Hamiltonian Simulation by Qubitization

Guang Hao Low; Isaac L. Chuang

doi:10.22331/q-2019-07-12-163

Hamiltonian Simulation by Qubitization

Guang Hao Low¹ and Isaac L. Chuang²

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
²Department of Electrical Engineering and Computer Science, Department of Physics, Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Published:	2019-07-12, volume 3, page 163
Eprint:	arXiv:1610.06546v3
Doi:	https://doi.org/10.22331/q-2019-07-12-163
Citation:	Quantum 3, 163 (2019).

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Abstract

We present the problem of approximating the time-evolution operator $e^{-i\hat{H}t}$ to error $\epsilon$, where the Hamiltonian $\hat{H}=(\langle G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}})$ is the projection of a unitary oracle $\hat{U}$ onto the state $|G\rangle$ created by another unitary oracle. Our algorithm solves this with a query complexity $\mathcal{O}\big(t+\log({1/\epsilon})\big)$ to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are $d$-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where $\hat{H}$ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large class of operator functions of $\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\hat{H}t}$ is a special case.

► BibTeX data

@article{Low2019hamiltonian,
  doi = {10.22331/q-2019-07-12-163},
  url = {https://doi.org/10.22331/q-2019-07-12-163},
  title = {Hamiltonian {S}imulation by {Q}ubitization},
  author = {Low, Guang Hao and Chuang, Isaac L.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {163},
  month = jul,
  year = {2019}
}

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[60] Minh C. Tran, Yuan Su, Daniel Carney, and Jacob M. Taylor, "Faster Digital Quantum Simulation by Symmetry Protection", PRX Quantum 2 1, 010323 (2021).

[61] Andrew Litteken, Lennart Maximilian Seifert, Jason D. Chadwick, Natalia Nottingham, Tanay Roy, Ziqian Li, David Schuster, Frederic T. Chong, and Jonathan M. Baker, Proceedings of the 50th Annual International Symposium on Computer Architecture 1 (2023) ISBN:9798400700958.

[62] Lin Lin and Yu Tong, "Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems", Quantum 4, 361 (2020).

[63] Connor T. Hann, Gideon Lee, S.M. Girvin, and Liang Jiang, "Resilience of Quantum Random Access Memory to Generic Noise", PRX Quantum 2 2, 020311 (2021).

[64] Nhat A. Nghiem and Tzu-Chieh Wei, "Quantum algorithm for estimating largest eigenvalues", Physics Letters A 488, 129138 (2023).

[65] Nhung H. Nguyen, Minh C. Tran, Yingyue Zhu, Alaina M. Green, C. Huerta Alderete, Zohreh Davoudi, and Norbert M. Linke, "Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions", PRX Quantum 3 2, 020324 (2022).

[66] Chan Gu Kang and Hakjoo Oh, "Modular Component-Based Quantum Circuit Synthesis", Proceedings of the ACM on Programming Languages 7 OOPSLA1, 348 (2023).

[67] Marcela Carena, Erik J. Gustafson, Henry Lamm, Ying-Ying Li, and Wanqiang Liu, "Gauge theory couplings on anisotropic lattices", Physical Review D 106 11, 114504 (2022).

[68] Thais L. Silva, Márcio M. Taddei, Stefano Carrazza, and Leandro Aolita, "Fragmented imaginary-time evolution for early-stage quantum signal processors", Scientific Reports 13 1, 18258 (2023).

[69] Kyle DeBry, Jasmine Sinanan-Singh, Colin D. Bruzewicz, David Reens, May E. Kim, Matthew P. Roychowdhury, Robert McConnell, Isaac L. Chuang, and John Chiaverini, "Experimental Quantum Channel Discrimination Using Metastable States of a Trapped Ion", Physical Review Letters 131 17, 170602 (2023).

[70] Jeremy Hartse and Alessandro Roggero, "Faster spectral density calculation using energy moments", The European Physical Journal A 59 3, 41 (2023).

[71] Mostafizur Rahaman Laskar, Subhadeep Mondal, and Amit Kumar Dutta, "A Low-Complexity Quantum Simulation Framework for Toeplitz-Structured Matrix and Its Application in Signal Processing", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[72] Yihui Quek and Patrick Rebentrost, "Fast algorithm for quantum polar decomposition and applications", Physical Review Research 4 1, 013144 (2022).

[73] Sitan Chen, Jordan Cotler, Hsin-Yuan Huang, and Jerry Li, "The complexity of NISQ", Nature Communications 14 1, 6001 (2023).

[74] Matthew B. Hastings, "Duality in Quantum Quenches and Classical Approximation Algorithms: Pretty Good or Very Bad", Quantum 3, 201 (2019).

[75] Alessandro Roggero and Joseph Carlson, "Dynamic linear response quantum algorithm", Physical Review C 100 3, 034610 (2019).

[76] Le Hu and Andrew N. Jordan, "Quantum state driving along arbitrary trajectories", Physical Review Research 5 3, 033045 (2023).

[77] Alicia B. Magann, Matthew D. Grace, Herschel A. Rabitz, and Mohan Sarovar, "Digital quantum simulation of molecular dynamics and control", Physical Review Research 3 2, 023165 (2021).

[78] Luuk Coopmans, Yuta Kikuchi, and Marcello Benedetti, "Predicting Gibbs-State Expectation Values with Pure Thermal Shadows", PRX Quantum 4 1, 010305 (2023).

[79] Roman Novak, "Quantum Algorithms in Electromagnetic Propagation Modelling for Telecommunications", IEEE Access 11, 111545 (2023).

[80] Bruno Senjean, Saad Yalouz, and Matthieu Saubanère, "Toward density functional theory on quantum computers?", SciPost Physics 14 3, 055 (2023).

[81] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush, "Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction", PRX Quantum 2 3, 030305 (2021).

[82] Xi-Ning Zhuang, Zhao-Yun Chen, Yu-Chun Wu, and Guo-Ping Guo, "Quantum computational quantitative trading: high-frequency statistical arbitrage algorithm", New Journal of Physics 24 7, 073036 (2022).

[83] Alessandro Roggero, Andy C. Y. Li, Joseph Carlson, Rajan Gupta, and Gabriel N. Perdue, "Quantum computing for neutrino-nucleus scattering", Physical Review D 101 7, 074038 (2020).

[84] Chenyi Zhang, Jiaqi Leng, and Tongyang Li, "Quantum algorithms for escaping from saddle points", Quantum 5, 529 (2021).

[85] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, and Shuchen Zhu, "Theory of Trotter Error with Commutator Scaling", Physical Review X 11 1, 011020 (2021).

[86] Jinzhao Sun, Suguru Endo, Huiping Lin, Patrick Hayden, Vlatko Vedral, and Xiao Yuan, "Perturbative Quantum Simulation", Physical Review Letters 129 12, 120505 (2022).

[87] Yuval R. Sanders, Dominic W. Berry, Pedro C.S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Gidney, Hartmut Neven, and Ryan Babbush, "Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization", PRX Quantum 1 2, 020312 (2020).

[88] Dominic W. Berry, Andrew M. Childs, Yuan Su, Xin Wang, and Nathan Wiebe, "Time-dependent Hamiltonian simulation withL1-norm scaling", Quantum 4, 254 (2020).

[89] Ignacio Loaiza, Alireza Marefat Khah, Nathan Wiebe, and Artur F Izmaylov, "Reducing molecular electronic Hamiltonian simulation cost for linear combination of unitaries approaches", Quantum Science and Technology 8 3, 035019 (2023).

[90] Fabian Langkabel and Annika Bande, "Quantum-Compute Algorithm for Exact Laser-Driven Electron Dynamics in Molecules", Journal of Chemical Theory and Computation 18 12, 7082 (2022).

[91] William M. Kirby and Peter J. Love, "Contextuality Test of the Nonclassicality of Variational Quantum Eigensolvers", Physical Review Letters 123 20, 200501 (2019).

[92] Yasunari Suzuki, Suguru Endo, Keisuke Fujii, and Yuuki Tokunaga, "Quantum Error Mitigation as a Universal Error Reduction Technique: Applications from the NISQ to the Fault-Tolerant Quantum Computing Eras", PRX Quantum 3 1, 010345 (2022).

[93] Yuan Su, "Framework for Hamiltonian simulation and beyond: standard-form encoding, qubitization, and quantum signal processing", Quantum Views 3, 21 (2019).

[94] Shi-Jie Pan, Lin-Chun Wan, Hai-Ling Liu, Qing-Le Wang, Su-Juan Qin, Qiao-Yan Wen, and Fei Gao, "Improved quantum algorithm for A-optimal projection", Physical Review A 102 5, 052402 (2020).

[95] Anthony Ciavarella, "Algorithm for quantum computation of particle decays", Physical Review D 102 9, 094505 (2020).

[96] Andrew M. Childs and Yuan Su, "Nearly Optimal Lattice Simulation by Product Formulas", Physical Review Letters 123 5, 050503 (2019).

[97] Yatian Wang, Hua Xiang, and Songling Zhang, "Quantum algorithm for matrix logarithm by integral formula", Quantum Information Processing 22 1, 76 (2023).

[98] Anandu Kalleri Madhu, Alexey A. Melnikov, Leonid E. Fedichkin, Alexander P. Alodjants, and Ray-Kuang Lee, "Quantum walk processes in quantum devices", Heliyon 9 3, e13416 (2023).

[99] Modjtaba Shokrian Zini, Alain Delgado, Roberto dos Reis, Pablo Antonio Moreno Casares, Jonathan E. Mueller, Arne-Christian Voigt, and Juan Miguel Arrazola, "Quantum simulation of battery materials using ionic pseudopotentials", Quantum 7, 1049 (2023).

[100] I. Y. Dodin and E. A. Startsev, "On applications of quantum computing to plasma simulations", Physics of Plasmas 28 9, 092101 (2021).

[101] Benjamin A. Cordier, Nicolas P. D. Sawaya, Gian Giacomo Guerreschi, and Shannon K. McWeeney, "Biology and medicine in the landscape of quantum advantages", Journal of The Royal Society Interface 19 196, 20220541 (2022).

[102] Grecia Castelazo, Quynh T. Nguyen, Giacomo De Palma, Dirk Englund, Seth Lloyd, and Bobak T. Kiani, "Quantum algorithms for group convolution, cross-correlation, and equivariant transformations", Physical Review A 106 3, 032402 (2022).

[103] Daniel Bultrini and Oriol Vendrell, "Mixed quantum-classical dynamics for near term quantum computers", Communications Physics 6 1, 328 (2023).

[104] Michael R. Geller, Zoë Holmes, Patrick J. Coles, and Andrew Sornborger, "Experimental quantum learning of a spectral decomposition", Physical Review Research 3 3, 033200 (2021).

[105] Ryan Shaffer, Hang Ren, Emiliia Dyrenkova, Christopher G. Yale, Daniel S. Lobser, Ashlyn D. Burch, Matthew N. H. Chow, Melissa C. Revelle, Susan M. Clark, and Hartmut Häffner, "Sample-efficient verification of continuously-parameterized quantum gates for small quantum processors", Quantum 7, 997 (2023).

[106] Leonardo Novo, "Bridging gaps between random approaches to quantum simulation", Quantum Views 4, 33 (2020).

[107] Lawrence Z. Cohen, Isaac H. Kim, Stephen D. Bartlett, and Benjamin J. Brown, "Low-overhead fault-tolerant quantum computing using long-range connectivity", Science Advances 8 20, eabn1717 (2022).

[108] Sevag Gharibian and François Le Gall, Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing 19 (2022) ISBN:9781450392648.

[109] Quynh T. Nguyen, Bobak T. Kiani, and Seth Lloyd, "Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra", Quantum 6, 876 (2022).

[110] Marcela Carena, Henry Lamm, Ying-Ying Li, and Wanqiang Liu, "Lattice renormalization of quantum simulations", Physical Review D 104 9, 094519 (2021).

[111] Yanbing Zhang, Tingting Song, and Zhihao Wu, "An improved algorithm for computing hitting probabilities of quantum walks", Physica A: Statistical Mechanics and its Applications 594, 127009 (2022).

[112] Ruizhe Zhang, Guoming Wang, and Peter Johnson, "Computing Ground State Properties with Early Fault-Tolerant Quantum Computers", Quantum 6, 761 (2022).

[113] Mohammadhossein Mohammadisiahroudi and Tamás Terlaky, Encyclopedia of Optimization 1 (2023) ISBN:978-3-030-54621-2.

[114] A. Roggero, "Spectral-density estimation with the Gaussian integral transform", Physical Review A 102 2, 022409 (2020).

[115] Jules Tilly, Hongxiang Chen, Shuxiang Cao, Dario Picozzi, Kanav Setia, Ying Li, Edward Grant, Leonard Wossnig, Ivan Rungger, George H. Booth, and Jonathan Tennyson, "The Variational Quantum Eigensolver: A review of methods and best practices", Physics Reports 986, 1 (2022).

[116] Guru-Vamsi Policharla and Sai Vinjanampathy, "Algorithmic Primitives for Quantum-Assisted Quantum Control", Physical Review Letters 127 22, 220504 (2021).

[117] Yong-Mei Li, Hai-Ling Liu, Shi-Jie Pan, Su-Juan Qin, Fei Gao, and Qiao-Yan Wen, "General quantum matrix exponential dimensionality-reduction framework based on block encoding", Physical Review A 108 4, 042418 (2023).

[118] Andrew M. Childs, Aaron Ostrander, and Yuan Su, "Faster quantum simulation by randomization", Quantum 3, 182 (2019).

[119] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Benjamin Commeau, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, and Andrew Sornborger, "Long-time simulations for fixed input states on quantum hardware", npj Quantum Information 8 1, 135 (2022).

[120] Bin Cheng, Xiu-Hao Deng, Xiu Gu, Yu He, Guangchong Hu, Peihao Huang, Jun Li, Ben-Chuan Lin, Dawei Lu, Yao Lu, Chudan Qiu, Hui Wang, Tao Xin, Shi Yu, Man-Hong Yung, Junkai Zeng, Song Zhang, Youpeng Zhong, Xinhua Peng, Franco Nori, and Dapeng Yu, "Noisy intermediate-scale quantum computers", Frontiers of Physics 18 2, 21308 (2023).

[121] Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf, "Quantum SDP-Solvers: Better upper and lower bounds", Quantum 4, 230 (2020).

[122] Lindsay Bassman, Miroslav Urbanek, Mekena Metcalf, Jonathan Carter, Alexander F Kemper, and Wibe A de Jong, "Simulating quantum materials with digital quantum computers", Quantum Science and Technology 6 4, 043002 (2021).

[123] Zane M. Rossi and Isaac L. Chuang, "Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle", Quantum 6, 811 (2022).

[124] Dong An, Di Fang, and Lin Lin, "Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation", Quantum 6, 690 (2022).

[125] Lin Lin and Yu Tong, "Near-optimal ground state preparation", Quantum 4, 372 (2020).

[126] Jessica Lemieux, Guillaume Duclos-Cianci, David Sénéchal, and David Poulin, "Resource estimate for quantum many-body ground-state preparation on a quantum computer", Physical Review A 103 5, 052408 (2021).

[127] Marcela Carena, Henry Lamm, Ying-Ying Li, and Wanqiang Liu, "Improved Hamiltonians for Quantum Simulations of Gauge Theories", Physical Review Letters 129 5, 051601 (2022).

[128] Abtin Ameri, Erika Ye, Paola Cappellaro, Hari Krovi, and Nuno F. Loureiro, 2023 IEEE International Conference on Quantum Computing and Engineering (QCE) 56 (2023) ISBN:979-8-3503-4323-6.

[129] Lin Lin and Yu Tong, "Heisenberg-Limited Ground-State Energy Estimation for Early Fault-Tolerant Quantum Computers", PRX Quantum 3 1, 010318 (2022).

[130] Hai‐Ling Liu, Lin‐Chun Wan, Chao‐Hua Yu, Shi‐Jie Pan, Su‐Juan Qin, Fei Gao, and Qiao‐Yan Wen, "A Quantum Algorithm for Solving Eigenproblem of the Laplacian Matrix of a Fully Connected Weighted Graph", Advanced Quantum Technologies 6 7, 2300031 (2023).

[131] Thomas E. O’Brien, Lev B. Ioffe, Yuan Su, David Fushman, Hartmut Neven, Ryan Babbush, and Vadim Smelyanskiy, "Quantum Computation of Molecular Structure Using Data from Challenging-To-Classically-Simulate Nuclear Magnetic Resonance Experiments", PRX Quantum 3 3, 030345 (2022).

[132] Rolando D Somma, "Quantum eigenvalue estimation via time series analysis", New Journal of Physics 21 12, 123025 (2019).

[133] Paul K. Faehrmann, Mark Steudtner, Richard Kueng, Maria Kieferova, and Jens Eisert, "Randomizing multi-product formulas for Hamiltonian simulation", Quantum 6, 806 (2022).

[134] Dominik Hangleiter and Jens Eisert, "Computational advantage of quantum random sampling", Reviews of Modern Physics 95 3, 035001 (2023).

[135] Valentina Amitrano, Alessandro Roggero, Piero Luchi, Francesco Turro, Luca Vespucci, and Francesco Pederiva, "Trapped-ion quantum simulation of collective neutrino oscillations", Physical Review D 107 2, 023007 (2023).

[136] Cristian L. Cortes and Stephen K. Gray, "Quantum Krylov subspace algorithms for ground- and excited-state energy estimation", Physical Review A 105 2, 022417 (2022).

[137] Matthew Hagan and Nathan Wiebe, "Composite Quantum Simulations", Quantum 7, 1181 (2023).

[138] Zane M. Rossi and Isaac L. Chuang, "Quantum hypothesis testing with group structure", Physical Review A 104 1, 012425 (2021).

[139] Xiao-Ming Zhang, Tongyang Li, and Xiao Yuan, "Quantum State Preparation with Optimal Circuit Depth: Implementations and Applications", Physical Review Letters 129 23, 230504 (2022).

[140] Mostafizur Rahaman Laskar and Amit Kumar Dutta, "A Complexity-Efficient Quantum Architecture and Simulation for Eigen Spectrum Estimation of Vandermonde System in a Large Antenna Array", IEEE Transactions on Circuits and Systems I: Regular Papers 70 5, 2106 (2023).

[141] William M. Kirby, Sultana Hadi, Michael Kreshchuk, and Peter J. Love, "Quantum simulation of second-quantized Hamiltonians in compact encoding", Physical Review A 104 4, 042607 (2021).

[142] Tatiana A. Bespalova and Oleksandr Kyriienko, "Hamiltonian Operator Approximation for Energy Measurement and Ground-State Preparation", PRX Quantum 2 3, 030318 (2021).

[143] Chen He, Jiazhen Li, Weiqi Liu, Jinye Peng, and Z. Jane Wang, "A Low-Complexity Quantum Principal Component Analysis Algorithm", IEEE Transactions on Quantum Engineering 3, 1 (2022).

[144] Andrés Gómez and Javier Mas, "Hermitian matrix definiteness from quantum phase estimation", Quantum Information Processing 21 6, 213 (2022).

[145] Min-Quan He, Dan-Bo Zhang, and Z. D. Wang, "Quantum Gaussian filter for exploring ground-state properties", Physical Review A 106 3, 032420 (2022).

[146] Weiyuan Gong, Si Jiang, and Dong-Ling Deng, "No-go theorem and a universal decomposition strategy for quantum channel compilation", Physical Review Research 5 1, 013060 (2023).

[147] Pedro C.S. Costa, Dong An, Yuval R. Sanders, Yuan Su, Ryan Babbush, and Dominic W. Berry, "Optimal Scaling Quantum Linear-Systems Solver via Discrete Adiabatic Theorem", PRX Quantum 3 4, 040303 (2022).

[148] Dominic W. Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, and Ryan Babbush, "Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization", Quantum 3, 208 (2019).

[149] Margarite L. LaBorde and Mark M. Wilde, "Quantum Algorithms for Testing Hamiltonian Symmetry", Physical Review Letters 129 16, 160503 (2022).

[150] Thomas E. Baker, "Lanczos recursion on a quantum computer for the Green's function and ground state", Physical Review A 103 3, 032404 (2021).

[151] Alessandro Roggero, Chenyi Gu, Alessandro Baroni, and Thomas Papenbrock, "Preparation of excited states for nuclear dynamics on a quantum computer", Physical Review C 102 6, 064624 (2020).

[152] Scott E. Smart and David A. Mazziotti, "Many-fermion simulation from the contracted quantum eigensolver without fermionic encoding of the wave function", Physical Review A 105 6, 062424 (2022).

[153] Mario Motta, Tanvi P. Gujarati, Julia E. Rice, Ashutosh Kumar, Conner Masteran, Joseph A. Latone, Eunseok Lee, Edward F. Valeev, and Tyler Y. Takeshita, "Quantum simulation of electronic structure with a transcorrelated Hamiltonian: improved accuracy with a smaller footprint on the quantum computer", Physical Chemistry Chemical Physics 22 42, 24270 (2020).

[154] Rawad Mezher, Ana Filipa Carvalho, and Shane Mansfield, "Solving graph problems with single photons and linear optics", Physical Review A 108 3, 032405 (2023).

[155] Shang Gao and Yu-Guang Yang, "A novel quantum recommender system", Physica Scripta 98 1, 010001 (2023).

[156] M. Sohaib Alam, Stuart Hadfield, Henry Lamm, and Andy C. Y. Li, "Primitive quantum gates for dihedral gauge theories", Physical Review D 105 11, 114501 (2022).

[157] Riley W. Chien, Sha Xue, Tarini S. Hardikar, Kanav Setia, and James D. Whitfield, "Analysis of superfast encoding performance for electronic structure simulations", Physical Review A 100 3, 032337 (2019).

[158] Olivia Di Matteo, Anna McCoy, Peter Gysbers, Takayuki Miyagi, R. M. Woloshyn, and Petr Navrátil, "Improving Hamiltonian encodings with the Gray code", Physical Review A 103 4, 042405 (2021).

[159] Duarte Magano and Miguel Murça, "Simplifying a classical-quantum algorithm interpolation with quantum singular value transformations", Physical Review A 106 6, 062419 (2022).

[160] Erik J. Gustafson, Henry Lamm, Felicity Lovelace, and Damian Musk, "Primitive quantum gates for an SU(2) discrete subgroup: Binary tetrahedral", Physical Review D 106 11, 114501 (2022).

[161] William Kirby, Bryce Fuller, Charles Hadfield, and Antonio Mezzacapo, "Second-Quantized Fermionic Operators with Polylogarithmic Qubit and Gate Complexity", PRX Quantum 3 2, 020351 (2022).

[162] Kaito Mizukami and Akihisa Koga, "Quantum algorithm for the microcanonical thermal pure quantum state method", Physical Review A 108 1, 012404 (2023).

[163] Ze-Tong Li, Fan-Xu Meng, Xu-Tao Yu, and Zai-Chen Zhang, "Quantum algorithm for Laplacian eigenmap via Rayleigh quotient iteration", Quantum Information Processing 21 1, 11 (2022).

[164] Karuna Kadian, Sunita Garhwal, and Ajay Kumar, "Quantum walk and its application domains: A systematic review", Computer Science Review 41, 100419 (2021).

[165] Alexander Engel, Graeme Smith, and Scott E. Parker, "Quantum algorithm for the Vlasov equation", Physical Review A 100 6, 062315 (2019).

[166] I. Novikau, I.Y. Dodin, and E.A. Startsev, "Simulation of Linear Non-Hermitian Boundary-Value Problems with Quantum Singular-Value Transformation", Physical Review Applied 19 5, 054012 (2023).

[167] Priyanka Mukhopadhyay, "Composability of global phase invariant distance and its application to approximation error management", Journal of Physics Communications 5 11, 115017 (2021).

[168] Zane M. Rossi, Jeffery Yu, Isaac L. Chuang, and Sho Sugiura, "Quantum advantage for noisy channel discrimination", Physical Review A 105 3, 032401 (2022).

[169] Michael Kreshchuk, William M. Kirby, Gary Goldstein, Hugo Beauchemin, and Peter J. Love, "Quantum simulation of quantum field theory in the light-front formulation", Physical Review A 105 3, 032418 (2022).

[170] Tong Ning, Youlong Yang, and Zhenye Du, "Quantum algorithm for twin extreme learning machine", Physica Scripta 98 8, 085110 (2023).

[171] Alexis Ralli, Tim Weaving, Andrew Tranter, William M. Kirby, Peter J. Love, and Peter V. Coveney, "Unitary partitioning and the contextual subspace variational quantum eigensolver", Physical Review Research 5 1, 013095 (2023).

[172] Shantanav Chakraborty, Leonardo Novo, and Jérémie Roland, "Finding a marked node on any graph via continuous-time quantum walks", Physical Review A 102 2, 022227 (2020).

[173] Isaac H. Kim, Ye-Hua Liu, Sam Pallister, William Pol, Sam Roberts, and Eunseok Lee, "Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules", Physical Review Research 4 2, 023019 (2022).

[174] Matthew Thibodeau and Bryan K. Clark, "Nearly-frustration-free ground state preparation", Quantum 7, 1084 (2023).

[175] Ryan Babbush, Dominic W. Berry, Robin Kothari, Rolando D. Somma, and Nathan Wiebe, "Exponential Quantum Speedup in Simulating Coupled Classical Oscillators", Physical Review X 13 4, 041041 (2023).

[176] Yu Tong, Dong An, Nathan Wiebe, and Lin Lin, "Fast inversion, preconditioned quantum linear system solvers, fast Green's-function computation, and fast evaluation of matrix functions", Physical Review A 104 3, 032422 (2021).

[177] Wenjun Yu, Jinzhao Sun, Zeyao Han, and Xiao Yuan, "Robust and Efficient Hamiltonian Learning", Quantum 7, 1045 (2023).

[178] Oriel Kiss, Michele Grossi, and Alessandro Roggero, "Importance sampling for stochastic quantum simulations", Quantum 7, 977 (2023).

[179] Brandon Augustino and Giacomo Nannicini, Encyclopedia of Optimization 1 (2023) ISBN:978-3-030-54621-2.

[180] Earl T Campbell, "Early fault-tolerant simulations of the Hubbard model", Quantum Science and Technology 7 1, 015007 (2022).

[181] Mark Steudtner, Sam Morley-Short, William Pol, Sukin Sim, Cristian L. Cortes, Matthias Loipersberger, Robert M. Parrish, Matthias Degroote, Nikolaj Moll, Raffaele Santagati, and Michael Streif, "Fault-tolerant quantum computation of molecular observables", Quantum 7, 1164 (2023).

[182] Yuanye Zhu, "Quantum-Solving Algorithm for d’Alembert Solutions of the Wave Equation", Entropy 25 1, 62 (2022).

[183] Jacob Bringewatt and Zohreh Davoudi, "Parallelization techniques for quantum simulation of fermionic systems", Quantum 7, 975 (2023).

[184] Annika Bande, Chemical Modelling 91 (2022) ISBN:978-1-83916-741-6.

[185] Aleksei V. Ivanov, Christoph Sünderhauf, Nicole Holzmann, Tom Ellaby, Rachel N. Kerber, Glenn Jones, and Joan Camps, "Quantum computation for periodic solids in second quantization", Physical Review Research 5 1, 013200 (2023).

[186] Zongping Gong and Ryusuke Hamazaki, "Bounds in nonequilibrium quantum dynamics", International Journal of Modern Physics B 36 31, 2230007 (2022).

[187] Giorgio Tosti Balducci, Boyang Chen, Matthias Möller, Marc Gerritsma, and Roeland De Breuker, "Review and perspectives in quantum computing for partial differential equations in structural mechanics", Frontiers in Mechanical Engineering 8, 914241 (2022).

[188] Matthew Otten, Byeol Kang, Dmitry Fedorov, Joo-Hyoung Lee, Anouar Benali, Salman Habib, Stephen K. Gray, and Yuri Alexeev, "QREChem: quantum resource estimation software for chemistry applications", Frontiers in Quantum Science and Technology 2, 1232624 (2023).

[189] Guang Hao Low, Yuan Su, Yu Tong, and Minh C. Tran, "Complexity of Implementing Trotter Steps", PRX Quantum 4 2, 020323 (2023).

[190] Alexander M. Dalzell, B. David Clader, Grant Salton, Mario Berta, Cedric Yen-Yu Lin, David A. Bader, Nikitas Stamatopoulos, Martin J. A. Schuetz, Fernando G. S. L. Brandão, Helmut G. Katzgraber, and William J. Zeng, "End-To-End Resource Analysis for Quantum Interior-Point Methods and Portfolio Optimization", PRX Quantum 4 4, 040325 (2023).

[191] Kenneth Choi, Dean Lee, Joey Bonitati, Zhengrong Qian, and Jacob Watkins, "Rodeo Algorithm for Quantum Computing", Physical Review Letters 127 4, 040505 (2021).

[192] Andrew M. Childs, Jiaqi Leng, Tongyang Li, Jin-Peng Liu, and Chenyi Zhang, "Quantum simulation of real-space dynamics", Quantum 6, 860 (2022).

[193] Abhijeet Alase, Owen Doty, and David L. Feder, "Matrix permanent and determinant from a spin system", Physical Review A 108 1, 012207 (2023).

[194] Matthew B. Hastings, "Classical and Quantum Algorithms for Tensor Principal Component Analysis", Quantum 4, 237 (2020).

[195] Michael P Kaicher, Simon B Jäger, and Frank K Wilhelm, "Algorithm for initializing a generalized fermionic Gaussian state on a quantum computer", Journal of Physics A: Mathematical and Theoretical 54 39, 395303 (2021).

[196] Xiu Gu, Jorge Fernández-Pendás, Pontus Vikstål, Tahereh Abad, Christopher Warren, Andreas Bengtsson, Giovanna Tancredi, Vitaly Shumeiko, Jonas Bylander, Göran Johansson, and Anton Frisk Kockum, "Fast Multiqubit Gates through Simultaneous Two-Qubit Gates", PRX Quantum 2 4, 040348 (2021).

[197] Javier Argüello-Luengo, Tao Shi, and Alejandro González-Tudela, "Engineering analog quantum chemistry Hamiltonians using cold atoms in optical lattices", Physical Review A 103 4, 043318 (2021).

[198] Yulong Dong, Xiang Meng, K. Birgitta Whaley, and Lin Lin, "Efficient phase-factor evaluation in quantum signal processing", Physical Review A 103 4, 042419 (2021).

[199] Nicholas C. Rubin, Dominic W. Berry, Fionn D. Malone, Alec F. White, Tanuj Khattar, A. Eugene DePrince, Sabrina Sicolo, Michael Küehn, Michael Kaicher, Joonho Lee, and Ryan Babbush, "Fault-Tolerant Quantum Simulation of Materials Using Bloch Orbitals", PRX Quantum 4 4, 040303 (2023).

[200] Mostafizur Rahaman Laskar and Amit Kumar Dutta, "A Proposed Quantum Framework for Low-Complexity Quantum Simulation and Spectrum Estimation of Hankel-Patterned Systems", IEEE Transactions on Quantum Engineering 4, 1 (2023).

[201] Yutaka Shikano, Hiroshi C. Watanabe, Ken M. Nakanishi, and Yu-ya Ohnishi, "Post-Hartree–Fock method in quantum chemistry for quantum computer", The European Physical Journal Special Topics 230 4, 1037 (2021).

[202] Dong An, Di Fang, and Lin Lin, "Time-dependent unbounded Hamiltonian simulation with vector norm scaling", Quantum 5, 459 (2021).

[203] John S. Van Dyke, George S. Barron, Nicholas J. Mayhall, Edwin Barnes, and Sophia E. Economou, "Preparing Bethe Ansatz Eigenstates on a Quantum Computer", PRX Quantum 2 4, 040329 (2021).

[204] Richard Meister, Simon C. Benjamin, and Earl T. Campbell, "Tailoring Term Truncations for Electronic Structure Calculations Using a Linear Combination of Unitaries", Quantum 6, 637 (2022).

[205] Andrew Zhao, Andrew Tranter, William M. Kirby, Shu Fay Ung, Akimasa Miyake, and Peter J. Love, "Measurement reduction in variational quantum algorithms", Physical Review A 101 6, 062322 (2020).

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Hamiltonian Simulation by Qubitization

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