{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T08:16:50Z","timestamp":1788509810238,"version":"build-2803163510"},"reference-count":99,"publisher":"Springer Science and Business Media LLC","issue":"10","license":[{"start":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T00:00:00Z","timestamp":1788480000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T00:00:00Z","timestamp":1788480000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Front. Comput. Sci."],"published-print":{"date-parts":[[2026,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Quantum simulation is a rapidly advancing field poised to revolutionize our understanding of complex quantum systems by harnessing the unique capabilities of quantum computers. In this review, we present a concise overview of key developments in quantum simulation algorithms, with a focus on time-independent digital Hamiltonian simulation of quantum dynamics. We review seminal methods\u2014from traditional Trotter formulas to techniques such as truncated Taylor series combined with the linear combination of unitaries, and quantum signal processing. Additionally, we examine error analyses and recent innovations that enhance the efficiency and accuracy of quantum simulations.<\/jats:p>","DOI":"10.1007\/s11704-026-51794-6","type":"journal-article","created":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T06:35:34Z","timestamp":1788503734000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Advances in time-independent Hamiltonian simulation algorithms"],"prefix":"10.1007","volume":"20","author":[{"given":"Yue","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wenjun","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tianfeng","family":"Feng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qi","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,9,4]]},"reference":[{"issue":"6\u20137","key":"51794_CR1","doi-asserted-by":"publisher","first-page":"467","DOI":"10.1007\/BF02650179","volume":"21","author":"R P Feynman","year":"1982","unstructured":"Feynman R P. Simulating physics with computers. International Journal of Theoretical Physics, 1982, 21(6\u20137): 467\u2013488","journal-title":"International Journal of Theoretical Physics"},{"issue":"5278","key":"51794_CR2","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1126\/science.273.5278.1073","volume":"273","author":"S Lloyd","year":"1996","unstructured":"Lloyd S. Universal quantum simulators. Science, 1996, 273(5278): 1073\u20131078","journal-title":"Science"},{"issue":"1","key":"51794_CR3","doi-asserted-by":"publisher","first-page":"011020","DOI":"10.1103\/PhysRevX.11.011020","volume":"11","author":"A M Childs","year":"2021","unstructured":"Childs A M, Su Y, Tran M C, Wiebe N, Zhu S. Theory of trotter error with commutator scaling. Physical Review X, 2021, 11(1): 011020","journal-title":"Physical Review X"},{"issue":"1\u20134","key":"51794_CR4","doi-asserted-by":"publisher","first-page":"432","DOI":"10.1016\/0378-4371(93)90375-E","volume":"194","author":"M Suzuki","year":"1993","unstructured":"Suzuki M. Quantum Monte Carlo methods \u2014 recent developments. Physica A: Statistical Mechanics and its Applications, 1993, 194(1\u20134): 432\u2013449","journal-title":"Physica A: Statistical Mechanics and its Applications"},{"issue":"9","key":"51794_CR5","doi-asserted-by":"publisher","first-page":"090502","DOI":"10.1103\/PhysRevLett.114.090502","volume":"114","author":"D W Berry","year":"2015","unstructured":"Berry D W, Childs A M, Cleve R, Kothari R, Somma R D. Simulating Hamiltonian dynamics with a truncated Taylor series. Physical Review Letters, 2015, 114(9): 090502","journal-title":"Physical Review Letters"},{"issue":"4","key":"51794_CR6","doi-asserted-by":"publisher","first-page":"1305","DOI":"10.1007\/s10773-010-0603-z","volume":"50","author":"G L Long","year":"2011","unstructured":"Long G L. Duality quantum computing and duality quantum information processing. International Journal of Theoretical Physics, 2011, 50(4): 1305\u20131318","journal-title":"International Journal of Theoretical Physics"},{"issue":"11\u201312","key":"51794_CR7","doi-asserted-by":"publisher","first-page":"901","DOI":"10.26421\/QIC12.11-12-1","volume":"12","author":"A M Childs","year":"2012","unstructured":"Childs A M, Wiebe N. Hamiltonian simulation using linear combinations of unitary operations. Quantum Information and Computation, 2012, 12(11\u201312): 901\u2013924","journal-title":"Quantum Information and Computation"},{"key":"51794_CR8","doi-asserted-by":"publisher","first-page":"163","DOI":"10.22331\/q-2019-07-12-163","volume":"3","author":"G H Low","year":"2019","unstructured":"Low G H, Chuang I L. Hamiltonian simulation by qubitization. Quantum, 2019, 3: 163","journal-title":"Quantum"},{"key":"51794_CR9","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1145\/3313276.3316366","volume-title":"Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing","author":"A Gily\u00e9n","year":"2019","unstructured":"Gily\u00e9n A, Su Y, Low G H, Wiebe N. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. 2019, 193\u2013204"},{"key":"51794_CR10","doi-asserted-by":"crossref","unstructured":"Martyn J M, Rossi Z M, Tan A K, Chuang I L. A grand unification of quantum algorithms. 2021, arXiv preprint arXiv: 2105.02859","DOI":"10.1103\/PRXQuantum.2.040203"},{"key":"51794_CR11","doi-asserted-by":"crossref","unstructured":"Berry D W, Wan K, Baczewski A D, Eklund E C, Tikku A, Babbush R. Quantum simulation of chemistry via quantum fast multipole method. 2025, arXiv preprint arXiv: 2510.07380v1","DOI":"10.1103\/b2vn-zf6c"},{"issue":"1","key":"51794_CR12","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1038\/s41534-024-00896-9","volume":"10","author":"D W Berry","year":"2024","unstructured":"Berry D W, Rubin N C, Elnabawy A O, Ahlers G, DePrince III A E, Lee J, Gogolin C, Babbush R. Quantum simulation of realistic materials in first quantization using non-local pseudopotentials. npj Quantum Information, 2024, 10(1): 130","journal-title":"npj Quantum Information"},{"key":"51794_CR13","doi-asserted-by":"publisher","first-page":"1460","DOI":"10.22331\/q-2024-09-04-1460","volume":"8","author":"M Ballarin","year":"2024","unstructured":"Ballarin M, Cataldi G, Magnifico G, Jaschke D, Liberto M D, Siloi I, Montangero S, Silvi P. Digital quantum simulation of lattice fermion theories with local encoding. Quantum, 2024, 8: 1460","journal-title":"Quantum"},{"issue":"7","key":"51794_CR14","doi-asserted-by":"publisher","first-page":"070503","DOI":"10.1103\/PhysRevLett.123.070503","volume":"123","author":"E Campbell","year":"2019","unstructured":"Campbell E. Random compiler for fast Hamiltonian simulation. Physical Review Letters, 2019, 123(7): 070503","journal-title":"Physical Review Letters"},{"issue":"4","key":"51794_CR15","doi-asserted-by":"publisher","first-page":"042306","DOI":"10.1103\/PhysRevA.95.042306","volume":"95","author":"E Campbell","year":"2017","unstructured":"Campbell E. Shorter gate sequences for quantum computing by mixing unitaries. Physical Review A, 2017, 95(4): 042306","journal-title":"Physical Review A"},{"issue":"5\u20136","key":"51794_CR16","doi-asserted-by":"publisher","first-page":"488","DOI":"10.26421\/QIC17.5-6-7","volume":"17","author":"M B Hastings","year":"2017","unstructured":"Hastings M B. Turning gate synthesis errors into incoherent errors. Quantum Information and Computation, 2017, 17(5\u20136): 488\u2013494","journal-title":"Quantum Information and Computation"},{"key":"51794_CR17","doi-asserted-by":"publisher","first-page":"182","DOI":"10.22331\/q-2019-09-02-182","volume":"3","author":"A M Childs","year":"2019","unstructured":"Childs A M, Ostrander A, Su Y. Faster quantum simulation by randomization. Quantum, 2019, 3: 182","journal-title":"Quantum"},{"key":"51794_CR18","unstructured":"Wang Y, Zhao Q. Faster quantum algorithms with \u201cfractional\u201d-truncated series. 2024, arXiv preprint arXiv: 2402.05595v2"},{"issue":"1\u20132","key":"51794_CR19","doi-asserted-by":"publisher","first-page":"29","DOI":"10.26421\/QIC12.1-2-4","volume":"12","author":"A M Childs","year":"2012","unstructured":"Childs A M, Berry D W. Black-box Hamiltonian simulation and unitary implementation. Quantum Information and Computation, 2012, 12(1\u20132): 29\u201362","journal-title":"Quantum Information and Computation"},{"key":"51794_CR20","first-page":"20","volume-title":"Proceedings of the 35th Annual ACM Symposium on Theory of Computing","author":"D Aharonov","year":"2003","unstructured":"Aharonov D, Ta-Shma A. Adiabatic quantum state generation and statistical zero knowledge. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing. 2003, 20\u201329"},{"issue":"2","key":"51794_CR21","doi-asserted-by":"publisher","first-page":"022314","DOI":"10.1103\/PhysRevA.70.022314","volume":"70","author":"A M Childs","year":"2004","unstructured":"Childs A M, Goldstone J. Spatial search by quantum walk. Physical Review A, 2004, 70(2): 022314","journal-title":"Physical Review A"},{"issue":"2","key":"51794_CR22","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1007\/s00220-006-0150-x","volume":"270","author":"D W Berry","year":"2007","unstructured":"Berry D W, Ahokas G, Cleve R, Sanders B C. Efficient quantum algorithms for simulating sparse Hamiltonians. Communications in Mathematical Physics, 2007, 270(2): 359\u2013371","journal-title":"Communications in Mathematical Physics"},{"issue":"2","key":"51794_CR23","doi-asserted-by":"publisher","first-page":"400","DOI":"10.1063\/1.529425","volume":"32","author":"M Suzuki","year":"1991","unstructured":"Suzuki M. General theory of fractal path integrals with applications to many-body theories and statistical physics. Journal of Mathematical Physics, 1991, 32(2): 400\u2013407","journal-title":"Journal of Mathematical Physics"},{"issue":"6","key":"51794_CR24","doi-asserted-by":"publisher","first-page":"062202","DOI":"10.1063\/1.4952761","volume":"57","author":"R D Somma","year":"2016","unstructured":"Somma R D. A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation. Journal of Mathematical Physics, 2016, 57(6): 062202","journal-title":"Journal of Mathematical Physics"},{"issue":"5","key":"51794_CR25","doi-asserted-by":"publisher","first-page":"050503","DOI":"10.1103\/PhysRevLett.123.050503","volume":"123","author":"A M Childs","year":"2019","unstructured":"Childs A M, Su Y. Nearly optimal lattice simulation by product formulas. Physical Review Letters, 2019, 123(5): 050503","journal-title":"Physical Review Letters"},{"issue":"38","key":"51794_CR26","doi-asserted-by":"publisher","first-page":"9456","DOI":"10.1073\/pnas.1801723115","volume":"115","author":"A M Childs","year":"2018","unstructured":"Childs A M, Maslov D, Nam Y, Ross N J, Su Y. Toward the first quantum simulation with quantum speedup. Proceedings of the National Academy of Sciences of the United States of America, 2018, 115(38): 9456\u20139461","journal-title":"Proceedings of the National Academy of Sciences of the United States of America"},{"issue":"27","key":"51794_CR27","doi-asserted-by":"publisher","first-page":"270502","DOI":"10.1103\/PhysRevLett.129.270502","volume":"129","author":"Q Zhao","year":"2022","unstructured":"Zhao Q, Zhou Y, Shaw A F, Li T, Childs A M. Hamiltonian simulation with random inputs. Physical Review Letters, 2022, 129(27): 270502","journal-title":"Physical Review Letters"},{"issue":"2","key":"51794_CR28","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1007\/s00220-023-04912-5","volume":"405","author":"C F Chen","year":"2024","unstructured":"Chen C F, Brand\u00e3o F G S L. Average-case speedup for product formulas. Communications in Mathematical Physics, 2024, 405(2): 32","journal-title":"Communications in Mathematical Physics"},{"issue":"1","key":"51794_CR29","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1038\/s41534-021-00451-w","volume":"7","author":"B \u015eahino\u01e7lu","year":"2021","unstructured":"\u015eahino\u01e7lu B, Somma R D. Hamiltonian simulation in the low-energy subspace. npj Quantum Information, 2021, 7(1): 119","journal-title":"npj Quantum Information"},{"issue":"8","key":"51794_CR30","doi-asserted-by":"publisher","first-page":"1338","DOI":"10.1038\/s41567-025-02945-2","volume":"21","author":"Q Zhao","year":"2025","unstructured":"Zhao Q, Zhou Y, Childs A M. Entanglement accelerates quantum simulation. Nature Physics, 2025, 21(8): 1338\u20131345","journal-title":"Nature Physics"},{"issue":"4","key":"51794_CR31","doi-asserted-by":"publisher","first-page":"eaau8342","DOI":"10.1126\/sciadv.aau8342","volume":"5","author":"M Heyl","year":"2019","unstructured":"Heyl M, Hauke P, Zoller P. Quantum localization bounds trotter errors in digital quantum simulation. Science Advances, 2019, 5(4): eaau8342","journal-title":"Science Advances"},{"issue":"1","key":"51794_CR32","doi-asserted-by":"publisher","first-page":"78","DOI":"10.1038\/s41534-019-0192-5","volume":"5","author":"L M Sieberer","year":"2019","unstructured":"Sieberer L M, Olsacher T, Elben A, Heyl M, Hauke P, Haake F, Zoller P. Digital quantum simulation, Trotter errors, and quantum chaos of the kicked top. npj Quantum Information, 2019, 5(1): 78","journal-title":"npj Quantum Information"},{"issue":"1","key":"51794_CR33","doi-asserted-by":"publisher","first-page":"6507","DOI":"10.1038\/s41467-024-50750-x","volume":"15","author":"R Trivedi","year":"2024","unstructured":"Trivedi R, Franco Rubio A, Cirac J I. Quantum advantage and stability to errors in analogue quantum simulators. Nature Communications, 2024, 15(1): 6507","journal-title":"Nature Communications"},{"issue":"1","key":"51794_CR34","doi-asserted-by":"publisher","first-page":"340","DOI":"10.1038\/s42005-025-02260-5","volume":"8","author":"W Yu","year":"2025","unstructured":"Yu W, Xu J, Zhao Q. Observable-driven speed-ups in quantum simulations. Communications Physics, 2025, 8(1): 340","journal-title":"Communications Physics"},{"key":"51794_CR35","unstructured":"Feng T, Cao Y, Zhao Q. Trotterization, operator scrambling, and entanglement. 2025, arXiv preprint arXiv: 2506.23345"},{"issue":"2","key":"51794_CR36","doi-asserted-by":"publisher","first-page":"020330","DOI":"10.1103\/PRXQuantum.5.020330","volume":"5","author":"K Nakaji","year":"2024","unstructured":"Nakaji K, Bagherimehrab M, Aspuru-Guzik A. High-order randomized compiler for Hamiltonian simulation. PRX Quantum, 2024, 5(2): 020330","journal-title":"PRX Quantum"},{"issue":"1","key":"51794_CR37","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1038\/s41534-024-00877-y","volume":"10","author":"E Granet","year":"2024","unstructured":"Granet E, Dreyer H. Hamiltonian dynamics on digital quantum computers without discretization error. npj Quantum Information, 2024, 10(1): 82","journal-title":"npj Quantum Information"},{"key":"51794_CR38","unstructured":"Low G H, Kliuchnikov V, Wiebe N. Well-conditioned multiproduct Hamiltonian simulation. 2019, arXiv preprint arXiv: 1907.11679"},{"key":"51794_CR39","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.22331\/q-2023-07-25-1067","volume":"7","author":"A Carrera Vazquez","year":"2023","unstructured":"Carrera Vazquez A, Egger D J, Ochsner D, Woerner S. Well-conditioned multi-product formulas for hardware-friendly Hamiltonian simulation. Quantum, 2023, 7: 1067","journal-title":"Quantum"},{"issue":"3","key":"51794_CR40","doi-asserted-by":"publisher","first-page":"033309","DOI":"10.1103\/PhysRevResearch.6.033309","volume":"6","author":"S Zhuk","year":"2024","unstructured":"Zhuk S, Robertson N F, Bravyi S. Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation. Physical Review Research, 2024, 6(3): 033309","journal-title":"Physical Review Research"},{"issue":"1","key":"51794_CR41","doi-asserted-by":"publisher","first-page":"010359","DOI":"10.1103\/PRXQuantum.6.010359","volume":"6","author":"P Zeng","year":"2025","unstructured":"Zeng P, Sun J, Jiang L, Zhao Q. Simple and high-precision Hamiltonian simulation by compensating trotter error with linear combination of unitary operations. PRX Quantum, 2025, 6(1): 010359","journal-title":"PRX Quantum"},{"issue":"1","key":"51794_CR42","doi-asserted-by":"publisher","first-page":"012334","DOI":"10.1103\/PhysRevA.99.012334","volume":"99","author":"S Endo","year":"2019","unstructured":"Endo S, Zhao Q, Li Y, Benjamin S, Yuan X. Mitigating algorithmic errors in a Hamiltonian simulation. Physical Review A, 2019, 99(1): 012334","journal-title":"Physical Review A"},{"issue":"3","key":"51794_CR43","doi-asserted-by":"publisher","first-page":"030325","DOI":"10.1103\/kw39-yxq5","volume":"6","author":"J D Watson","year":"2025","unstructured":"Watson J D, Watkins J. Exponentially reduced circuit depths using trotter error mitigation. PRX Quantum, 2025, 6(3): 030325","journal-title":"PRX Quantum"},{"key":"51794_CR44","unstructured":"Watson J D. Randomly compiled quantum simulation with exponentially reduced circuit depths. 2025, arXiv preprint arXiv: 2411.04240"},{"issue":"1","key":"51794_CR45","doi-asserted-by":"publisher","first-page":"010501","DOI":"10.1103\/PhysRevLett.118.010501","volume":"118","author":"G H Low","year":"2017","unstructured":"Low G H, Chuang I L. Optimal Hamiltonian simulation by quantum signal processing. Physical Review Letters, 2017, 118(1): 010501","journal-title":"Physical Review Letters"},{"key":"51794_CR46","first-page":"792","volume-title":"Proceedings of the 56th Annual Symposium on Foundations of Computer Science","author":"D W Berry","year":"2015","unstructured":"Berry D W, Childs A M, Kothari R. Hamiltonian simulation with nearly optimal dependence on all parameters. In: Proceedings of the 56th Annual Symposium on Foundations of Computer Science. 2015, 792\u2013809"},{"issue":"6","key":"51794_CR47","doi-asserted-by":"publisher","first-page":"1000","DOI":"10.1109\/TCAD.2005.855930","volume":"25","author":"V V Shende","year":"2006","unstructured":"Shende V V, Bullock S S, Markov I L. Synthesis of quantum-logic circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2006, 25(6): 1000\u20131010","journal-title":"IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems"},{"key":"51794_CR48","doi-asserted-by":"publisher","first-page":"534","DOI":"10.22331\/q-2021-08-31-534","volume":"5","author":"Q Zhao","year":"2021","unstructured":"Zhao Q, Yuan X. Exploiting anticommutation in Hamiltonian simulation. Quantum, 2021, 5: 534","journal-title":"Quantum"},{"issue":"4","key":"51794_CR49","doi-asserted-by":"publisher","first-page":"041067","DOI":"10.1103\/PhysRevX.6.041067","volume":"6","author":"G H Low","year":"2016","unstructured":"Low G H, Yoder T J, Chuang I L. Methodology of resonant equiangular composite quantum gates. Physical Review X, 2016, 6(4): 041067","journal-title":"Physical Review X"},{"key":"51794_CR50","first-page":"59","volume-title":"Proceedings of the 35th Annual ACM Symposium on Theory of Computing","author":"A M Childs","year":"2003","unstructured":"Childs A M, Cleve R, Deotto E, Farhi E, Gutmann S, Spielman D A. Exponential algorithmic speedup by a quantum walk. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing. 2003, 59\u201368"},{"key":"51794_CR51","unstructured":"Low G H, Chuang I L. Hamiltonian simulation by uniform spectral amplification. 2017, arXiv preprint arXiv: 1707.05391"},{"issue":"6","key":"51794_CR52","doi-asserted-by":"publisher","first-page":"FOCS18-250","DOI":"10.1137\/18M1231511","volume":"52","author":"J Haah","year":"2023","unstructured":"Haah J, Hastings M B, Kothari R, Low G H. Quantum algorithm for simulating real time evolution of lattice Hamiltonians. SIAM Journal on Computing, 2023, 52(6): FOCS18-250\u2013FOCS18-284","journal-title":"SIAM Journal on Computing"},{"issue":"3","key":"51794_CR53","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/BF01645779","volume":"28","author":"E H Lieb","year":"1972","unstructured":"Lieb E H, Robinson D W. The finite group velocity of quantum spin systems. Communications in Mathematical Physics, 1972, 28(3): 251\u2013257","journal-title":"Communications in Mathematical Physics"},{"issue":"1","key":"51794_CR54","doi-asserted-by":"publisher","first-page":"012612","DOI":"10.1103\/PhysRevA.110.012612","volume":"110","author":"D W Berry","year":"2024","unstructured":"Berry D W, Motlagh D, Pantaleoni G, Wiebe N. Doubling the efficiency of Hamiltonian simulation via generalized quantum signal processing. Physical Review A, 2024, 110(1): 012612","journal-title":"Physical Review A"},{"issue":"1","key":"51794_CR55","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1038\/s41534-022-00636-x","volume":"8","author":"Y Dong","year":"2022","unstructured":"Dong Y, Whaley K B, Lin L. A quantum Hamiltonian simulation benchmark. npj Quantum Information, 2022, 8(1): 131","journal-title":"npj Quantum Information"},{"key":"51794_CR56","doi-asserted-by":"publisher","first-page":"190","DOI":"10.22331\/q-2019-10-07-190","volume":"3","author":"J Haah","year":"2019","unstructured":"Haah J. Product decomposition of periodic functions in quantum signal processing. Quantum, 2019, 3: 190","journal-title":"Quantum"},{"key":"51794_CR57","unstructured":"Chao R, Ding D, Gilyen A, Huang C, Szegedy M. Finding angles for quantum signal processing with machine precision. 2020, arXiv preprint arXiv: 2003.02831v2"},{"issue":"4","key":"51794_CR58","doi-asserted-by":"publisher","first-page":"042419","DOI":"10.1103\/PhysRevA.103.042419","volume":"103","author":"Y Dong","year":"2021","unstructured":"Dong Y, Meng X, Whaley K B, Lin L. Efficient phase-factor evaluation in quantum signal processing. Physical Review A, 2021, 103(4): 042419","journal-title":"Physical Review A"},{"key":"51794_CR59","doi-asserted-by":"publisher","first-page":"020368","DOI":"10.1103\/PRXQuantum.5.020368","volume":"5","author":"D Motlagh","year":"2024","unstructured":"Motlagh D, Wiebe N. Generalized quantum signal processing. PRX Quantum, 2024, 5: 020368","journal-title":"PRX Quantum"},{"issue":"23","key":"51794_CR60","doi-asserted-by":"publisher","first-page":"230504","DOI":"10.1103\/PhysRevLett.129.230504","volume":"129","author":"X M Zhang","year":"2022","unstructured":"Zhang X M, Li T, Yuan X. Quantum state preparation with optimal circuit depth: implementations and applications. Physical Review Letters, 2022, 129(23): 230504","journal-title":"Physical Review Letters"},{"key":"51794_CR61","doi-asserted-by":"publisher","first-page":"1228","DOI":"10.22331\/q-2024-01-15-1228","volume":"8","author":"Z Zhang","year":"2024","unstructured":"Zhang Z, Wang Q, Ying M. Parallel quantum algorithm for Hamiltonian simulation. Quantum, 2024, 8: 1228","journal-title":"Quantum"},{"issue":"4","key":"51794_CR62","doi-asserted-by":"publisher","first-page":"041015","DOI":"10.1103\/PhysRevX.8.041015","volume":"8","author":"R Babbush","year":"2018","unstructured":"Babbush R, Gidney C, Berry D W, Wiebe N, McClean J, Paler A, Fowler A, Neven H. Encoding electronic spectra in quantum circuits with linear T complexity. Physical Review X, 2018, 8(4): 041015","journal-title":"Physical Review X"},{"key":"51794_CR63","doi-asserted-by":"publisher","first-page":"380","DOI":"10.22331\/q-2021-01-12-380","volume":"5","author":"K Wan","year":"2021","unstructured":"Wan K. Exponentially faster implementations of Select(H) for fermionic Hamiltonians. Quantum, 2021, 5: 380","journal-title":"Quantum"},{"key":"51794_CR64","doi-asserted-by":"publisher","first-page":"637","DOI":"10.22331\/q-2022-02-02-637","volume":"6","author":"R Meister","year":"2022","unstructured":"Meister R, Benjamin S C, Campbell E T. Tailoring term truncations for electronic structure calculations using a linear combination of unitaries. Quantum, 2022, 6: 637","journal-title":"Quantum"},{"issue":"1","key":"51794_CR65","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1038\/s41534-025-01003-2","volume":"11","author":"J M Martyn","year":"2025","unstructured":"Martyn J M, Rall P. Halving the cost of quantum algorithms with randomization. npj Quantum Information, 2025, 11(1): 47","journal-title":"npj Quantum Information"},{"issue":"1","key":"51794_CR66","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1038\/s41534-023-00762-0","volume":"9","author":"Y Kikuchi","year":"2023","unstructured":"Kikuchi Y, Mc Keever C, Coopmans L, Lubasch M, Benedetti M. Realization of quantum signal processing on a noisy quantum computer. npj Quantum Information, 2023, 9(1): 93","journal-title":"npj Quantum Information"},{"issue":"2","key":"51794_CR67","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1364\/ON.11.2.000011","volume":"11","author":"R P Feynman","year":"1985","unstructured":"Feynman R P. Quantum mechanical computers. Optics News, 1985, 11(2): 11\u201320","journal-title":"Optics News"},{"issue":"5","key":"51794_CR68","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1137\/S0097539796300921","volume":"26","author":"E Bernstein","year":"1997","unstructured":"Bernstein E, Vazirani U. Quantum complexity theory. SIAM Journal on Computing, 1997, 26(5): 1411\u20131473","journal-title":"SIAM Journal on Computing"},{"key":"51794_CR69","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/047","volume-title":"Classical and Quantum Computation","author":"A Y Kitaev","year":"2002","unstructured":"Kitaev A Y, Shen A H, Vyalyi M N. Classical and Quantum Computation. Providence: American Mathematical Society, 2002"},{"key":"51794_CR70","unstructured":"Wocjan P, Zhang S. Several natural BQP-complete problems. 2006, arXiv preprint arXiv: quant-ph\/0606179"},{"key":"51794_CR71","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804090","volume-title":"Computational Complexity: A Modern Approach","author":"S Arora","year":"2009","unstructured":"Arora S, Barak B. Computational Complexity: A Modern Approach. New York: Cambridge University Press, 2009"},{"issue":"18","key":"51794_CR72","doi-asserted-by":"publisher","first-page":"187902","DOI":"10.1103\/PhysRevLett.92.187902","volume":"92","author":"M Christandl","year":"2004","unstructured":"Christandl M, Datta N, Ekert A, Landahl A J. Perfect state transfer in quantum spin networks. Physical Review Letters, 2004, 92(18): 187902","journal-title":"Physical Review Letters"},{"issue":"4","key":"51794_CR73","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1142\/S0219749910006514","volume":"8","author":"A Kay","year":"2010","unstructured":"Kay A. Perfect, efficient, state transfer and its application as a constructive tool. International Journal of Quantum Information, 2010, 8(4): 641\u2013676","journal-title":"International Journal of Quantum Information"},{"key":"51794_CR74","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198787488.001.0001","volume-title":"Quantum Information Science","author":"R Manenti","year":"2023","unstructured":"Manenti R, Motta M. Quantum Information Science. Oxford: Oxford University Press, 2023"},{"key":"51794_CR75","volume-title":"Lecture Notes on Quantum Algorithms","author":"A M Childs","year":"2025","unstructured":"Childs A M. Lecture Notes on Quantum Algorithms. University of Maryland, 2025. Available at: https:\/\/www.cs.umd.edu\/~amchilds\/qa\/qa.pdf"},{"issue":"1","key":"51794_CR76","doi-asserted-by":"publisher","first-page":"015004","DOI":"10.1088\/2058-9565\/2\/1\/015004","volume":"2","author":"D Hangleiter","year":"2017","unstructured":"Hangleiter D, Kliesch M, Schwarz M, Eisert J. Direct certification of a class of quantum simulations. Quantum Science and Technology, 2017, 2(1): 015004","journal-title":"Quantum Science and Technology"},{"issue":"2","key":"51794_CR77","doi-asserted-by":"publisher","first-page":"021010","DOI":"10.1103\/PhysRevX.8.021010","volume":"8","author":"J Bermejo-Vega","year":"2018","unstructured":"Bermejo-Vega J, Hangleiter D, Schwarz M, Raussendorf R, Eisert J. Architectures for quantum simulation showing a quantum speedup. Physical Review X, 2018, 8(2): 021010","journal-title":"Physical Review X"},{"issue":"25","key":"51794_CR78","doi-asserted-by":"publisher","first-page":"250501","DOI":"10.1103\/PhysRevLett.125.250501","volume":"125","author":"J Haferkamp","year":"2020","unstructured":"Haferkamp J, Hangleiter D, Bouland A, Fefferman B, Eisert J, Bermejo-Vega J. Closing gaps of a quantum advantage with short-time Hamiltonian dynamics. Physical Review Letters, 2020, 125(25): 250501","journal-title":"Physical Review Letters"},{"issue":"1","key":"51794_CR79","doi-asserted-by":"publisher","first-page":"010341","DOI":"10.1103\/PRXQuantum.6.010341","volume":"6","author":"Z Liu","year":"2025","unstructured":"Liu Z, Devulapalli D, Hangleiter D, Liu Y K, Koll\u00e1r A J, Gorshkov A V, Childs A M. Efficiently verifiable quantum advantage on near-term analog quantum simulators. PRX Quantum, 2025, 6(1): 010341","journal-title":"PRX Quantum"},{"key":"51794_CR80","unstructured":"Quek Y. Quantum advantage from random geometrically-two-local Hamiltonian dynamics. 2025, arXiv preprint arXiv: 2510.06321"},{"issue":"24","key":"51794_CR81","doi-asserted-by":"publisher","first-page":"5442","DOI":"10.1103\/PhysRevLett.81.5442","volume":"81","author":"E Farhi","year":"1998","unstructured":"Farhi E, Goldstone J, Gutmann S, Sipser M. Limit on the speed of quantum computation in determining parity. Physical Review Letters, 1998, 81(24): 5442","journal-title":"Physical Review Letters"},{"issue":"4","key":"51794_CR82","doi-asserted-by":"publisher","first-page":"778","DOI":"10.1145\/502090.502097","volume":"48","author":"R Beals","year":"2001","unstructured":"Beals R, Buhrman H, Cleve R, Mosca M, de Wolf R. Quantum lower bounds by polynomials. Journal of the ACM, 2001, 48(4): 778\u2013797","journal-title":"Journal of the ACM"},{"key":"51794_CR83","first-page":"283","volume-title":"Proceedings of the 46th Annual ACM Symposium on Theory of Computing","author":"D W Berry","year":"2014","unstructured":"Berry D W, Childs A M, Cleve R, Kothari R, Somma R D. Exponential improvement in precision for simulating sparse Hamiltonians. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing. 2014, 283\u2013292"},{"issue":"1","key":"51794_CR84","doi-asserted-by":"publisher","first-page":"1572","DOI":"10.1038\/s41467-017-01637-7","volume":"8","author":"Y Atia","year":"2017","unstructured":"Atia Y, Aharonov D. Fast-forwarding of Hamiltonians and exponentially precise measurements. Nature Communications, 2017, 8(1): 1572","journal-title":"Nature Communications"},{"key":"51794_CR85","first-page":"33","volume-title":"Proceedings of the 38th Computational Complexity Conference","author":"N H Chia","year":"2023","unstructured":"Chia N H, Chung K M, Hsieh Y C, Lin H H, Lin Y T, Shen Y C. On the impossibility of general parallel fast-forwarding of Hamiltonian simulation. In: Proceedings of the 38th Computational Complexity Conference. 2023, 33"},{"key":"51794_CR86","doi-asserted-by":"publisher","first-page":"1409","DOI":"10.22331\/q-2024-07-15-1409","volume":"8","author":"W Gong","year":"2024","unstructured":"Gong W, Zhou S, Li T. Complexity of digital quantum simulation in the low-energy subspace: applications and a lower bound. Quantum, 2024, 8: 1409","journal-title":"Quantum"},{"key":"51794_CR87","doi-asserted-by":"publisher","first-page":"1449","DOI":"10.22331\/q-2024-08-27-1449","volume":"8","author":"A Zlokapa","year":"2024","unstructured":"Zlokapa A, Somma R D. Hamiltonian simulation for low-energy states with optimal time dependence. Quantum, 2024, 8: 1449","journal-title":"Quantum"},{"key":"51794_CR88","doi-asserted-by":"publisher","first-page":"577","DOI":"10.22331\/q-2021-11-15-577","volume":"5","author":"S Gu","year":"2021","unstructured":"Gu S, Somma R D, \u015eahino\u01e7lu B. Fast-forwarding quantum evolution. Quantum, 2021, 5: 577","journal-title":"Quantum"},{"key":"51794_CR89","unstructured":"Feng T, Xu J, Yu W, Ye Z, Yao P, Zhao Q. Distributed quantum simulation. 2024, arXiv preprint arXiv: 2411.02881"},{"key":"51794_CR90","series-title":"Dissertation","volume-title":"Quantum communication","author":"I Kremer","year":"1995","unstructured":"Kremer I. Quantum communication. Hebrew University, Dissertation, 1995"},{"key":"51794_CR91","series-title":"Dissertation","volume-title":"Quantum computing and communication complexity","author":"R de Wolf","year":"2001","unstructured":"de Wolf R. Quantum computing and communication complexity. University of Amsterdam, Dissertation, 2001"},{"key":"51794_CR92","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1007\/978-3-031-20624-5_39","volume-title":"Proceedings of the 15th Latin American Symposium on LATIN 2022: Theoretical Informatics","author":"F Le Gall","year":"2022","unstructured":"Le Gall F, Suruga D. Bounds on oblivious multiparty quantum communication complexity. In: Proceedings of the 15th Latin American Symposium on LATIN 2022: Theoretical Informatics. 2022, 641\u2013657"},{"key":"51794_CR93","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1145\/3313276.3316386","volume-title":"Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing","author":"G H Low","year":"2019","unstructured":"Low G H. Hamiltonian simulation with nearly optimal dependence on spectral norm. In: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing. 2019, 491\u2013502"},{"key":"51794_CR94","doi-asserted-by":"publisher","first-page":"191","DOI":"10.22331\/q-2019-10-07-191","volume":"3","author":"X Yuan","year":"2019","unstructured":"Yuan X, Endo S, Zhao Q, Li Y, Benjamin S C. Theory of variational quantum simulation. Quantum, 2019, 3: 191","journal-title":"Quantum"},{"issue":"12","key":"51794_CR95","doi-asserted-by":"publisher","first-page":"120505","DOI":"10.1103\/PhysRevLett.129.120505","volume":"129","author":"J Sun","year":"2022","unstructured":"Sun J, Endo S, Lin H, Hayden P, Vedral V, Yuan X. Perturbative quantum simulation. Physical Review Letters, 2022, 129(12): 120505","journal-title":"Physical Review Letters"},{"issue":"1","key":"51794_CR96","doi-asserted-by":"publisher","first-page":"2690","DOI":"10.1038\/s41467-025-57451-z","volume":"16","author":"R D Somma","year":"2025","unstructured":"Somma R D, King R, Kothari R, O\u2019Brien T E, Babbush R. Shadow Hamiltonian simulation. Nature Communications, 2025, 16(1): 2690","journal-title":"Nature Communications"},{"key":"51794_CR97","doi-asserted-by":"publisher","first-page":"1266","DOI":"10.22331\/q-2024-02-26-1266","volume":"8","author":"G Rendon","year":"2024","unstructured":"Rendon G, Watkins J, Wiebe N. Improved accuracy for trotter simulations using Chebyshev interpolation. Quantum, 2024, 8: 1266","journal-title":"Quantum"},{"key":"51794_CR98","doi-asserted-by":"crossref","unstructured":"Zhang Y, Zhang X, Sun J, Lin H, Huang Y, Lv D, Yuan X. Fault-tolerant quantum algorithms for quantum molecular systems: a survey. 2025, arXiv preprint arXiv: 2502.02139","DOI":"10.1002\/wcms.70020"},{"issue":"1","key":"51794_CR99","doi-asserted-by":"publisher","first-page":"015003","DOI":"10.1103\/RevModPhys.92.015003","volume":"92","author":"S McArdle","year":"2020","unstructured":"McArdle S, Endo S, Aspuru-Guzik A, Benjamin S C, Yuan X. Quantum computational chemistry. Reviews of Modern Physics, 2020, 92(1): 015003","journal-title":"Reviews of Modern Physics"}],"container-title":["Frontiers of Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11704-026-51794-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11704-026-51794-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11704-026-51794-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T08:03:08Z","timestamp":1788508988000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11704-026-51794-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,4]]},"references-count":99,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2026,10]]}},"alternative-id":["51794"],"URL":"https:\/\/doi.org\/10.1007\/s11704-026-51794-6","relation":{},"ISSN":["2095-2228","2095-2236"],"issn-type":[{"value":"2095-2228","type":"print"},{"value":"2095-2236","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,4]]},"assertion":[{"value":"31 October 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 September 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no competing interests or financial conflicts to disclose.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"2010909"}}