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Progress towards practical quantum variational algorithms
Open accessAbstract
The preparation of quantum states using short quantum circuits is one of the most promising near-term applications of small quantum computers, especially if the circuit is short enough and the fidelity of gates high enough that it can be executed without quantum error correction. Such quantum state preparation can be used in variational approaches, optimizing parameters in the circuit to minimize the energy of the constructed quantum state for a given problem Hamiltonian. For this purpose we propose a simple-to-implement class of quantum states motivated by adiabatic state preparation. We test its accuracy and determine the required circuit depth for a Hubbard model on ladders with up to 12 sites (24 spin orbitals), and for small molecules. We find that this ansatz converges faster than previously proposed schemes based on unitary coupled clusters. While the required number of measurements is astronomically large for quantum chemistry applications to molecules, applying the variational approach to the Hubbard model (and related models) is found to be far less demanding and potentially practical on small quantum computers. We also discuss another application of quantum state preparation using short quantum circuits, to prepare trial ground states of models faster than using adiabatic state preparation.
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“Several QML algorithms based on PQC have been proposed [21–27], demonstrating superior performance and computational abilities compared to conventional algorithms [28–31].”
From Medical image diagnosis based on adaptive Hybrid Quantum CNN (2023) · cites this paper as [21]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Motivated by adiabatic state preparation, the Variational Hamiltonian Ansatz (VHA) was developed to reduce the number of parameters and accelerate the convergence (McClean et al. , 2016; Wecker et al. , 2015).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Wecker et al., 2015
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In the citing paper
“To this end, VQE (McClean et al. , 2016; Peruzzo et al. , 2014) has been proposed, to find the ground state of a Hamiltonian H in a manner that is suited for NISQ devices (Wecker et al. , 2015).”
From Noisy intermediate-scale quantum algorithms (2022) · cites this paper as Wecker et al. , 2015
In the citing paper
“The AQC model has been shown to be computationally equivalent to the gate model 62 , meaning that algorithms written in either model can be translated to ones in the other model without incurring prohibitive computational overhead.”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [62]
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Citations
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Performance comparison of optimization methods on variational quantum algorithms
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Measuring the Rényi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computer
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Molecular Quantum Dynamics: A Quantum Computing Perspective
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Parallel quantum simulation of large systems on small NISQ computers
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Does provable absence of barren plateaus imply classical simulability?
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“The arrival of variational quantum algorithms effectively democratized the world of quantum computing.”
Quantum Power Method by a Superposition of Time-Evolved States
Seki, K., & Yunoki, S. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.010333 · Cited by 109
“[36] D.”
MethodOptimizing quantum optimization algorithms via faster quantum gradient computation
Gilyén, A., Arunachalam, S., & Wiebe, N. (2019) · Society for Industrial and Applied Mathematics eBooks · doi:10.1137/1.9781611975482.87 · Cited by 134
“Thus, from the perspective of our algorithm, QAOA looks exactly like VQE.”
BackgroundMethodInfluentialBauman, N. P., Bylaska, E. J., Krishnamoorthy, S., Low, G. H., Wiebe, N., Granade, C., Roetteler, M., Troyer, M., & Kowalski, K. (2019) · The Journal of Chemical Physics · doi:10.1063/1.5094643 · Cited by 103
Self-citationKoh, D. E., & Grewal, S. (2022) · Quantum · doi:10.22331/q-2022-08-16-776 · Cited by 130
“However, estimation is often the major bottleneck in many of these applications, where the number of measurements required is often too large for the algorithms to achieve the desired accuracy on useful instances using nearterm quantum hardware [9, 10].”
BackgroundNearly Optimal Measurement Scheduling for Partial Tomography of Quantum States
Bonet-Monroig, X., Babbush, R., & O’Brien, T. E. (2020) · Physical Review X · doi:10.1103/physrevx.10.031064 · Cited by 121
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Jastrow-type Decomposition in Quantum Chemistry for Low-Depth Quantum Circuits
Matsuzawa, Y., & Kurashige, Y. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.9b00963 · Cited by 104
A comprehensive review of quantum machine learning: from NISQ to fault tolerance
Wang, Y., & Liu, J. (2024) · Reports on Progress in Physics · doi:10.1088/1361-6633/ad7f69 · Cited by 145
“The concept of Variational Quantum Algorithms (VQA) was initially introduced to address certain quantum chemistry challenges, notably through the well-known Variational Quantum Eigensolver (VQE) [108, 120, 155].”
A Full Quantum Eigensolver for Quantum Chemistry Simulations
Wei, S., Li, H., & Long, G. (2020) · Research · doi:10.34133/2020/1486935 · Cited by 97
Calculation of the Green's function on near-term quantum computers
Endo, S., Kurata, I., & Nakagawa, Y. O. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033281 · Cited by 106
García-Pérez, G., Rossi, M. A. C., Sokolov, B., Tacchino, F., Barkoutsos, P. K., Mazzola, G., Tavernelli, I., & Maniscalco, S. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.040342 · Cited by 105
“, wave-function collapses, used to estimate term k [32].”
MethodMedical image diagnosis based on adaptive Hybrid Quantum CNN
Ajlouni, N., Özyavaş, A., Takaoğlu, M., Takaoğlu, F., & Ajlouni, F. (2023) · BMC Medical Imaging · doi:10.1186/s12880-023-01084-5 · Cited by 97
“Several QML algorithms based on PQC have been proposed [21–27], demonstrating superior performance and computational abilities compared to conventional algorithms [28–31].”
MethodConstructing a virtual two-qubit gate by sampling single-qubit operations
Mitarai, K., & Fujii, K. (2020) · New Journal of Physics · doi:10.1088/1367-2630/abd7bc · Cited by 120
A functional architecture for scalable quantum computing
Sete, E. A., Zeng, W. J., & Rigetti, C. T. (2016) · IEEE International Conference on Rebooting Computing (ICRC) · doi:10.1109/icrc.2016.7738703 · Cited by 90
“In machine learning and quantum simulation, particularly for catalysts [13] and high temperature superconductivity [9], scalable quantum computers promise performance unrivaled by classical supercomputers.”
BackgroundNeural predictor based quantum architecture search
Zhang, S.-X., Hsieh, C.-Y., Zhang, S., & Yao, H. (2021) · Machine Learning: Science and Technology · doi:10.1088/2632-2153/ac28dd · Cited by 88
“Some famous PQCs include QAOA ansatz for combinatorial problems [10], hardware efficient ansatz [28] and unitary coupled clusters (UCC) ansatz [4, 5, 29] for VQE.”
MethodBiology and medicine in the landscape of quantum advantages
Cordier, B. A., Sawaya, N. P. D., Guerreschi, G. G., & McWeeney, S. K. (2022) · Journal of The Royal Society Interface · doi:10.1098/rsif.2022.0541 · Cited by 87
“One barrier is the number of measurements required by VQS algorithms to attain chemical accuracy competitive with existing classical simulation techniques [182, 183].”
MethodWecker, D., Hastings, M. B., & Troyer, M. (2016) · Physical Review A · doi:10.1103/physreva.94.022309 · Cited by 88
Self-citationQuantum Variational Optimization of Ramsey Interferometry and Atomic Clocks
Kaubruegger, R., Vasilyev, D. V., Schulte, M., Hammerer, K., & Zoller, P. (2021) · Physical Review X · doi:10.1103/physrevx.11.041045 · Cited by 102
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
“Numerous variational forms Û(θ) have been explored in the literature [18, 23–28].”
BackgroundMethodInfluentialBabbush, 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
Error 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 92
Real-Time Evolution for Ultracompact Hamiltonian Eigenstates on Quantum Hardware
Klymko, K., Mejuto-Zaera, C., Cotton, S. J., Wudarski, F., Urbánek, M., Hait, D., Head‐Gordon, M., Whaley, K. B., Moussa, J. E., Wiebe, N., et al. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.020323 · Cited by 98
Adaptive Variational Quantum Imaginary Time Evolution Approach for Ground State Preparation
Gomes, N., Mukherjee, A., Zhang, F., Iadecola, T., Wang, C., Ho, K., Orth, P. P., & Yao, Y. (2021) · Advanced Quantum Technologies · doi:10.1002/qute.202100114 · Cited by 99
Gujju, Y., Matsuo, A., & Raymond, R. (2024) · Physical Review Applied · doi:10.1103/physrevapplied.21.067001 · Cited by 104
Characterizing barren plateaus in quantum ansätze with the adjoint representation
Fontana, E., Herman, D., Chakrabarti, S., Kumar, N., Yalovetzky, R., Heredge, J., Sureshbabu, S. H., & Pistoia, M. (2024) · Nature Communications · doi:10.1038/s41467-024-49910-w · Cited by 96
“In this work, we present a general framework for diagnosing the BP phenomenoninLiealgebrasupportedansätze,whichincludespopular PQCs, such as HVA, QAOA, and various equivariant QNNs.”
Using models to improve optimizers for variational quantum algorithms
Sung, K. J., Yao, J., Harrigan, M. P., Rubin, N. C., Jiang, Z., Lin, L., Babbush, R., & McClean, J. R. (2020) · Quantum Science and Technology · doi:10.1088/2058-9565/abb6d9 · Cited by 92
“The ansatz is similar to the one used in reference [16] but corresponds to a different ordering of terms.”
BackgroundMethodInfluentialSimulating 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 83
Variational benchmarks for quantum many-body problems
Wu, D., Rossi, R., Vicentini, F., Astrakhantsev, N., Becca, F., Cao, X., Carrasquilla, J., Ferrari, F., Georges, A., Hibat-Allah, M., et al. (2024) · Science · doi:10.1126/science.adg9774 · Cited by 79
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 79
Resource Estimation for Quantum Variational Simulations of the Hubbard Model
Cai, Z. (2020) · Physical Review Applied · doi:10.1103/physrevapplied.14.014059 · Cited by 78
“com A huge number of circuit runs are needed to run VQE [3].”
BackgroundMethodResultInfluentialIzmaylov, A. F., Yen, T.-C., & Ryabinkin, I. G. (2019) · Chemical Science · doi:10.1039/c8sc05592k · Cited by 78
Efficient evaluation of quantum observables using entangled measurements
Hamamura, I., & Imamichi, T. (2020) · npj Quantum Information · doi:10.1038/s41534-020-0284-2 · Cited by 80
Efficient Step-Merged Quantum Imaginary Time Evolution Algorithm for Quantum Chemistry
Gomes, N., Zhang, F., Berthusen, N., Wang, C., Ho, K., Orth, P. P., & Yao, Y. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.0c00666 · Cited by 72
Improving Hamiltonian encodings with the Gray code
Matteo, O. D., McCoy, A. E., Gysbers, P., Miyagi, T., Woloshyn, R. M., & Navrátil, P. (2021) · Physical Review A · doi:10.1103/physreva.103.042405 · Cited by 77
“[32] D.”
BackgroundProspects of quantum computing for molecular sciences
Liu, H., Low, G. H., Steiger, D. S., Häner, T., Reiher, M., & Troyer, M. (2022) · Materials Theory · doi:10.1186/s41313-021-00039-z · Cited by 73
“Furthermore, the 1011 measurements per optimization step (Wecker et al. 2015) further increase the cost.”
BackgroundSelf-citationMinimizing Estimation Runtime on Noisy Quantum Computers
Wang, G., Koh, D. E., Johnson, P. D., & Cao, Y. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.010346 · Cited by 77
“However, in practice, the inefficiency of this method makes the VQE impractical for many problems of interest [2].”
BackgroundMethodNearly Optimal Measurement Scheduling for Partial Tomography of Quantum States
Bonet-Monroig, X., Babbush, R., & O’Brien, T. E. (2020) · Leiden Repository (Leiden University) · Cited by 74
Purification-based quantum error mitigation of pair-correlated electron simulations
O’Brien, T. E., Anselmetti, G.-L., Gkritsis, F., Elfving, V. E., Polla, S., Huggins, W. J., Oumarou, O., Kechedzhi, K., Abanin, D. A., Acharya, R., et al. (2023) · Nature Physics · doi:10.1038/s41567-023-02240-y · Cited by 100
Early fault-tolerant simulations of the Hubbard model
Campbell, E. T. (2022) · Quantum Science and Technology · doi:10.1088/2058-9565/ac3110 · Cited by 81
“For any hopping Hamiltonian, it is possible to exactly diagonalize e i sH h using a product of O ( L 4 ) Givens rotations [33–35], and while this is formally efficient, O ( L 4 ) is very large in practice.”
Mitigating algorithmic errors in a Hamiltonian simulation
Endo, S., Zhao, Q., Li, Y., Benjamin, S., & Yuan, X. (2019) · Physical Review A · doi:10.1103/physreva.99.012334 · Cited by 72
Synergistic pretraining of parametrized quantum circuits via tensor networks
Rudolph, M. S., Miller, J., Motlagh, D., Chen, J., Acharya, A., & Perdomo‐Ortiz, A. (2023) · Nature Communications · doi:10.1038/s41467-023-43908-6 · Cited by 118
“Given the practical nature of the problem, and the decades of classical computational techniques towards solving such high-value problems, gave rise to highly specific quantum circuit ansätze and parameter initialization [26, 27].”
Accuracy and Resource Estimations for Quantum Chemistry on a Near-Term Quantum Computer
Kühn, M., Zanker, S., Deglmann, P., Marthaler, M., & Weiß, H. (2019) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.9b00236 · Cited by 71
Cao, C., Hu, J., Zhang, W., Xu, X., Chen, D., Yu, F., Li, J., Hu, H., Lv, D., & Yung, M. (2022) · Physical Review A · doi:10.1103/physreva.105.062452 · Cited by 84
Experimental quantum computational chemistry with optimized unitary coupled cluster ansatz
Guo, S., Sun, J., Qian, H., Gong, M., Zhang, Y., Chen, F., Ye, Y., Wu, Y., Cao, S., Liu, K., et al. (2024) · Nature Physics · doi:10.1038/s41567-024-02530-z · Cited by 95
“There has been considerable progress in reducing the number of operators in the circuit ansatz, such as using classical pre-calculation to screen out the terms that are small enough [10] or only choosing the important excitation operators [11], for instance, paired double excitation operators in ansatz [12].”
BackgroundPrecise measurement of quantum observables with neural-network estimators
Torlai, G., Mazzola, G., Carleo, G., & Mezzacapo, A. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.022060 · Cited by 80
“4, where we also include an upper bound pMax often referenced in literature [6, 9].”
BackgroundMethodInfluentialOptimizing quantum heuristics with meta-learning
Wilson, M., Stromswold, R., Wudarski, F., Hadfield, S., Tubman, N. M., & Rieffel, E. G. (2021) · Quantum Machine Intelligence · doi:10.1007/s42484-020-00022-w · Cited by 79
“Nelder-Mead and L-BFGS-B are gradient-free and gradient-based approaches, respectively, which are standard local optimizers [28–31].”
BackgroundMachine learning phase transitions with a quantum processor
Uvarov, A. V., Kardashin, A. S., & Biamonte, J. D. (2020) · Physical Review A · doi:10.1103/physreva.102.012415 · Cited by 69
Toward scalable simulations of lattice gauge theories on quantum computers
Mathis, S. V., Mazzola, G., & Tavernelli, I. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.102.094501 · Cited by 65
“In the VQE approach the expectation value of the Hamiltonian needs to be computed as sum of all the Pauli operators expectation values, therefore this scaling readily determines the timecomplexity of the algorithm [83–85].”
MethodNear-term quantum algorithms for linear systems of equations with regression loss functions
Huang, H.-Y., Bharti, K., & Rebentrost, P. (2021) · New Journal of Physics · doi:10.1088/1367-2630/ac325f · Cited by 193
Hamiltonian variational ansatz without barren plateaus
Park, C.-Y., & Killoran, N. (2024) · Quantum · doi:10.22331/q-2024-02-01-1239 · Cited by 90
“The HVA [7, 9] is a natural ansatz for solving quantum many-body Hamiltonians.”
BackgroundInfluentialAdaptive pruning-based optimization of parameterized quantum circuits
Sim, S., Romero, J., Gonthier, J. F., & Kunitsa, A. A. (2021) · Quantum Science and Technology · doi:10.1088/2058-9565/abe107 · Cited by 75
Experimental Evaluation of Quantum Machine Learning Algorithms
Simoes, R. D. M., Huber, P., Meier, N., Smailov, N., Füchslin, R. M., & Stockinger, K. (2023) · IEEE Access · doi:10.1109/access.2023.3236409 · Cited by 74
“These steps comprise a so-called variational quantum classifier [31] leveraging parameterized quantum circuits.”
BackgroundSoftware engineering for quantum programming: How far are we?
Stefano, M. D., Pecorelli, F., Nucci, D. D., Palomba, F., & Lucia, A. D. (2022) · Journal of Systems and Software · doi:10.1016/j.jss.2022.111326 · Cited by 81
Digitized Counterdiabatic Quantum Algorithm for Protein Folding
Chandarana, P., Hegade, N. N., Montalban, I., Solano, E., & Chen, X. (2023) · Physical Review Applied · doi:10.1103/physrevapplied.20.014024 · Cited by 68
“Some examples of problem-inspired ansatz are the Quantum Approximate Optimization Algorithm (QAOA) [2], the unitary coupled-cluster ansatz [3] or the Hamiltonian variational ansatz [4, 5].”
MethodExperimental Quantum State Measurement with Classical Shadows
Zhang, T., Sun, J., Fang, X.-X., Zhang, X.-M., Yuan, X., & Lu, H. (2021) · Physical Review Letters · doi:10.1103/physrevlett.127.200501 · Cited by 73
Quantum singular value decomposer
Bravo-Prieto, C., García-Martín, D., & Latorre, J. I. (2020) · Physical Review A · doi:10.1103/physreva.101.062310 · Cited by 64
Rossmannek, M., Barkoutsos, P. K., Ollitrault, P. J., & Tavernelli, I. (2021) · The Journal of Chemical Physics · doi:10.1063/5.0029536 · Cited by 73
Jnane, H., Undseth, B., Cai, Z., Benjamin, S. C., & Koczor, B. (2022) · Physical Review Applied · doi:10.1103/physrevapplied.18.044064 · Cited by 71
Simulating Many-Body Systems with a Projective Quantum Eigensolver
Stair, N. H., & Evangelista, F. A. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.030301 · Cited by 59
“These issues are compounded by the sizable number of total measurements needed for operator averaging [57].”
Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz
Mele, A. A., Mbeng, G. B., Santoro, G. E., Collura, M., & Torta, P. (2022) · Physical Review A · doi:10.1103/physreva.106.l060401 · Cited by 59
Towards near-term quantum simulation of materials
Clinton, L., Cubitt, T. S., Flynn, B., Gambetta, F. M., Klassen, J., Montanaro, A., Piddock, S., Santos, R. A., & Sheridan, E. (2024) · Nature Communications · doi:10.1038/s41467-023-43479-6 · Cited by 79
“The Hamiltonian variational ansatz is a sequence of circuit layers l of the form Q k e it lk h k , with t lk variational parameters 19 .”
Solving quantum statistical mechanics with variational autoregressive networks and quantum circuits
Liu, J.-G., Mao, L., Zhang, P., & Wang, L. (2021) · Machine Learning: Science and Technology · doi:10.1088/2632-2153/aba19d · Cited by 57
Lang, R. A., Ryabinkin, I. G., & Izmaylov, A. F. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.0c00170 · Cited by 62
Variational Quantum-Neural Hybrid Eigensolver
Zhang, S., Wan, Z., Lee, C., Hsieh, C., Zhang, S., & Yao, H. (2022) · Physical Review Letters · doi:10.1103/physrevlett.128.120502 · Cited by 65
“For example, the probably most famous VQE ansatz, unitary coupled cluster (UCC) ansatz [10, 11, 30, 31] is inspired by the coupled cluster method, a post Hartree-Fock approach.”
MethodQuantum inverse iteration algorithm for programmable quantum simulators
Kyriienko, O. (2020) · npj Quantum Information · doi:10.1038/s41534-019-0239-7 · Cited by 66
“It can use a chemically inspired ansatz [23], Hamiltonian variational ansatz [24], or rely on the variational imaginary time evolution [25].”
MethodSugisaki, K., Yamamoto, S., Nakazawa, S., Toyota, K., Sato, K., Shiomi, D., & Takui, T. (2016) · The Journal of Physical Chemistry A · doi:10.1021/acs.jpca.6b04932 · Cited by 55
Variational quantum eigensolver for the Heisenberg antiferromagnet on the kagome lattice
Kattemölle, J., & van Wezel, J. (2022) · Physical review. B./Physical review. B · doi:10.1103/physrevb.106.214429 · Cited by 65
“KVQEG/K uses the Hamiltonian Variational Ansatz (HVA) [20].”
InfluentialOverlapped grouping measurement: A unified framework for measuring quantum states
Wu, B., Sun, J., Huang, Q. J., & Yuan, X. (2023) · Quantum · doi:10.22331/q-2023-01-13-896 · Cited by 71
“Without introducing additional entangling circuits, three types of advanced measurement schemes have been proposed to reduce the measurement cost by exploiting different features of the to-be-measured observables [11, 15, 16, 27, 31, 33, 38, 62, 63, 65, 67].”
BackgroundEfficient quantum computation of molecular forces and other energy gradients
O’Brien, T. E., Streif, M., Rubin, N. C., Santagati, R., Su, Y., Huggins, W. J., Goings, J. J., Moll, N., Kyoseva, E. S., Degroote, M., et al. (2022) · Physical Review Research · doi:10.1103/physrevresearch.4.043210 · Cited by 71
Analytic Theory for the Dynamics of Wide Quantum Neural Networks
Liu, J., Najafi, K., Sharma, K., Tacchino, F., Jiang, L., & Mezzacapo, A. (2023) · Physical Review Letters · doi:10.1103/physrevlett.130.150601 · Cited by 68
“[49] for these problems and numerically observed that the computational phase transition (or overparam-eterization) takes place when the number of parameters is much less than D 2 .”
BackgroundQuantum Computing: An Overview Across the System StackPreprint
Resch, S., & Karpuzcu, U. R. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1905.07240 · Cited by 94
“ons in quantum chemistry [31, 91, 103–105]. Additional theory related to VQE can be found in [106]. There is a concern about the number of samples required by VQE, which can lead to excessive runtime [92]. The authors of [33] proposed a new algorithm, α-VQE, which attempts to compromise between the larger number of sa…”
BackgroundMethodOperator Sampling for Shot-frugal Optimization in Variational Algorithms
Arrasmith, A., Cincio, L., Somma, R. D., & Coles, P. J. (2020) · arXiv · doi:10.48550/arxiv.2004.06252 · Cited by 53
“To improve the shot frugality of reaching a given precision, it has been proposed [40, 54] that the number of shots si allocated to the measurement of each operator hi should be proportional to the magnitude of the coefficients ci.”
BackgroundSimulating Periodic Systems on a Quantum Computer Using Molecular Orbitals
Liu, J., Wan, L., Li, Z., & Yang, J. (2020) · Journal of Chemical Theory and Computation · doi:10.1021/acs.jctc.0c00881 · Cited by 54
Variational Quantum Simulation for Quantum Chemistry
Li, Y., Hu, J., Zhang, X., Song, Z., & Yung, M. (2019) · Advanced Theory and Simulations · doi:10.1002/adts.201800182 · Cited by 56
Kreshchuk, M., Jia, S., Kirby, W. M., Goldstein, G. R., Vary, J. P., & Love, P. J. (2021) · Physical Review A · doi:10.1103/physreva.103.062601 · Cited by 62
“[67] Dave Wecker, Matthew B Hastings, and Matthias Troyer.”
BackgroundAdaptive variational quantum eigensolvers for highly excited states
Zhang, F., Gomes, N., Yao, Y., Orth, P. P., & Iadecola, T. (2021) · Physical review. B./Physical review. B · doi:10.1103/physrevb.104.075159 · Cited by 58
An Image Classification Algorithm Based on Hybrid Quantum Classical Convolutional Neural Network
Li, W., Chu, P.-C., Liu, G.-Z., Tian, Y., Qiu, T., & Wang, S. (2022) · Quantum Engineering · doi:10.1155/2022/5701479 · Cited by 54
Quantum computing with and for many-body physics
Ayral, T., Besserve, P., Denis Lacroix, A., & Guzman, E. A. R. (2023) · The European Physical Journal A · doi:10.1140/epja/s10050-023-01141-1 · Cited by 71
“2) and the Quantum Approximate Optimization Algorithm circuit (QAOA [200]), which is ubiquitous in quantum combinatorial optimization, the Hamiltonian Variational Ansatz (HVA [201]) state reads”
BackgroundReiner, J.-M., Wilhelm, F. K., Schön, G., & Marthaler, M. (2019) · Quantum Science and Technology · doi:10.1088/2058-9565/ab1e85 · Cited by 55
Physical-Layer Supervised Learning Assisted by an Entangled Sensor Network
Zhuang, Q., & Zhang, Z. (2019) · Physical Review X · doi:10.1103/physrevx.9.041023 · Cited by 56
“In this regard, quantum variational schemes, in conjunction with classical optimization, are widely applicable to tasks including quantum state preparation [6], variational eigensolvers [7, 8], state diagonalization [9], and machine learning [10–20].”
MethodQuantum computation of molecular response properties
Cai, X., Fang, W., Fan, H., & Li, Z. (2020) · Physical Review Research · doi:10.1103/physrevresearch.2.033324 · Cited by 51
“Since the number of measurements is proportional to the number of terms [29,52] in ˆ z ˆ z and ˆ A † ( ω ) ˆ A ( ω ), the cost scales polynomially in N .”
BackgroundChallenges of variational quantum optimization with measurement shot noise
Scriva, G., Astrakhantsev, N., Pilati, S., & Mazzola, G. (2024) · Physical Review A · doi:10.1103/physreva.109.032408 · Cited by 64
“Note that the issue of shot noise is well-known in VQE for genuine many-body quantum Hamiltonians, for example, in chemistry [57].”
BackgroundqTorch: The quantum tensor contraction handler
Fried, E., Sawaya, N. P. D., Cao, Y., Kivlichan, I., Romero, J., & Aspuru‐Guzik, A. (2018) · PLoS ONE · doi:10.1371/journal.pone.0208510 · Cited by 48
Quantum algorithms for quantum dynamics: A performance study on the spin-boson model
Miessen, A., Ollitrault, P. J., & Tavernelli, I. (2021) · Physical Review Research · doi:10.1103/physrevresearch.3.043212 · Cited by 51
“Instead of the PT, here we employ a Variational Hamiltonian Ansatz (VHA) [55].”
MethodQuantum computation: Algorithms and Applications
Cho, C., Chen, C.-Y., Chen, K.-C., Huang, T., Hsu, M.-C., Cao, N., Zeng, B., Tan, S.-G., & Chang, C. (2021) · Chinese Journal of Physics · doi:10.1016/j.cjph.2021.05.001 · Cited by 54
Error Propagation in NISQ Devices for Solving Classical Optimization Problems
González-García, G., Trivedi, R., & Cirac, J. I. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.040326 · Cited by 58
Building spatial symmetries into parameterized quantum circuits for faster training
Sauvage, F., Larocca, M., Coles, P. J., & Cerezo, M. (2023) · Quantum Science and Technology · doi:10.1088/2058-9565/ad152e · Cited by 60
“This includes the Hamiltonian Variational Ansatz (HVA) [10–13] and the quantum alternating operator ansatz [2, 14].”
BackgroundMethodAncilla-free implementation of generalized measurements for qubits embedded in a qudit space
Fischer, L. E., Miller, D., Tacchino, F., Barkoutsos, P. K., Egger, D. J., & Tavernelli, I. (2022) · Physical Review Research · doi:10.1103/physrevresearch.4.033027 · Cited by 52
“, an estimate for the Fe2S2 complex predicts up to O(1013) required measurements per energy evaluation [25].”
BackgroundSubspace methods for electronic structure simulations on quantum computers
Motta, M., Kirby, W., Liepuoniute, I., Sung, K. J., Cohn, J. F., Mezzacapo, A., Klymko, K., Nguyen, N., Yoshioka, N., & Rice, J. E. (2024) · Electronic Structure · doi:10.1088/2516-1075/ad3592 · Cited by 61
Asleep at the wheel? Responsible Innovation in quantum computing
Inglesant, P., Holter, C. T., Jirotka, M. D., & Williams, R. (2021) · Technology Analysis and Strategic Management · doi:10.1080/09537325.2021.1988557 · Cited by 57
Berthusen, N., Trevisan, T. V., Iadecola, T., & Orth, P. P. (2022) · Physical Review Research · doi:10.1103/physrevresearch.4.023097 · Cited by 52
“Here, we choose a fixed ansatz that is inspired by the Hamiltonian variational ansatz (HVA) [34, 35] and that takes a layered form, |ψ(ϑ)〉 = U(ϑ) |ψ0〉 = ∏̀ l=1 N∏ i=1 e−iϑl,iAi |ψ0〉 .”
MethodNonunitary Operations for Ground-State Calculations in Near-Term Quantum Computers
Mazzola, G., Ollitrault, P. J., Barkoutsos, P. K., & Tavernelli, I. (2019) · Physical Review Letters · doi:10.1103/physrevlett.123.130501 · Cited by 43
“This methods relies on measurements reweighting and allows for an exact implementation of the Jas-trow correlation operator, while it does not increase the number of Pauli operators to be measured, that already constitutes a drawback of standard VQE[12].”
MethodHybrid classical-quantum linear solver using Noisy Intermediate-Scale Quantum machines
Chen, C.-C., Shiau, S.-Y., Wu, M.-F., & Wu, Y.-R. (2019) · Scientific Reports · doi:10.1038/s41598-019-52275-6 · Cited by 42
A Variational Algorithm for Quantum Neural Networks
Macaluso, A., Clissa, L., Lodi, S., & Sartori, C. (2020) · Lecture notes in computer science · doi:10.1007/978-3-030-50433-5_45 · Cited by 45
Quantifying the effect of gate errors on variational quantum eigensolvers for quantum chemistry
Dalton, K., Long, C. K., Yordanov, Y. S., Smith, C. G., Barnes, C. H. W., Mertig, N., & Arvidsson-Shukur, D. R. M. (2024) · npj Quantum Information · doi:10.1038/s41534-024-00808-x · Cited by 43
“Ultimately, it remains an open question whether the unfavorable scaling of error mitigation prevents its use in realistic quantum-chemistry applications.”
Background
Metadata
- Authors
- Wecker, D., Hastings, M. B., & Troyer, M.
- Affiliations
- Microsoft (United States) (US)
- ETH Zurich (CH)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2015-10-02
- Volume
- 92
- Issue
- 4
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2254754114
- Cited by
- 716
- References
- 15
- Citations by year
- 2026: 48 · 2025: 108 · 2024: 101 · 2023: 97 · 2022: 100 · 2021: 110 · 2020: 69 · 2019: 40 · 2018: 21 · 2017: 11 · 2016: 11
- FWCI
- 14.03
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Microsoft Research
- National Science Foundation (PHY-1066293)
- 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 state preparation, adiabatic state preparation, variational ansatz, Hubbard model, unitary coupled cluster, quantum chemistry, circuit depth, NISQ devices
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