Yung, M., & Aspuru‐Guzik, A. (2012) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1111758109 · Cited by 147 · see citing works
Self-citationAjagekar, A., & You, F. (2022) · Renewable and Sustainable Energy Reviews · doi:10.1016/j.rser.2022.112493 · Cited by 155 · see citing works
Faster quantum chemistry simulation on fault-tolerant quantum computers
Cody Jones, N., Whitfield, J. D., McMahon, P. L., Yung, M.-H., Meter, R. V., Aspuru-Guzik, A., & Yamamoto, Y. (2012) · New Journal of Physics · doi:10.1088/1367-2630/14/11/115023 · Cited by 145 · see citing works
“0567v1 [ qu a n t - ph ] 3 A p r 2012 problems such as calculating the eigenvalues of a molecular Hamiltonian [4–7].”
BackgroundMethodInfluentialSelf-citationScaffCC: Scalable compilation and analysis of quantum programs
Javadi-Abhari, A., Patil, S. R., Kudrow, D., Heckey, J., Lvov, A., Chong, F. T., & Martonosi, M. (2014) · Parallel Computing · doi:10.1016/j.parco.2014.12.001 · Cited by 144 · see citing works
Improved techniques for preparing eigenstates of fermionic Hamiltonians
Berry, D. W., Kieferová, M., Scherer, A., Sanders, Y. R., Low, G. H., Wiebe, N., Gidney, C., & Babbush, R. (2018) · npj Quantum Information · doi:10.1038/s41534-018-0071-5 · Cited by 142 · see citing works
Exponentially more precise quantum simulation of fermions in second quantization
Babbush, R., Berry, D. W., Kivlichan, I. D., Wei, A. Y., Love, P. J., & Aspuru-Guzik, A. (2016) · New Journal of Physics · doi:10.1088/1367-2630/18/3/033032 · Cited by 135 · see citing works
Self-citationImproving quantum algorithms for quantum chemistry
Hastings, M. B., Wecker, D., Bauer, B., & Troyer, M. (2015) · Quantum Information and Computation · doi:10.26421/qic15.1-2-1 · Cited by 132 · see citing works
“Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA”
BackgroundExploiting Locality in Quantum Computation for Quantum Chemistry
McClean, J. R., Babbush, R., Love, P. J., & Aspuru-Guzik, A. (2014) · The Journal of Physical Chemistry Letters · doi:10.1021/jz501649m · Cited by 125 · see citing works
Self-citationJohnson, T. H., Clark, S. R., & Jaksch, D. (2014) · EPJ Quantum Technology · doi:10.1140/epjqt10 · Cited by 127 · see citing works
Electron-Phonon Systems on a Universal Quantum Computer
Macridin, A., Spentzouris, P., Amundson, J. F., & Harnik, R. (2018) · Physical Review Letters · doi:10.1103/physrevlett.121.110504 · Cited by 125 · see citing works
Tranter, A., Love, P. J., Mintert, F., & Coveney, P. V. (2018) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.8b00450 · Cited by 120 · see citing works
Quantum Computing and Simulations for Energy Applications: Review and Perspective
Paudel, H. P., Syamlal, M., Crawford, S. E., Lee, Y., Shugayev, R., Lu, P., Ohodnicki, P. R., Mollot, D., & Duan, Y. (2022) · ACS Engineering Au · doi:10.1021/acsengineeringau.1c00033 · Cited by 132 · see citing works
Fault-tolerant quantum simulations of chemistry in first quantization
Su, Y., Berry, D. W., Wiebe, N., Rubin, N., & Babbush, R. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.040332 · Cited by 123 · see citing works
Low-depth circuit ansatz for preparing correlated fermionic states on a quantum computer
Dallaire-Demers, P.-L., Romero, J., Veis, L., Sim, S., & Aspuru-Guzik, A. (2019) · Quantum Science and Technology · doi:10.1088/2058-9565/ab3951 · Cited by 119 · see citing works
Self-citationMeasurement reduction in variational quantum algorithms
Zhao, A., Tranter, A., Kirby, W. M., Ung, S. F., Miyake, A., & Love, P. J. (2020) · Physical Review A · doi:10.1103/physreva.101.062322 · Cited by 121 · see citing works
The Dawn of Quantum Natural Language Processing
Sipio, R. D., Huang, J.-H., Chen, S. Y.-C., Mangini, S., & Worring, M. (2022) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp43922.2022.9747675 · Cited by 115 · see citing works
(2014) · doi:10.1145/2597917.2597939 · Cited by 152 · see citing works
“The benchmark is parameterized here by the molecular weight (M ), but could also be parameterized by precision [24].”
BackgroundBravyi-Kitaev Superfast simulation of electronic structure on a quantum computer
Setia, K., & Whitfield, J. D. (2018) · The Journal of Chemical Physics · doi:10.1063/1.5019371 · Cited by 103 · see citing works
Self-citationTime-dependent Hamiltonian simulation with L 1 -norm scaling
Berry, D. W., Childs, A. M., Su, Y., Wang, X., & Wiebe, N. (2020) · Quantum · doi:10.22331/q-2020-04-20-254 · Cited by 109 · see citing works
“Quantum computers can simulate various physical systems, including condensed matter physics [3], quantum field theory [28], and quantum chemistry [2, 14, 37, 48].”
BackgroundAutomated distribution of quantum circuits via hypergraph partitioning
Andrés-Martínez, P., & Heunen, C. (2019) · Physical Review A · doi:10.1103/physreva.100.032308 · Cited by 112 · see citing works
Efficient quantum circuits for quantum computational chemistry
Yordanov, Y. S., Arvidsson-Shukur, D. R. M., & Barnes, C. H. W. (2020) · Physical Review A · doi:10.1103/physreva.102.062612 · Cited by 105 · see citing works
$O(N^3)$ Measurement Cost for Variational Quantum Eigensolver on Molecular Hamiltonians
Gokhale, P., Angiuli, O., Ding, Y., Gui, K., Tomesh, T., Suchara, M., Martonosi, M., & Chong, F. T. (2020) · IEEE Transactions on Quantum Engineering · doi:10.1109/tqe.2020.3035814 · Cited by 104 · see citing works
Quantum simulation of chemistry with sublinear scaling in basis size
Babbush, R., Berry, D. W., McClean, J. R., & Neven, H. (2019) · npj Quantum Information · doi:10.1038/s41534-019-0199-y · Cited by 105 · see citing works
Fermion-to-qubit mappings with varying resource requirements for quantum simulation
Steudtner, M., & Wehner, S. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aac54f · Cited by 106 · see citing works
Calculating energy derivatives for quantum chemistry on a quantum computer
O’Brien, T. E., Senjean, B., Sagastizabal, R., Bonet-Monroig, X., Dutkiewicz, A., Buda, F., DiCarlo, L., & Visscher, L. (2019) · npj Quantum Information · doi:10.1038/s41534-019-0213-4 · Cited by 100 · see citing works
Electronic Structure Calculations and the Ising Hamiltonian
Xia, R., Bian, T., & Kais, S. (2017) · The Journal of Physical Chemistry B · doi:10.1021/acs.jpcb.7b10371 · Cited by 99 · see citing works
Operator locality in the quantum simulation of fermionic models
Havlíček, V., Troyer, M., & Whitfield, J. D. (2017) · Physical Review A · doi:10.1103/physreva.95.032332 · Cited by 97 · see citing works
“However, systems where the electronelectron interaction cannot be integrated out are called strongly correlated and represent a new frontier for electronic structure in both theoretical chemistry [4] and strongly correlated materials, such as high-temperature superconductors [5].”
BackgroundSelf-citationA Full Quantum Eigensolver for Quantum Chemistry Simulations
Wei, S., Li, H., & Long, G. (2020) · Research · doi:10.34133/2020/1486935 · Cited by 95 · see citing works
“[21] James D Whitfield, Jacob Biamonte, and Alán Aspuru-Guzik.”
BackgroundQubit-excitation-based adaptive variational quantum eigensolver
Yordanov, Y. S., Armaos, V., Barnes, C. H., & Arvidsson-Shukur, D. R. M. (2021) · Apollo (University of Cambridge) · doi:10.17863/cam.75907 · Cited by 95 · see citing works
Quantum computing for neutrino-nucleus scattering
Roggero, A., Li, A. C. Y., Carlson, J., Gupta, R., & Perdue, G. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.101.074038 · Cited by 91 · see citing works
High-fidelity quantum state evolution in imperfect photonic integrated circuits
Mower, J., Harris, N. C., Steinbrecher, G. R., Lahini, Y., & Englund, D. R. (2015) · Physical Review A · doi:10.1103/physreva.92.032322 · Cited by 93 · see citing works
Digital quantum computation of fermion-boson interacting systems
Macridin, A., Spentzouris, P., Amundson, J. F., & Harnik, R. (2018) · Physical Review A · doi:10.1103/physreva.98.042312 · Cited by 92 · see citing works
Sugisaki, K., Nakazawa, S., Toyota, K., Sato, K., Shiomi, D., & Takui, T. (2018) · ACS Central Science · doi:10.1021/acscentsci.8b00788 · Cited by 88 · see citing works
Adiabatic state preparation study of methylene
Veis, L., & Pittner, J. (2014) · The Journal of Chemical Physics · doi:10.1063/1.4880755 · Cited by 90 · see citing works
Quantum-optimal-control-inspired ansatz for variational quantum algorithms
Choquette, A., Di Paolo, A., Barkoutsos, P. K., Sénéchal, D., Tavernelli, I., & Blais, A. (2021) · Physical Review Research · doi:10.1103/physrevresearch.3.023092 · Cited by 86 · see citing works
“We also show the circuit compiled to one-and two-qubit (CNOT) gates. b shows a decomposition of the multi-qubit gates based on a conventional approach to decompose exponentials of Pauli strings into circuits of CNOTs described in [62].”
MethodBabbush, R., Berry, D. W., Sanders, Y. R., Kivlichan, I., Scherer, A., Wei, A. Y., Love, P. J., & Aspuru‐Guzik, A. (2017) · Quantum Science and Technology · doi:10.1088/2058-9565/aa9463 · Cited by 85 · see citing works
Self-citationError Mitigation via Verified Phase Estimation
O’Brien, T. E., Polla, S., Rubin, N. C., Huggins, W. J., McArdle, S., Boixo, S., McClean, J. R., & Babbush, R. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.020317 · Cited by 82 · see citing works
“H s depending on the type of H s chosen (bottom right).”
BackgroundYalouz, S., Senjean, B., Günther, J., Buda, F., O’Brien, T. E., & Visscher, L. (2020) · Quantum Science and Technology · doi:10.1088/2058-9565/abd334 · Cited by 82 · see citing works
Fermionic quantum processing with programmable neutral atom arrays
González-Cuadra, D., Bluvstein, D., Kalinowski, M., Kaubruegger, R., Maskara, N., Naldesi, P., Zache, T. V., Kaufman, A. M., Lukin, M. D., Pichler, H., et al. (2023) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2304294120 · Cited by 86 · see citing works
“· U (t) i,k j,l ( √ 2 , 2 2− 4 , ), [9]”
Simulating quantum materials with digital quantum computers
Oftelie, L. B., Urbánek, M., Metcalf, M., Carter, J. T., Kemper, A. F., & de Jong, W. A. (2021) · Quantum Science and Technology · doi:10.1088/2058-9565/ac1ca6 · Cited by 81 · see citing works
Metcalf, M., Bauman, N. P., Kowalski, K., & de Jong, W. A. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.0c00421 · Cited by 78 · see citing works
An Introduction to Quantum Programming in Quipper
Green, A. S., Lumsdaine, P. L., Ross, N. J., Selinger, P., & Valiron, B. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-38986-3_10 · Cited by 78 · see citing works
“Quipper’s development was guided by the goal of implementing seven non-trivial quantum algorithms from the literature [3,5,11,12,14,17,18].”
MethodConti, I., Cerullo, G., Nenov, A., & Garavelli, M. (2020) · Journal of the American Chemical Society · doi:10.1021/jacs.0c04952 · Cited by 76 · see citing works
Local spin operators for fermion simulations
Whitfield, J. D., Havlíček, V., & Troyer, M. (2016) · Physical Review A · doi:10.1103/physreva.94.030301 · Cited by 73 · see citing works
Self-citationQuantum-circuit design for efficient simulations of many-body quantum dynamics
Raeisi, S., Wiebe, N., & Sanders, B. C. (2012) · New Journal of Physics · doi:10.1088/1367-2630/14/10/103017 · Cited by 72 · see citing works
The Trotter step size required for accurate quantum simulation of quantum chemistry
Poulin, D., Hastings, M. B., Wecker, D., Wiebe, N., Doberty, A. C., & Troyer, M. (2015) · Quantum Information and Computation · doi:10.26421/qic15.5-6-1 · Cited by 159 · see citing works
“We note that this simulation of free Hamiltonians is exact, in contrast to previous approaches [11, 12, 13, 1, 2] that rely on TS approximations.”
MethodInfluential
- Authors
- Whitfield, J. D., Biamonte, J., & Aspuru-Guzik, A.
- Affiliations
- Harvard University (US)
- Corresponding author
- Whitfield, James D.
- Biamonte, Jacob
- ORCID
- Biamonte, Jacob
- Venue
- Molecular Physics
- Publisher
- Informa UK Limited
- Published
- 2011-03-10
- Volume
- 109
- Issue
- 5
- Pages
- 735-750
- ISSN
- 0026-8976 (ISSN-L)
- 1362-3028
- OpenAlex
- W2103373613
- Cited by
- 472
- References
- 36
- Citations by year
- 2026: 33 · 2025: 54 · 2024: 54 · 2023: 64 · 2022: 49 · 2021: 36 · 2020: 43 · 2019: 23 · 2018: 29 · 2017: 13 · 2016: 14 · 2015: 14 · 2014: 19 · 2013: 11 · 2012: 12
- FWCI
- 13.55
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Molecular Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Engineering and Physical Sciences Research Council (EP/G003017/1)
- 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 phase estimation, quantum circuits, molecular integrals, adiabatic state preparation, molecular hydrogen, quantum chemistry, quantum computing
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