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Universal Quantum Simulators
Abstract
Feynman's 1982 conjecture, that quantum computers can be programmed to simulate any local quantum system, is shown to be correct.
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In the citing paper
“The algorithm runs in time polynomial in the size of the system being simulated (the number of qubits) and the desired evolution time, giving an exponential speedup over the best general classical algorithms known.”
From Quantum algorithms: an overview (2016) · cites this paper as [44]
In the citing paper
“A general-purpose quantum computer can indeed ef fi ciently simulate quantum mechanics in this sense for many physically realistic cases, such as systems with locality restrictions on their interactions. 44”
From Quantum algorithms: an overview (2016) · cites this paper as [44]
In the citing paper
“It is noteworthy that we can program quantum computers to simulate localized quantum systems [68].”
From Quantum Computing in the Next-Generation Computational Biology Landscape: From Protein Folding to Molecular Dynamics (2024) · cites this paper as [68]
In the citing paper
“With this universal approach, a wide range of quantum problems can be simulated to the desired accuracy with a polynomial increase in quantum resources only (Lloyd, 1996).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Lloyd, 1996
In the citing paper
“The belief in this quantum approach was indeed supported in the 90's by the first efficient quantum algorithms [316], which were rapidly followed by breakthrough discoveries applicable to quantum chemistry [3,624]”
From Photonic quantum information processing: a review (2019) · cites this paper as [3]
In the citing paper
“The quantum computational cost of these algorithms has continued to be reduced 47-50 .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [50]
In the citing paper
“Later, it was also demonstrated that the Trotter errors depend on the maximum nuclear charge rather than the number of spin-orbitals 50 .”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [50]
In the citing paper
“In the general case, it would require a number of elementary gates of 2 20 ≃ 10 6 [2]. With our method, the number of gates can be as small as 10 per Trotter step, consisting of Mølmer-Sørensen gates (dark blue and green), local exp( iφ ′ σ y 2 ) gates (red), exp[ ± i ( π/ 4) ∑ i σ y i ] (yellow) and exp[ ± i ( π/ 4) ∑ i σ x i ] (cyan) gates (see Fig. 1b) [35].”
From Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions (2012) · cites this paper as [2]
In the citing paper
“We map products of fermionic operators onto nonlocal spin operators and decompose the resulting dynamics in efficient steps with Trotter methods, yielding an overall protocol that employs only polynomial resources.”
From Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions (2012) · cites this paper as [2]
In the citing paper
“Note that for bounded Hamiltonians, the Trotter expansion associated with the exponential of the polynomial sum of efficiently implementable nonlocal terms is also efficient [2, 29, 30], i.e., it only requires polynomial resources [35].”
From Quantum Simulation of Interacting Fermion Lattice Models in Trapped Ions (2012) · cites this paper as [2]
In the citing paper
“On a fault-tolerant computer, one may directly measure the full-CI energy by first preparing a wave function ansatz on the quantum hardware (as in VQE) and then applying the Quantum Phase Estimation algorithm, which projects a prepared state into an energy eigenstate of the Hamiltonian.[111,[132][133][134][135][136]”
From Quantum Computing for Molecular Biology** (2023) · cites this paper as [111]
In the citing paper
“A proposed solution is to use highly controlled laboratory quantum systems themselves to perform such simulations [1-3].”
From Quantum Simulation of the Klein Paradox with Trapped Ions (2011) · cites this paper as [2]
In the citing paper
“Simulating quantum mechanics using conventional computers rapidly becomes intractable as the physical systems to which it is applied grow larger.”
From Quantum Simulation of the Klein Paradox with Trapped Ions (2011) · cites this paper as [2]
In the citing paper
“Indeed, this fact is related to correlation and entanglement spreading according to the Lieb-Robinson theorem that states that, after a sufficiently large time, states can become strongly entangled ([91-96], see also [97]).”
From Can one trust quantum simulators? (2012) · cites this paper as [96]
In the citing paper
“or that the quantum dynamics is in general computationally hard [96, 97].”
From Can one trust quantum simulators? (2012) · cites this paper as [96]
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“First, one could exploit the natural decoherence of the simulator, as first suggested in (Lloyd, 1996) and investigated experimentally in (Tseng et al., 2000) (see also the discussion in Section V.”
BackgroundMethodInfluentialManipulating quantum entanglement with atoms and photons in a cavity
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Superconducting Circuits for Quantum Information: An Outlook
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A quantum engineer's guide to superconducting qubits
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Noisy intermediate-scale quantum algorithms
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“Among the further applications of a quantum computer which have so far been envisioned are a data base with faster access to unsorted data [2], and a universal simulator of quantum systems [3, 4].”
BackgroundQuantum simulations with trapped ions
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Quantum computational chemistry
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Quantum Chemistry in the Age of Quantum Computing
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“itrary subsets, H = P jHj. The criteria for grouping can dier depending on the target Hamiltonian as shown below. The variational ansatz is dened as j#TAB#()i= YS b=1 UH j (b j):::UH 1( b 1) ! j 0i; (37) where j 0ican be chosen to match the desired symmetry properties of the target ground 84 state, UH k (b k) = exp b…”
BackgroundDicke quantum phase transition with a superfluid gas in an optical cavity
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Superconducting Qubits: Current State of Play
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“Furthermore quantum dynamics becomes efficiently simulatable on a quantum computer [3].”
BackgroundSimulated Quantum Computation of Molecular Energies
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Many-body physics with individually controlled Rydberg atoms
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“Some work has been done on simulating Hamiltonians on quantum computers [1, 24, 33], but I do not believe this question has been completely addressed yet.”
Quantum computation and decision trees
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Quantum computing with trapped ions
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“In particular, the study of quantum mechanical many body systems would benefit considerably from such a device (Feynman, 1982; Lloyd, 1996).”
BackgroundQuantum Chemistry in the Age of Quantum Computing
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Ultimate physical limits to computation
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Self-citation14-Qubit Entanglement: Creation and Coherence
Monz, T., Schindler, P., Barreiro, J. T., Chwalla, M., Nigg, D., Coish, W. A., Harlander, M., Hänsel, W., Hennrich, M., & Blatt, R. (2011) · Physical Review Letters · doi:10.1103/physrevlett.106.130506 · Cited by 1,110
An open-system quantum simulator with trapped ions
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“(38)] is simulated by a digital expansion (Lloyd, 1996) by interleaving the different interactions.”
BackgroundSuppressing quantum errors by scaling a surface code logical qubit
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“A general-purpose quantum computer can indeed e(cid:14)ciently simulate quantum mechanics in this sense for many physically realistic cases, such as systems with locality restrictions on their interactions [64].”
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MethodThe Variational Quantum Eigensolver: A review of methods and best practices
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Quantum Computation Over Continuous Variables
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“First of all, quantum computers can do other things besides factoring, and some of these other tasks (in particular quantum simulation — Lloyd 1996) might be accomplished with a less reliable or smaller device. Furthermore, our estimate of the accuracy threshold might be too conservative for a number of reasons. For e…”
Quantum information processing with superconducting circuits: a review
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Ensemble quantum computing by NMR spectroscopy
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Power of One Bit of Quantum Information
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Effective Quantum Spin Systems with Trapped Ions
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“With the experimental method presented here but with a more sophisticated analysis [9], the density matrix of arbitrary spin states that spread over the whole Bloch sphere can be reconstructed.”
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“For purely coherent dynamics governed by the Hamiltonian, this is the familiar‘digital quantum simulato”
MethodQuantum Algorithms for Quantum Chemistry and Quantum Materials Science
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Hamiltonian Simulation by Qubitization
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“The first explicit quantum algorithms were discovered by Lloyd [31] for local interactions, and then generalized by Aharonov and Ta-Shma [3] to sparse Hamiltonians.”
MethodQuantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors
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Self-citationExponential algorithmic speedup by a quantum walk
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Real-time dynamics of lattice gauge theories with a few-qubit quantum computer
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“This provides the tools for universal digital quantum simulation [16], where the time evolution of a quantum system is approximated as a stroboscopic sequence of quantum gates [17].”
MethodScalable Quantum Simulation of Molecular Energies
O’Malley, P. J. J., Babbush, R., Kivlichan, I. D., Romero, J., McClean, J. R., Barends, R., Kelly, J., Roushan, P., Tranter, A., Ding, N., et al. (2016) · Physical Review X · doi:10.1103/physrevx.6.031007 · Cited by 1,059
“Universal and efficient simulation of physical systems [1] is among the most compelling applications of quantum”
BackgroundProgrammable quantum simulations of spin systems with trapped ions
Monroe, C. R., Campbell, W. C., Duan, L.-M., Gong, Z., Gorshkov, A. V., Hess, P., Islam, R., Kim, K., Linke, N. M., Pagano, G., et al. (2021) · Reviews of Modern Physics · doi:10.1103/revmodphys.93.025001 · Cited by 891
Programmable Quantum Simulations of Spin Systems with Trapped Ions
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Towards quantum chemistry on a quantum computer
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“The challenge of first-principles calculations arises because the computational resources required to obtain exact (that is, full configuration interaction) solutions of the Schrodinger equation on a conventional computer generally increase exponentially with the number of atoms involve”
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Optimal Hamiltonian Simulation by Quantum Signal Processing
Low, G. H., & Chuang, I. L. (2017) · Physical Review Letters · doi:10.1103/physrevlett.118.010501 · Cited by 955
“[11] S. Lloyd, Science 273, 1073 (1996).”
BackgroundPhotonic quantum information processing: a review
Flamini, F., Spagnolo, N., & Sciarrino, F. (2019) · Reports on Progress in Physics · doi:10.1088/1361-6633/aad5b2 · Cited by 770
Simulation of Many-Body Fermi Systems on a Universal Quantum Computer
Abrams, D. S., & Lloyd, S. (1997) · Physical Review Letters · doi:10.1103/physrevlett.79.2586 · Cited by 745
Self-citationElucidating reaction mechanisms on quantum computers
Reiher, M., Wiebe, N., Svore, K. M., Wecker, D., & Troyer, M. (2017) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1619152114 · Cited by 745
“[5] Seth Lloyd, “Universal quantum simulators,” Science 273, 1073 (1996).”
MethodQuantum error correction below the surface code threshold
Collaborators, G. Q. A. A., Acharya, R., Abanin, D. A., Aghababaie-Beni, L., Aleiner, I., Andersen, T. I., Ansmann, M., Arute, F., Arya, K., Asfaw, A., et al. (2025) · Nature · doi:10.1038/s41586-024-08449-y · Cited by 778
“Quantum computing promises computational speedups in quantum chemistry [5], quantum simulation [6], cryptography [7], and optimization [8].”
Simulating Hamiltonian Dynamics with a Truncated Taylor Series
Berry, D. W., Childs, A. M., Cleve, R., Kothari, R., & Somma, R. D. (2015) · Physical Review Letters · doi:10.1103/physrevlett.114.090502 · Cited by 742
Matrix product states represent ground states faithfully
Verstraete, F., & Cirac, J. I. (2006) · Physical Review B · doi:10.1103/physrevb.73.094423 · Cited by 730
Motta, M., Sun, C., Tan, A. T. K., O’Rourke, M. J., Ye, E., Minnich, A. J., Brandão, F. G. S. L., & Chan, G. K.-L. (2020) · Nature Physics · doi:10.1038/s41567-019-0704-4 · Cited by 709
Efficient Quantum Algorithms for Simulating Sparse Hamiltonians
Berry, D. W., Ahokas, G., Cleve, R., & Sanders, B. C. (2007) · Communications in Mathematical Physics · doi:10.1007/s00220-006-0150-x · Cited by 692
“Here we apply the higher-order integrators of Suzuki [12, 13] to reduce the temporal scaling from t 3 / 2 [5] or t 2 [4] to the slightly superlinear scaling t 1+1 / 2 k , where k is the order of the integrator and may be an arbitrarily large integer.”
BackgroundMethodInfluentialEfficient Variational Quantum Simulator Incorporating Active Error Minimization
Li, Y., & Benjamin, S. C. (2017) · Physical Review X · doi:10.1103/physrevx.7.021050 · Cited by 686
“However, there are still many problems that cannot be solved using these powerful classical methods, so it is hoped that quantum computers can extend their reach [18,19,32].”
BackgroundPractical quantum advantage in quantum simulation
Daley, A. J., Bloch, I., Kokail, C., Flannigan, S., Pearson, N., Troyer, M., & Zoller, P. (2022) · Nature · doi:10.1038/s41586-022-04940-6 · Cited by 783
“Many of the algorithms used on a classical computer can be modified for use on a fault-tolerant digital quantum computer, and vice versa 16 .”
BackgroundMethodTopological quantum computation
Freedman, M. H., Kitaev, A., Larsen, M., & Wang, Z. (2002) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-02-00964-3 · Cited by 771
“In this context, there are three areas where remarkable algorithms have been found: searching a data base [15], abelian groups (factoring and discrete logarithm) [19], [27], and simulating physical systems [5], [ 21 ].”
The quantum technologies roadmap: a European community view
Acín, A., Bloch, I., Buhrman, H., Calarco, T., Eichler, C., Eisert, J., Esteve, D., Gisin, N., Glaser, S. J., Jelezko, F., et al. (2018) · New Journal of Physics · doi:10.1088/1367-2630/aad1ea · Cited by 648
Quantum optimization using variational algorithms on near-term quantum devices
Moll, N., Barkoutsos, P., Bishop, L. S., Chow, J. M., Cross, A., Egger, D. J., Filipp, S., Fuhrer, A., Gambetta, J. M., Ganzhorn, M., et al. (2018) · Quantum Science and Technology · doi:10.1088/2058-9565/aab822 · Cited by 744
Hamiltonian Simulation by Qubitization
Low, G. H., & Chuang, I. L. (2019) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 714
Contextuality supplies the ‘magic’ for quantum computation
Howard, M., Wallman, J. J., Veitch, V., & Emerson, J. (2014) · Nature · doi:10.1038/nature13460 · Cited by 661
“Quantum information enables dramatic new advantages for computation, such as Shor’s factoring algorithm [1] and quantum simulation algorithms [2].”
High-dimensional one-way quantum processing implemented on d-level cluster states
Reimer, C., Sciara, S., Roztocki, P., Islam, M., Romero Cortés, L., Zhang, Y., Fischer, B., Loranger, S., Kashyap, R., Cino, A., et al. (2019) · Nature Physics · doi:10.1038/s41567-018-0347-x · Cited by 617
Universal Digital Quantum Simulation with Trapped Ions
Lanyon, B. P., Hempel, C., Nigg, D., Müller, M., Gerritsma, R., Zähringer, F., Schindler, P., Barreiro, J. T., Rambach, M., Kirchmair, G., et al. (2011) · Science · doi:10.1126/science.1208001 · Cited by 613
Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator
Hempel, C., Maier, C., Romero, J., McClean, J., Monz, T., Shen, H., Jurcevic, P., Lanyon, B. P., Love, P., Babbush, R., et al. (2018) · Physical Review X · doi:10.1103/physrevx.8.031022 · Cited by 629
“This can be done in a freely programmable way [7] that is amenable to quantum error correction [22, 23].”
BackgroundInfluentialDefining and detecting quantum speedup
Rønnow, T. F., Wang, Z., Job, J., Boixo, S., Isakov, S. V., Wecker, D., Martinis, J. M., Lidar, D. A., & Troyer, M. (2014) · Science · doi:10.1126/science.1252319 · Cited by 601
Simulating a quantum magnet with trapped ions
Friedenauer, A., Schmitz, H., Glueckert, J. T., Porras, D., & Schaetz, T. (2008) · Nature Physics · doi:10.1038/nphys1032 · Cited by 602
“Decoherence in the simulator might even mimic the influence of the natural environmen”
BackgroundSelf-verifying variational quantum simulation of lattice models
Kokail, C., Maier, C., van Bijnen, R., Brydges, T., Joshi, M. K., Jurcevic, P., Muschik, C. A., Silvi, P., Blatt, R., Roos, C. F., et al. (2019) · Nature · doi:10.1038/s41586-019-1177-4 · Cited by 594
Experimental Quantum Error Correction
Cory, D. G., Price, M. D., Maas, W., Knill, E., Laflamme, R., Zurek, W. H., Havel, T. F., & Somaroo, S. S. (1998) · Physical Review Letters · doi:10.1103/physrevlett.81.2152 · Cited by 581
Topological Quantum Computation
Wang, Z. (2010) · Regional conference series in mathematics · doi:10.1090/cbms/112 · Cited by 605
A Modular Functor Which is Universal¶for Quantum Computation
Freedman, M. H., Larsen, M., & Wang, Z. (2002) · Communications in Mathematical Physics · doi:10.1007/s002200200645 · Cited by 573
Computational Capacity of the Universe
Lloyd, S. (2002) · Physical Review Letters · doi:10.1103/physrevlett.88.237901 · Cited by 571
Self-citationThe cold atom Hubbard toolboxPreprint
Jaksch, D., & Zoller, P. (2004) · Cited by 627
Hybrid Quantum-Classical Algorithms and Quantum Error Mitigation
Endo, S., Cai, Z., Benjamin, S. C., & Yuan, X. (2021) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.90.032001 · Cited by 566
Toward the first quantum simulation with quantum speedup
Childs, A. M., Maslov, D., Nam, Y., Ross, N. J., & Su, Y. (2018) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1801723115 · Cited by 551
“Efficient quantum algorithms for simulating quantum dynamics have been known for over two decades (12).”
BackgroundThe Bravyi-Kitaev transformation for quantum computation of electronic structure
Seeley, J. T., Richard, M. J., & Love, P. J. (2012) · The Journal of Chemical Physics · doi:10.1063/1.4768229 · Cited by 585
Quantum computing with neutral atoms
Henriet, L., Beguin, L., Signoles, A., Lahaye, T., Browaeys, A., Reymond, G.-O., & Jurczak, C. (2020) · Quantum · doi:10.22331/q-2020-09-21-327 · Cited by 550
“For those models, the solution consists in using quantum circuits made of gates [53].”
MethodExperimental Ten-Photon Entanglement
Wang, X., Chen, L.-K., Li, W.-Y., Huang, H.-L., Liu, C., Chen, C., Luo, Y.-H., Su, Z.-E., Wu, D., Li, Z.-D., et al. (2016) · Physical Review Letters · doi:10.1103/physrevlett.117.210502 · Cited by 562
“[6] Type-II PPKTP, 20mm Pulsed, 160fs 521: 775+1590 ~ 334 ~44% (775 nm) ~7% (1590 nm) 91%”
Ising machines as hardware solvers of combinatorial optimization problems
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“Quantum simulation was in fact one of the early motivations for realizing a quantum computer, based on Feynman’s conjecture that a quantum computer should be able to simulate quantum systems more efficiently than classical computers 14 — proven true later by Lloyd 17 .”
BackgroundTheory of Trotter Error with Commutator Scaling
Childs, A. M., Su, Y., Tran, M. C., Wiebe, N., & Zhu, S. (2021) · Physical Review X · doi:10.1103/physrevx.11.011020 · Cited by 646
Quantum Computation by Adiabatic Evolution
Farhi, E., Goldstone, J., Gutmann, S., & Sipser, M. (2000) · arXiv · doi:10.48550/arxiv.quant-ph/0001106 · Cited by 536
“In any case, our algorithm can be recast within the conventional quantum computing paradigm using the technique introduced by Lloyd [5].”
MethodHigh-fidelity readout and control of a nuclear spin qubit in silicon
Pla, J. J., Tan, K. Y., Dehollain, J. P., Lim, W. H., Morton, J. J. L., Zwanenburg, F. A., Jamieson, D. N., Dzurak, A. S., & Morello, A. (2013) · Nature · doi:10.1038/nature12011 · Cited by 527
“30(1) MHz is close to, but not identical with, the bulk value of 117.”
Background(2016) · doi:10.1103/physrevlett.117.060505 · Cited by 540
Quantum Algorithms for Quantum Field Theories
Jordan, S. P., Lee, K. S. M., & Preskill, J. P. (2012) · Science · doi:10.1126/science.1217069 · Cited by 511
The physical nature of information
Landauer, R. W. (1996) · Physics Letters A · doi:10.1016/0375-9601(96)00453-7 · Cited by 526
A scalable quantum computer with ions in an array of microtraps
Cirac, J. I., & Zoller, P. (2000) · Nature · doi:10.1038/35007021 · Cited by 516
“Finally, we would like to stress that apart from using it for quantum computations, our scheme can emulate Ising (and similar) model”
MethodDeterministic Preparation of a Tunable Few-Fermion System
Serwane, F., Zürn, G., Lompe, T., Ottenstein, T. B., Wenz, A. N., & Jochim, S. (2011) · Science · doi:10.1126/science.1201351 · Cited by 509
Noise tailoring for scalable quantum computation via randomized compiling
Wallman, J. J., & Emerson, J. (2016) · Physical Review A · doi:10.1103/physreva.94.052325 · Cited by 535
“The rich complexity of quantum states and processes enables powerful new protocols for processing and communicating quantum information, as illustrated by Shor’s factoring algorithm [1] and quantum simulation algorithms [2].”
Metadata
- Authors
- Lloyd, S.
- Affiliations
- Massachusetts Institute of Technology (US)
- Corresponding author
- Lloyd, Seth
- ORCID
- Lloyd, Seth
- Venue
- Science
- Published
- 1996-08-23
- Volume
- 273
- Issue
- 5278
- Pages
- 1073-1078
- ISSN
- 0036-8075 (ISSN-L)
- 1095-9203
- OpenAlex
- W2040792108
- Cited by
- 3,088
- References
- 54
- Citations by year
- 2026: 183 · 2025: 317 · 2024: 287 · 2023: 268 · 2022: 221 · 2021: 251 · 2020: 208 · 2019: 150 · 2018: 117 · 2017: 134 · 2016: 104 · 2015: 82 · 2014: 97 · 2013: 72 · 2012: 75
- FWCI
- 30.78
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Science (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Keywords
- quantum simulation, quantum computing hardware, quantum computing
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