Variational methods are highly valuable computational tools for solving high-dimensional quantum systems. In this paper, we explore the effectiveness of three variational methods: density matrix renormalization group (DMRG), Boltzmann machine learning, and variational quantum eigensolver (VQE). We apply these methods to solve two different quantum systems: the Fermi-Hubbard model in condensed matter physics and the Schwinger model in high energy physics. To facilitate the computations on quantum computers, we map each model to a spin 1/2 system using the Jordan-Wigner transformation. This transformation allows us to take advantage of the capabilities of quantum computing. We calculate the ground state of both quantum systems and compare the results obtained using three variational methods. Our aim is to demonstrate the power and effectiveness of these variational approaches in tackling complex quantum systems.
References
[1]
Feynman, R.P. (1954) Atomic Theory of the Two-Fluid Model of Liquid Helium. Physical Review, 94, 262-277. https://doi.org/10.1103/physrev.94.262
[2]
Feynman, R.P. and Cohen, M. (1956) Energy Spectrum of the Excitations in Liquid Helium. Physical Review, 102, 1189-1204. https://doi.org/10.1103/physrev.102.1189
[3]
Tsui, D.C., Stormer, H.L. and Gossard, A.C. (1982) Two-Dimensional Magnetotransport in the Extreme Quantum Limit. Physical Review Letters, 48, 1559-1562. https://doi.org/10.1103/physrevlett.48.1559
[4]
Laughlin, R.B. (1983) Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations. Physical Review Letters, 50, 1395-1398. https://doi.org/10.1103/physrevlett.50.1395
[5]
Haldane, F.D.M. (1983) Continuum Dynamics of the 1-D Heisenberg Antiferromagnet: Identification with the O(3) Nonlinear Sigma Model. Physics Letters A, 93, 464-468. https://doi.org/10.1016/0375-9601(83)90631-x
[6]
Haldane, F.D.M. (1983) Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State. Physical Review Letters, 50, 1153-1156. https://doi.org/10.1103/physrevlett.50.1153
[7]
Anderson, P.W. (1973) Resonating Valence Bonds: A New Kind of Insulator? Materials Research Bulletin, 8, 153-160. https://doi.org/10.1016/0025-5408(73)90167-0
[8]
Bednorz, J.G. and Muller, K.A. (1986) Possible Hight C Superconductivity in the Ba-La-Cu-O System. Zeitschrift für Physik B Condensed Matter, 64, 189-193. https://doi.org/10.1007/bf01303701
[9]
Li, D. (2016) Fermion Bag Approach for the Massive Thirring Model at Finite Density. Physical Review D, 94, Article 114501. https://doi.org/10.1103/physrevd.94.114501
[10]
Hastings, M.B. (2007) An Area Law for One-Dimensional Quantum Systems. Journal of Statistical Mechanics: Theory and Experiment, 2007, P08024. https://doi.org/10.1088/1742-5468/2007/08/p08024
[11]
White, S.R. (1992) Density Matrix Formulation for Quantum Renormalization Groups. Physical Review Letters, 69, 2863-2866. https://doi.org/10.1103/physrevlett.69.2863
[12]
Schollwöck, U. (2011) The Density-Matrix Renormalization Group in the Age of Matrix Product States. Annals of Physics, 326, 96-192. https://doi.org/10.1016/j.aop.2010.09.012
[13]
White, S.R. and Feiguin, A.E. (2004) Real-Time Evolution Using the Density Matrix Renormalization Group. Physical Review Letters, 93, Article 076401. https://doi.org/10.1103/physrevlett.93.076401
[14]
Zaletel, M.P., Mong, R.S.K., Karrasch, C., Moore, J.E. and Pollmann, F. (2015) Time-Evolving a Matrix Product State with Long-Ranged Interactions. Physical Review B, 91, Article 165112. https://doi.org/10.1103/physrevb.91.165112
[15]
Haegeman, J., Cirac, J.I., Osborne, T.J., Pižorn, I., Verschelde, H. and Verstraete, F. (2011) Time-Dependent Variational Principle for Quantum Lattices. Physical Review Letters, 107, Article 070601. https://doi.org/10.1103/physrevlett.107.070601
[16]
Haegeman, J., Lubich, C., Oseledets, I., Vandereycken, B. and Verstraete, F. (2016) Unifying Time Evolution and Optimization with Matrix Product States. Physical Review B, 94, Article 165116. https://doi.org/10.1103/physrevb.94.165116
Khemani, V., Pollmann, F. and Sondhi, S.L. (2016) Obtaining Highly Excited Eigenstates of Many-Body Localized Hamiltonians by the Density Matrix Renormalization Group Approach. Physical Review Letters, 116, Article 247204. https://doi.org/10.1103/physrevlett.116.247204
[19]
Yu, X., Pekker, D. and Clark, B.K. (2017) Finding Matrix Product State Representations of Highly Excited Eigenstates of Many-Body Localized Hamiltonians. Physical Review Letters, 118, Article 017201. https://doi.org/10.1103/physrevlett.118.017201
[20]
Liang, S. and Pang, H. (1994) Approximate Diagonalization Using the Density Matrix Renormalization-Group Method: A Two-Dimensional-Systems Perspective. Physical Review B, 49, 9214-9217. https://doi.org/10.1103/physrevb.49.9214
[21]
McCulloch, I.P. and Gulácsi, M. (2002) The Non-Abelian Density Matrix Renormalization Group Algorithm. Europhysics Letters (EPL), 57, 852-858. https://doi.org/10.1209/epl/i2002-00393-0
[22]
Singh, S., Pfeifer, R.N.C. and Vidal, G. (2010) Tensor Network Decompositions in the Presence of a Global Symmetry. Physical Review A, 82, Article 050301. https://doi.org/10.1103/physreva.82.050301
[23]
Singh, S., Pfeifer, R.N.C. and Vidal, G. (2011) Tensor Network States and Algorithms in the Presence of a Global U(1) Symmetry. Physical Review B, 83, Article 115125. https://doi.org/10.1103/physrevb.83.115125
[24]
Singh, S. and Vidal, G. (2012) Tensor Network States and Algorithms in the Presence of a Global SU(2) Symmetry. Physical Review B, 86, Article 195114. https://doi.org/10.1103/physrevb.86.195114
[25]
Weichselbaum, A. (2012) Non-abelian Symmetries in Tensor Networks: A Quantum Symmetry Space Approach. Annals of Physics, 327, 2972-3047. https://doi.org/10.1016/j.aop.2012.07.009
[26]
White, S.R. (2005) Density Matrix Renormalization Group Algorithms with a Single Center Site. Physical Review B, 72, Article 180403. https://doi.org/10.1103/physrevb.72.180403
[27]
Hubig, C., McCulloch, I.P., Schollwöck, U. and Wolf, F.A. (2015) Strictly Single-Site DMRG Algorithm with Subspace Expansion. Physical Review B, 91, Article 155115. https://doi.org/10.1103/physrevb.91.155115
[28]
Motruk, J., Zaletel, M.P., Mong, R.S.K. and Pollmann, F. (2016) Density Matrix Renormalization Group on a Cylinder in Mixed Real and Momentum Space. Physical Review B, 93, Article 155139. https://doi.org/10.1103/physrevb.93.155139
[29]
Ehlers, G., White, S.R. and Noack, R.M. (2017) Hybrid-Space Density Matrix Renormalization Group Study of the Doped Two-Dimensional Hubbard Model. Physical Review B, 95, Article 125125. https://doi.org/10.1103/physrevb.95.125125
[30]
Stoudenmire, E.M. and White, S.R. (2013) Real-Space Parallel Density Matrix Renormalization Group. Physical Review B, 87, Article 155137. https://doi.org/10.1103/physrevb.87.155137
[31]
Verstraete, F. and Cirac, J.I. (2010) Continuous Matrix Product States for Quantum Fields. Physical Review Letters, 104, Article 190405. https://doi.org/10.1103/physrevlett.104.190405
[32]
Haegeman, J., Cirac, J.I., Osborne, T.J. and Verstraete, F. (2013) Calculus of Continuous Matrix Product States. Physical Review B, 88, Article 085118. https://doi.org/10.1103/physrevb.88.085118
[33]
Rincón, J., Ganahl, M. and Vidal, G. (2015) Lieb-Liniger Model with Exponentially Decaying Interactions: A Continuous Matrix Product State Study. Physical Review B, 92, Article 115107. https://doi.org/10.1103/physrevb.92.115107
[34]
Ganahl, M., Rincón, J. and Vidal, G. (2017) Continuous Matrix Product States for Quantum Fields: An Energy Minimization Algorithm. Physical Review Letters, 118, Article 220402. https://doi.org/10.1103/physrevlett.118.220402
[35]
Chung, S.S., Bauman, S., Sun, K. and Bolech, C.J. (2016) On the New Continuous Matrix Product Ansatz. Journal of Physics: Conference Series, 702, Article 012004. https://doi.org/10.1088/1742-6596/702/1/012004
[36]
Draxler, D., Haegeman, J., Verstraete, F. and Rizzi, M. (2017) Continuous Matrix Product States with Periodic Boundary Conditions and an Application to Atomtronics. Physical Review B, 95, Article 045145. https://doi.org/10.1103/physrevb.95.045145
[37]
Tilloy, A. (2021) Relativistic Continuous Matrix Product States for Quantum Fields without Cutoff. Physical Review D, 104, Article 096007. https://doi.org/10.1103/physrevd.104.096007
[38]
Carleo, G., Cirac, I., Cranmer, K., Daudet, L., Schuld, M., Tishby, N., et al. (2019) Machine Learning and the Physical Sciences. Reviews of Modern Physics, 91, Article 045002. https://doi.org/10.1103/revmodphys.91.045002
[39]
Carleo, G. and Troyer, M. (2017) Solving the Quantum Many-Body Problem with Artificial Neural Networks. Science, 355, 602-606. https://doi.org/10.1126/science.aag2302
[40]
Gao, X. and Duan, L. (2017) Efficient Representation of Quantum Many-Body States with Deep Neural Networks. Nature Communications, 8, Article No. 662. https://doi.org/10.1038/s41467-017-00705-2
[41]
Carleo, G., Nomura, Y. and Imada, M. (2018) Constructing Exact Representations of Quantum Many-Body Systems with Deep Neural Networks. Nature Communications, 9, Article No. 5322. https://doi.org/10.1038/s41467-018-07520-3
[42]
Freitas, N., Morigi, G. and Dunjko, V. (2018) Neural Network Operations and Susuki-Trotter Evolution of Neural Network States. International Journal of Quantum Information, 16, Article 1840008. https://doi.org/10.1142/s0219749918400087
[43]
Glasser, I., Pancotti, N., August, M., Rodriguez, I.D. and Cirac, J.I. (2018) Neural-Network Quantum States, String-Bond States, and Chiral Topological States. Physical Review X, 8, Article 011006. https://doi.org/10.1103/physrevx.8.011006
[44]
Nomura, Y., Darmawan, A.S., Yamaji, Y. and Imada, M. (2017) Restricted Boltzmann Machine Learning for Solving Strongly Correlated Quantum Systems. Physical Review B, 96, Article 205152. https://doi.org/10.1103/physrevb.96.205152
[45]
Vieijra, T., Casert, C., Nys, J., De Neve, W., Haegeman, J., Ryckebusch, J., et al. (2020) Restricted Boltzmann Machines for Quantum States with Non-Abelian or Anyonic Symmetries. Physical Review Letters, 124, Article 097201. https://doi.org/10.1103/physrevlett.124.097201
[46]
Golubeva, A. and Melko, R.G. (2022) Pruning a Restricted Boltzmann Machine for Quantum State Reconstruction. Physical Review B, 105, Article 125124. https://doi.org/10.1103/physrevb.105.125124
[47]
Pilati, S. and Pieri, P. (2020) Simulating Disordered Quantum Ising Chains via Dense and Sparse Restricted Boltzmann Machines. Physical Review E, 101, Article 063308. https://doi.org/10.1103/physreve.101.063308
[48]
Lu, S., Gao, X. and Duan, L.-. (2019) Efficient Representation of Topologically Ordered States with Restricted Boltzmann Machines. Physical Review B, 99, Article 155136. https://doi.org/10.1103/physrevb.99.155136
[49]
Nomura, Y., Yoshioka, N. and Nori, F. (2021) Purifying Deep Boltzmann Machines for Thermal Quantum States. Physical Review Letters, 127, Article 060601. https://doi.org/10.1103/physrevlett.127.060601
[50]
Vicentini, F., Biella, A., Regnault, N. and Ciuti, C. (2019) Variational Neural-Network Ansatz for Steady States in Open Quantum Systems. Physical Review Letters, 122, Article 250503. https://doi.org/10.1103/physrevlett.122.250503
[51]
Yoshioka, N. and Hamazaki, R. (2019) Constructing Neural Stationary States for Open Quantum Many-Body Systems. Physical Review B, 99, Article 214306. https://doi.org/10.1103/physrevb.99.214306
[52]
Nagy, A. and Savona, V. (2019) Variational Quantum Monte Carlo Method with a Neural-Network Ansatz for Open Quantum Systems. Physical Review Letters, 122, Article 250501. https://doi.org/10.1103/physrevlett.122.250501
[53]
Hartmann, M.J. and Carleo, G. (2019) Neural-Network Approach to Dissipative Quantum Many-Body Dynamics. Physical Review Letters, 122, Article 250502. https://doi.org/10.1103/physrevlett.122.250502
[54]
Nomura, Y. (2021) Helping Restricted Boltzmann Machines with Quantum-State Representation by Restoring Symmetry. Journal of Physics: Condensed Matter, 33, Article 174003. https://doi.org/10.1088/1361-648x/abe268
[55]
Park, C. and Kastoryano, M.J. (2022) Expressive Power of Complex-Valued Restricted Boltzmann Machines for Solving Nonstoquastic Hamiltonians. Physical Review B, 106, Article 134437. https://doi.org/10.1103/physrevb.106.134437
[56]
Park, C. and Kastoryano, M.J. (2020) Geometry of Learning Neural Quantum States. Physical Review Research, 2, Article 023232. https://doi.org/10.1103/physrevresearch.2.023232
[57]
Choo, K., Carleo, G., Regnault, N. and Neupert, T. (2018) Symmetries and Many-Body Excitations with Neural-Network Quantum States. Physical Review Letters, 121, Article 167204. https://doi.org/10.1103/physrevlett.121.167204
[58]
Torlai, G. and Melko, R.G. (2018) Latent Space Purification via Neural Density Operators. Physical Review Letters, 120, Article 240503. https://doi.org/10.1103/physrevlett.120.240503
[59]
Zen, R., My, L., Tan, R., Hébert, F., Gattobigio, M., Miniatura, C., et al. (2020) Transfer Learning for Scalability of Neural-Network Quantum States. Physical Review E, 101, Article 053301. https://doi.org/10.1103/physreve.101.053301
[60]
Cai, Z. and Liu, J. (2018) Approximating Quantum Many-Body Wave Functions Using Artificial Neural Networks. Physical Review B, 97, Article 035116. https://doi.org/10.1103/physrevb.97.035116
[61]
Luo, D. and Clark, B.K. (2019) Backflow Transformations via Neural Networks for Quantum Many-Body Wave Functions. Physical Review Letters, 122, Article 226401. https://doi.org/10.1103/physrevlett.122.226401
[62]
Sharir, O., Levine, Y., Wies, N., Carleo, G. and Shashua, A. (2020) Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems. Physical Review Letters, 124, Article 020503. https://doi.org/10.1103/physrevlett.124.020503
[63]
Kessler, J., Calcavecchia, F. and Kühne, T.D. (2021) Artificial Neural Networks as Trial Wave Functions for Quantum Monte Carlo. Advanced Theory and Simulations, 4, Article 200069. https://doi.org/10.1002/adts.202000269
[64]
Hibat-Allah, M., Ganahl, M., Hayward, L.E., Melko, R.G. and Carrasquilla, J. (2020) Recurrent Neural Network Wave Functions. Physical Review Research, 2, Article 023358. https://doi.org/10.1103/physrevresearch.2.023358
[65]
Roth, C. (2020) Iterative Retraining of Quantum Spin Models Using Recurrent Neural Networks.
[66]
Chen, Z., Newhouse, L., Chen, E., Luo, D. and Soljačić, M. (2023) ANTN: Bridging Autoregressive Neural Networks and Tensor Networks for Quantum Many-Body Simulation.
[67]
Schmitt, M. and Heyl, M. (2020) Quantum Many-Body Dynamics in Two Dimensions with Artificial Neural Networks. Physical Review Letters, 125, Article 100503. https://doi.org/10.1103/physrevlett.125.100503
[68]
Gutiérrez, I.L. and Mendl, C.B. (2022) Real Time Evolution with Neural-Network Quantum States. Quantum, 6, 627. https://doi.org/10.22331/q-2022-01-20-627
[69]
Yang, L., Leng, Z., Yu, G., Patel, A., Hu, W. and Pu, H. (2020) Deep Learning-Enhanced Variational Monte Carlo Method for Quantum Many-Body Physics. Physical Review Research, 2, Article 012039. https://doi.org/10.1103/physrevresearch.2.012039
[70]
Irikura, N. and Saito, H. (2020) Neural-Network Quantum States at Finite Temperature. Physical Review Research, 2, Article 013284. https://doi.org/10.1103/physrevresearch.2.013284
[71]
Vieijra, T. and Nys, J. (2021) Many-body Quantum States with Exact Conservation of Non-Abelian and Lattice Symmetries through Variational Monte Carlo. Physical Review B, 104, Article 045123. https://doi.org/10.1103/physrevb.104.045123
[72]
Liang, X., Liu, W., Lin, P., Guo, G., Zhang, Y. and He, L. (2018) Solving Frustrated Quantum Many-Particle Models with Convolutional Neural Networks. Physical Review B, 98, Article 104426. https://doi.org/10.1103/physrevb.98.104426
[73]
Liu, C. and Wang, D. (2021) Random Sampling Neural Network for Quantum Many-Body Problems. Physical Review B, 103, Article 205107. https://doi.org/10.1103/physrevb.103.205107
[74]
Saito, H. and Kato, M. (2018) Machine Learning Technique to Find Quantum Many-Body Ground States of Bosons on a Lattice. Journal of the Physical Society of Japan, 87, Article 014001. https://doi.org/10.7566/jpsj.87.014001
[75]
Roth, C. and MacDonald, A.H. (2021) Group Convolutional Neural Networks Improve Quantum State Accuracy.
[76]
Fu, C., Zhang, X., Zhang, H., Ling, H., Xu, S. and Ji, S. (2021) Lattice Convolutional Networks for Learning Ground States of Quantum Many-Body Systems.
[77]
Liang, X., Li, M., Xiao, Q., An, H., et al. (2022) 21296 Exponentially Complex Quantum Many-Body Simulation via Scalable Deep Learning Method.
[78]
Choo, K., Neupert, T. and Carleo, G. (2019) Two-Dimensional Frustrated J1-J2 Model Studied with Neural Network Quantum States. Physical Review B, 100, Article 125124. https://doi.org/10.1103/physrevb.100.125124
[79]
Mikolov, T., Kombrink, S., Burget, L., Cernocky, J. and Khudanpur, S. (2011) Extensions of Recurrent Neural Network Language Model. 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, 22-27 May 2011, 5528-5531. https://doi.org/10.1109/icassp.2011.5947611
[80]
Graves, A., Mohamed, A. and Hinton, G. (2013) Speech Recognition with Deep Recurrent Neural Networks. 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, Vancouver, 26-31 May 2013, 6645-6649. https://doi.org/10.1109/icassp.2013.6638947
[81]
Gregor, K., Danihelka, I., Graves, A., Rezende, D.J. and Wierstra, D. (2015) Draw: A Recurrent Neural Network for Image Generation.
[82]
Oord, A.V.D., Kalchbrenner, N. and Kavukcuoglu, K. (2016) Pixel Recurrent Neural Networks.
[83]
Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., et al. (2017) Attention Is All You Need. 2017 31st Conference on Neural Information Processing Systems, Long Beach, 4-9 December 2017, 1-11.
[84]
Devlin, J., Chang, M.W., Lee, K. and Toutanova, K. (2018) BERT: Pre-Training of Deep Bidirectional Transformers for Language Understanding.
[85]
Brown, T.B., Mann, B., Ryder, N., et al. (2020) Language Models Are Few-Shot Learners. Advances in Neural Information Processing Systems, 33, 1877-1901.
[86]
Dosovitskiy, A., Beyer, L., Kolesnikov, A., et al. (2020) An Image Is Worth 16x16 Words: Transformers for Image Recognition at Scale.
[87]
Jumper, J., Evans, R., Pritzel, A., et al. (2021) Highly Accurate Protein Structure Prediction with AlphaFold. Nature, 596, Article 583589.
[88]
Luo, D., Chen, Z., Hu, K., Zhao, Z., et al. (2021) Gauge Invariant and Anyonic Symmetric Transformer and RNN Quantum States for Quantum Lattice Models.
[89]
Luo, D., Chen, Z., Carrasquilla, J. and Clark, B.K. (2022) Autoregressive Neural Network for Simulating Open Quantum Systems via a Probabilistic Formulation. Physical Review Letters, 128, Article 090501. https://doi.org/10.1103/physrevlett.128.090501
[90]
Cha, P., Ginsparg, P., Wu, F., Carrasquilla, J., McMahon, P.L. and Kim, E. (2021) Attention-Based Quantum Tomography. Machine Learning: Science and Technology, 3, 01LT01. https://doi.org/10.1088/2632-2153/ac362b
[91]
Carrasquilla, J., Luo, D., Pérez, F., Milsted, A., Clark, B.K., Volkovs, M., et al. (2021) Probabilistic Simulation of Quantum Circuits Using a Deep-Learning Architecture. Physical Review A, 104, Article 032610. https://doi.org/10.1103/physreva.104.032610
[92]
Abrams, D.S. and Lloyd, S. (1999) Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors. Physical Review Letters, 83, 5162-5165. https://doi.org/10.1103/physrevlett.83.5162
[93]
Aspuru-Guzik, A., Dutoi, A.D., Love, P.J. and Head-Gordon, M. (2005) Simulated Quantum Computation of Molecular Energies. Science, 309, 1704-1707. https://doi.org/10.1126/science.1113479
[94]
Kandala, A., Temme, K., Córcoles, A.D., Mezzacapo, A., Chow, J.M. and Gambetta, J.M. (2019) Error Mitigation Extends the Computational Reach of a Noisy Quantum Processor. Nature, 567, 491-495. https://doi.org/10.1038/s41586-019-1040-7
[95]
Higgott, O., Wang, D. and Brierley, S. (2019) Variational Quantum Computation of Excited States. Quantum, 3, 156. https://doi.org/10.22331/q-2019-07-01-156
[96]
McClean, J.R., Kimchi-Schwartz, M.E., Carter, J. and de Jong, W.A. (2017) Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States. Physical Review A, 95, Article 042308. https://doi.org/10.1103/physreva.95.042308
[97]
Heya, K., Nakanishi, K.M., Mitarai, K. and Fujii, K. (2019) Subspace Variational Quantum Simulator.
[98]
Nakanishi, K.M., Mitarai, K. and Fujii, K. (2019) Subspace-Search Variational Quantum Eigensolver for Excited States. Physical Review Research, 1, Article 033062. https://doi.org/10.1103/physrevresearch.1.033062
[99]
Parrish, R.M., Hohenstein, E.G., McMahon, P.L. and Martínez, T.J. (2019) Quantum Computation of Electronic Transitions Using a Variational Quantum Eigensolver. Physical Review Letters, 122, Article 230401. https://doi.org/10.1103/physrevlett.122.230401
[100]
Garcia-Saez, A. and Latorre, J. (2018) Addressing Hard Classical Problems with Adiabatically Assisted Variational Quantum Eigensolvers.
[101]
Cerezo, M., Sharma, K., Arrasmith, A. and Coles, P.J. (2020) Variational Quantum State Eigensolver.
[102]
Wang, D., Higgott, O. and Brierley, S. (2019) Accelerated Variational Quantum Eigensolver. Physical Review Letters, 122, Article 140504. https://doi.org/10.1103/physrevlett.122.140504
[103]
Wang, G., Koh, D.E., Johnson, P.D. and Cao, Y. (2021) Minimizing Estimation Runtime on Noisy Quantum Computers. PRX Quantum, 2, Article 010346. https://doi.org/10.1103/prxquantum.2.010346
[104]
Wang, G., Koh, D.E., Johnson, P.D. and Cao, Y. (2006) Bayesian Inference with Engineered Likelihood Functions for Robust Amplitude Estimation.
[105]
Wang, Y. (2012) Quantum Computation and Quantum Information. Statistical Science, 27, 373-394. https://doi.org/10.1214/11-sts378
[106]
Li, Y. and Benjamin, S.C. (2017) Efficient Variational Quantum Simulator Incorporating Active Error Minimization. Physical Review X, 7, Article 021050. https://doi.org/10.1103/physrevx.7.021050
[107]
Yuan, X., Endo, S., Zhao, Q., Li, Y. and Benjamin, S.C. (2019) Theory of Variational Quantum Simulation. Quantum, 3, 191. https://doi.org/10.22331/q-2019-10-07-191
[108]
McArdle, S., Jones, T., Endo, S., Li, Y., Benjamin, S.C. and Yuan, X. (2019) Variational Ansatz-Based Quantum Simulation of Imaginary Time Evolution. NPJ Quantum Information, 5, Article No. 75. https://doi.org/10.1038/s41534-019-0187-2
[109]
Endo, S., Sun, J., Li, Y., Benjamin, S.C. and Yuan, X. (2020) Variational Quantum Simulation of General Processes. Physical Review Letters, 125, Article 010501. https://doi.org/10.1103/physrevlett.125.010501
Zhang, Z.J., Sun, J., Yuan, X. and Yung, M.H. (2020) Low-Depth Hamiltonian Simulation by Adaptive Product Formula.
[112]
Cîrstoiu, C., Holmes, Z., Iosue, J., Cincio, L., Coles, P.J. and Sornborger, A. (2020) Variational Fast Forwarding for Quantum Simulation beyond the Coherence Time. npj Quantum Information, 6, Article No. 82. https://doi.org/10.1038/s41534-020-00302-0
[113]
Gibbs, J., Gili, K., Holmes, Z., Commeau, B., Arrasmith, A., Cincio, L., et al. (2022) Long-Time Simulations for Fixed Input States on Quantum Hardware. npj Quantum Information, 8, Article No. 135. https://doi.org/10.1038/s41534-022-00625-0
[114]
Khatri, S., LaRose, R., Poremba, A., Cincio, L., Sornborger, A.T. and Coles, P.J. (2019) Quantum-Assisted Quantum Compiling. Quantum, 3, 140. https://doi.org/10.22331/q-2019-05-13-140
[115]
Commeau, B., Cerezo, M., Holmes, Z., Cincio, L., et al. (2020) Variational Hamiltonian Diagonalization for Dynamical Quantum Simulation.
[116]
Wecker, D., Hastings, M.B. and Troyer, M. (2015) Progress towards Practical Quantum Variational Algorithms. Physical Review A, 92, Article 042303. https://doi.org/10.1103/physreva.92.042303
[117]
Wiersema, R., Zhou, C., de Sereville, Y., Carrasquilla, J.F., Kim, Y.B. and Yuen, H. (2020) Exploring Entanglement and Optimization within the Hamiltonian Variational Ansatz. PRX Quantum, 1, Article 020319. https://doi.org/10.1103/prxquantum.1.020319
[118]
Ho, W.W. and Hsieh, T.H. (2019) Efficient Variational Simulation of Non-Trivial Quantum States. SciPost Physics, 6, 10-19. https://doi.org/10.21468/scipostphys.6.3.029
[119]
Hamer, C.J., Weihong, Z. and Oitmaa, J. (1997) Series Expansions for the Massive Schwinger Model in Hamiltonian Lattice Theory. Physical Review D, 56, 55-67. https://doi.org/10.1103/physrevd.56.55
[120]
Bañuls, M.C., Cichy, K., Cirac, J.I. and Jansen, K. (2013) The Mass Spectrum of the Schwinger Model with Matrix Product States. Journal of High Energy Physics, 2013, Article No. 158. https://doi.org/10.1007/jhep11(2013)158
[121]
Cohen, T. and Welling, M. (2016) Group Equivariant Convolutional Networks.
[122]
Zhang, Y. and Di Ventra, M. (2023) Transformer Quantum State: A Multipurpose Model for Quantum Many-Body Problems. Physical Review B, 107, Article 075147. https://doi.org/10.1103/physrevb.107.075147
