Block-encoding structured matrices for data input in quantum computing
1Riverlane, St. Andrews House, 59 St. Andrews Street, Cambridge CB2 3BZ, United Kingdom
2Dept. of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, United Kingdom
| Published: | 2024-01-11, volume 8, page 1226 |
| Eprint: | arXiv:2302.10949v2 |
| Doi: | https://doi.org/10.22331/q-2024-01-11-1226 |
| Citation: | Quantum 8, 1226 (2024). |
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
The cost of data input can dominate the run-time of quantum algorithms. Here, we consider data input of arithmetically structured matrices via $\textit{block encoding}$ circuits, the input model for the quantum singular value transform and related algorithms. We demonstrate how to construct block encoding circuits based on an arithmetic description of the sparsity and pattern of repeated values of a matrix. We present schemes yielding different subnormalisations of the block encoding; a comparison shows that the best choice depends on the specific matrix. The resulting circuits reduce flag qubit number according to sparsity, and data loading cost according to repeated values, leading to an exponential improvement for certain matrices. We give examples of applying our block encoding schemes to a few families of matrices, including Toeplitz and tridiagonal matrices.
Featured image: Block encoding circuit for the base scheme introduced in this article. The row and column oracles translate between a description of the matrix in terms of repeated elements (d: data index, m: multiplicity index) and a description in terms of row index i and row sparsity index s_r, or column index j and sparsity index s_c. Thereby, both sparsity and repetition of data elements can be harnessed to reduce circuit cost.
Popular summary
In this research article, we present a new set of schemes how data can be loaded into block encodings. Particularly, if the data matrices are structured, i.e. have a certain pattern and/or repeated data elements, our scheme shows how to make use of this structure in order to reduce the cost of data loading. We explain how to construct quantum circuits taking into account and optimising for such structured data. In the future, our work can help to load various data matrices into quantum computers for use in various quantum algorithms, making the most of the data’s structure to reduce the data loading bottleneck.
► BibTeX data
► References
[1] Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge University Press, Cambridge ; New York, 10th anniversary ed edition, 2010. ISBN 978-1-107-00217-3.
[2] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando G. S. L. Brandao, David A. Buell, Brian Burkett, Yu Chen, Zijun Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew P. Harrigan, Michael J. Hartmann, Alan Ho, Markus Hoffmann, Trent Huang, Travis S. Humble, Sergei V. Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul V. Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, David Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod R. McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John C. Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas C. Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin J. Sung, Matthew D. Trevithick, Amit Vainsencher, Benjamin Villalonga, Theodore White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, and John M. Martinis. Quantum supremacy using a programmable superconducting processor. Nature, 574 (7779), October 2019. ISSN 1476-4687. 10.1038/s41586-019-1666-5. URL https://www.nature.com/articles/s41586-019-1666-5.
https://doi.org/10.1038/s41586-019-1666-5
https://www.nature.com/articles/s41586-019-1666-5
[3] IBM. IBM Unveils Breakthrough 127-Qubit Quantum Processor, 2021. URL https://newsroom.ibm.com/2021-11-16-IBM-Unveils-Breakthrough-127-Qubit-Quantum-Processor.
https://newsroom.ibm.com/2021-11-16-IBM-Unveils-Breakthrough-127-Qubit-Quantum-Processor
[4] Yulin Wu, Wan-Su Bao, Sirui Cao, Fusheng Chen, Ming-Cheng Chen, Xiawei Chen, Tung-Hsun Chung, Hui Deng, Yajie Du, Daojin Fan, Ming Gong, Cheng Guo, Chu Guo, Shaojun Guo, Lianchen Han, Linyin Hong, He-Liang Huang, Yong-Heng Huo, Liping Li, Na Li, Shaowei Li, Yuan Li, Futian Liang, Chun Lin, Jin Lin, Haoran Qian, Dan Qiao, Hao Rong, Hong Su, Lihua Sun, Liangyuan Wang, Shiyu Wang, Dachao Wu, Yu Xu, Kai Yan, Weifeng Yang, Yang Yang, Yangsen Ye, Jianghan Yin, Chong Ying, Jiale Yu, Chen Zha, Cha Zhang, Haibin Zhang, Kaili Zhang, Yiming Zhang, Han Zhao, Youwei Zhao, Liang Zhou, Qingling Zhu, Chao-Yang Lu, Cheng-Zhi Peng, Xiaobo Zhu, and Jian-Wei Pan. Strong quantum computational advantage using a superconducting quantum processor. Physical Review Letters, 127 (18): 180501, October 2021. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.127.180501. URL http://arxiv.org/abs/2106.14734. arXiv:2106.14734 [quant-ph].
https://doi.org/10.1103/PhysRevLett.127.180501
arXiv:2106.14734
[5] Scott Aaronson. How Much Structure Is Needed for Huge Quantum Speedups?, September 2022. URL http://arxiv.org/abs/2209.06930. arXiv:2209.06930 [quant-ph].
arXiv:2209.06930
[6] Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong, Alexander M. Dalzell, Ashutosh Kumar, Phillip Helms, Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, Michael Kastoryano, Ryan Babbush, John Preskill, David R. Reichman, Earl T. Campbell, Edward F. Valeev, Lin Lin, and Garnet Kin-Lic Chan. Is there evidence for exponential quantum advantage in quantum chemistry?, November 2022. URL . arXiv:2208.02199 [physics, physics:quant-ph].
https://doi.org/10.1038/s41467-023-37587-6
arXiv:2208.02199
[7] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193–204, June 2019. 10.1145/3313276.3316366. URL http://arxiv.org/abs/1806.01838. arXiv: 1806.01838.
https://doi.org/10.1145/3313276.3316366
arXiv:1806.01838
[8] John M. Martyn, Zane M. Rossi, Andrew K. Tan, and Isaac L. Chuang. Grand Unification of Quantum Algorithms. PRX Quantum, 2 (4): 040203, December 2021. 10.1103/PRXQuantum.2.040203. URL https://link.aps.org/doi/10.1103/PRXQuantum.2.040203. Publisher: American Physical Society.
https://doi.org/10.1103/PRXQuantum.2.040203
[9] Scott Aaronson. Read the fine print. Nature Physics, 11 (4), April 2015. ISSN 1745-2481. 10.1038/nphys3272. URL https://www.nature.com/articles/nphys3272.
https://doi.org/10.1038/nphys3272
https://www.nature.com/articles/nphys3272
[10] B. David Clader, Alexander M. Dalzell, Nikitas Stamatopoulos, Grant Salton, Mario Berta, and William J. Zeng. Quantum Resources Required to Block-Encode a Matrix of Classical Data. arXiv, June 2022. URL . arXiv:2206.03505 [quant-ph].
https://doi.org/10.1109/TQE.2022.3231194
arXiv:2206.03505
[11] Shantanav Chakraborty, András Gilyén, and Stacey Jeffery. The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation. arXiv:1804.01973 [quant-ph], page 14 pages, 2019. 10.4230/LIPIcs.ICALP.2019.33. URL http://arxiv.org/abs/1804.01973. arXiv: 1804.01973.
https://doi.org/10.4230/LIPIcs.ICALP.2019.33
arXiv:1804.01973
[12] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum random access memory. Physical Review Letters, 100 (16): 160501, April 2008. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.100.160501. URL http://arxiv.org/abs/0708.1879. arXiv:0708.1879 [quant-ph].
https://doi.org/10.1103/PhysRevLett.100.160501
arXiv:0708.1879
[13] 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, April 2021. ISSN 2691-3399. 10.1103/PRXQuantum.2.020311. URL http://arxiv.org/abs/2012.05340. arXiv:2012.05340 [quant-ph].
https://doi.org/10.1103/PRXQuantum.2.020311
arXiv:2012.05340
[14] 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, December 2022. 10.22331/q-2022-12-13-876. URL https://quantum-journal.org/papers/q-2022-12-13-876/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.
https://doi.org/10.22331/q-2022-12-13-876
https://quantum-journal.org/papers/q-2022-12-13-876/
[15] Daan Camps, Lin Lin, Roel Van Beeumen, and Chao Yang. Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrice. arXiv:2203.10236 [quant-ph], March 2022. URL http://arxiv.org/abs/2203.10236. arXiv: 2203.10236.
arXiv:2203.10236
[16] Guang Hao Low and Isaac L. Chuang. Hamiltonian Simulation by Qubitization. Quantum, 3: 163, July 2019. ISSN 2521-327X. 10.22331/q-2019-07-12-163. URL http://arxiv.org/abs/1610.06546. arXiv: 1610.06546.
https://doi.org/10.22331/q-2019-07-12-163
arXiv:1610.06546
[17] Ryan Babbush, Craig Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven. Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity. Physical Review X, 8 (4): 041015, October 2018. 10.1103/PhysRevX.8.041015. URL https://link.aps.org/doi/10.1103/PhysRevX.8.041015. Publisher: American Physical Society.
https://doi.org/10.1103/PhysRevX.8.041015
[18] 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, December 2019. ISSN 2521-327X. 10.22331/q-2019-12-02-208. URL http://arxiv.org/abs/1902.02134. arXiv:1902.02134 [physics, physics:quant-ph].
https://doi.org/10.22331/q-2019-12-02-208
arXiv:1902.02134
[19] 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, July 2021. ISSN 2691-3399. 10.1103/PRXQuantum.2.030305. URL http://arxiv.org/abs/2011.03494. arXiv: 2011.03494.
https://doi.org/10.1103/PRXQuantum.2.030305
arXiv:2011.03494
[20] 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, October 2022. URL . arXiv:2210.02403 [quant-ph].
https://doi.org/10.1103/PhysRevResearch.5.013200
arXiv:2210.02403
[21] M. Szegedy. Quantum speed-up of Markov chain based algorithms. In 45th Annual IEEE Symposium on Foundations of Computer Science, pages 32–41, October 2004. 10.1109/FOCS.2004.53. ISSN: 0272-5428.
https://doi.org/10.1109/FOCS.2004.53
[22] Dominic W. Berry, Andrew M. Childs, and Robin Kothari. Hamiltonian simulation with nearly optimal dependence on all parameters. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 792–809, October 2015. 10.1109/FOCS.2015.54. URL http://arxiv.org/abs/1501.01715. arXiv:1501.01715 [quant-ph].
https://doi.org/10.1109/FOCS.2015.54
arXiv:1501.01715
[23] Yuta Kikuchi, Conor Mc Keever, Luuk Coopmans, Michael Lubasch, and Marcello Benedetti. Realization of quantum signal processing on a noisy quantum computer. npj Quantum Information, 9 (1), September 2023. ISSN 2056-6387. 10.1038/s41534-023-00762-0. URL http://dx.doi.org/10.1038/s41534-023-00762-0.
https://doi.org/10.1038/s41534-023-00762-0
[24] Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52 (4): R2493–R2496, October 1995. ISSN 1050-2947, 1094-1622. 10.1103/PhysRevA.52.R2493. URL https://link.aps.org/doi/10.1103/PhysRevA.52.R2493.
https://doi.org/10.1103/PhysRevA.52.R2493
[25] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. Surface codes: Towards practical large-scale quantum computation. Physical Review A, 86 (3): 032324, September 2012. 10.1103/PhysRevA.86.032324. URL https://link.aps.org/doi/10.1103/PhysRevA.86.032324. Publisher: American Physical Society.
https://doi.org/10.1103/PhysRevA.86.032324
[26] Sergei Bravyi and Alexei Kitaev. Universal Quantum Computation with ideal Clifford gates and noisy ancillas. arXiv:quant-ph/0403025, December 2004. 10.1103/PhysRevA.71.022316. URL http://arxiv.org/abs/quant-ph/0403025. arXiv: quant-ph/0403025.
https://doi.org/10.1103/PhysRevA.71.022316
arXiv:quant-ph/0403025
[27] Joe O'Gorman and Earl T. Campbell. Quantum computation with realistic magic state factories. Physical Review A, 95 (3): 032338, March 2017. ISSN 2469-9926, 2469-9934. 10.1103/PhysRevA.95.032338. URL http://arxiv.org/abs/1605.07197. arXiv:1605.07197 [quant-ph].
https://doi.org/10.1103/PhysRevA.95.032338
arXiv:1605.07197
[28] Earl T. Campbell, Barbara M. Terhal, and Christophe Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549 (7671): 172–179, September 2017. ISSN 0028-0836, 1476-4687. 10.1038/nature23460. URL http://arxiv.org/abs/1612.07330. arXiv: 1612.07330.
https://doi.org/10.1038/nature23460
arXiv:1612.07330
[29] Austin G. Fowler and Craig Gidney. Low overhead quantum computation using lattice surgery. arXiv:1808.06709 [quant-ph], August 2019. URL http://arxiv.org/abs/1808.06709. arXiv: 1808.06709.
arXiv:1808.06709
[30] Nick S. Blunt, Joan Camps, Ophelia Crawford, Róbert Izsák, Sebastian Leontica, Arjun Mirani, Alexandra E. Moylett, Sam A. Scivier, Christoph Sünderhauf, Patrick Schopf, Jacob M. Taylor, and Nicole Holzmann. Perspective on the Current State-of-the-Art of Quantum Computing for Drug Discovery Applications. Journal of Chemical Theory and Computation, 18 (12): 7001–7023, December 2022. ISSN 1549-9618. 10.1021/acs.jctc.2c00574. URL https://doi.org/10.1021/acs.jctc.2c00574. Publisher: American Chemical Society.
https://doi.org/10.1021/acs.jctc.2c00574
[31] Craig Gidney. Halving the cost of quantum addition. Quantum, 2: 74, June 2018. 10.22331/q-2018-06-18-74. URL https://quantum-journal.org/papers/q-2018-06-18-74/. Publisher: Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften.
https://doi.org/10.22331/q-2018-06-18-74
https://quantum-journal.org/papers/q-2018-06-18-74/
[32] 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, November 2020. 10.1103/PRXQuantum.1.020312. URL https://link.aps.org/doi/10.1103/PRXQuantum.1.020312. Publisher: American Physical Society.
https://doi.org/10.1103/PRXQuantum.1.020312
[33] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer. Trading T-gates for dirty qubits in state preparation and unitary synthesis, December 2018. URL http://arxiv.org/abs/1812.00954. arXiv:1812.00954 [quant-ph] type: article.
arXiv:1812.00954
[34] D.K Callebaut. Generalization of the cauchy-schwarz inequality. Journal of Mathematical Analysis and Applications, 12 (3): 491–494, 1965. ISSN 0022-247X. https://doi.org/10.1016/0022-247X(65)90016-8. URL https://www.sciencedirect.com/science/article/pii/0022247X65900168.
https://doi.org/10.1016/0022-247X(65)90016-8
https://www.sciencedirect.com/science/article/pii/0022247X65900168
[35] Thomas G. Draper. Addition on a Quantum Computer. arXiv:quant-ph/0008033, August 2000. URL http://arxiv.org/abs/quant-ph/0008033. arXiv: quant-ph/0008033.
arXiv:quant-ph/0008033
[36] Steven A. Cuccaro, Thomas G. Draper, Samuel A. Kutin, and David Petrie Moulton. A new quantum ripple-carry addition circuit. arXiv:quant-ph/0410184, October 2004. URL http://arxiv.org/abs/quant-ph/0410184. arXiv: quant-ph/0410184.
arXiv:quant-ph/0410184
[37] Lidia Ruiz-Perez and Juan Carlos Garcia-Escartin. Quantum arithmetic with the Quantum Fourier Transform. Quantum Information Processing, 16 (6): 152, June 2017. ISSN 1570-0755, 1573-1332. 10.1007/s11128-017-1603-1. URL http://arxiv.org/abs/1411.5949. arXiv:1411.5949 [quant-ph].
https://doi.org/10.1007/s11128-017-1603-1
arXiv:1411.5949
[38] A. Mahasinghe and J. B. Wang. Efficient quantum circuits for Toeplitz and Hankel matrices. Journal of Physics A: Mathematical and Theoretical, 49 (27): 275301, July 2016. ISSN 1751-8113, 1751-8121. 10.1088/1751-8113/49/27/275301. URL http://arxiv.org/abs/1605.07710. arXiv:1605.07710 [quant-ph].
https://doi.org/10.1088/1751-8113/49/27/275301
arXiv:1605.07710
[39] Daan Camps and Roel Van Beeumen. FABLE: Fast Approximate Quantum Circuits for Block-Encodings. April 2022. URL . arXiv:2205.00081 [quant-ph].
https://doi.org/10.1109/QCE53715.2022.00029
arXiv:2205.00081
[40] Mikko Mottonen, Juha J. Vartiainen, Ville Bergholm, and Martti M. Salomaa. Quantum Circuits for General Multiqubit Gates. Physical Review Letters, 93 (13): 130502, September 2004. ISSN 0031-9007, 1079-7114. 10.1103/PhysRevLett.93.130502. URL http://arxiv.org/abs/quant-ph/0404089. arXiv:quant-ph/0404089.
https://doi.org/10.1103/PhysRevLett.93.130502
arXiv:quant-ph/0404089
[41] Vivek V. Shende, Stephen S. Bullock, and Igor L. Markov. Synthesis of Quantum Logic Circuits. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 25 (6): 1000–1010, June 2006. ISSN 0278-0070, 1937-4151. 10.1109/TCAD.2005.855930. URL http://arxiv.org/abs/quant-ph/0406176. arXiv:quant-ph/0406176.
https://doi.org/10.1109/TCAD.2005.855930
arXiv:quant-ph/0406176
[42] Neil J. Ross and Peter Selinger. Optimal ancilla-free Clifford+T approximation of z-rotations, June 2016. URL http://arxiv.org/abs/1403.2975. arXiv:1403.2975 [quant-ph].
arXiv:1403.2975
[43] Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, and Matthias Troyer. Quantum computing enhanced computational catalysis. Physical Review Research, 3 (3), July 2021. ISSN 2643-1564. 10.1103/PhysRevResearch.3.033055. URL http://arxiv.org/abs/2007.14460. arXiv:2007.14460 [physics, physics:quant-ph].
https://doi.org/10.1103/PhysRevResearch.3.033055
arXiv:2007.14460
[44] Guang Hao Low. Halving the cost of quantum multiplexed rotations. arXiv:2110.13439 [quant-ph], October 2021. URL http://arxiv.org/abs/2110.13439. arXiv: 2110.13439.
arXiv:2110.13439
[45] Guang Hao Low and Isaac L. Chuang. Hamiltonian Simulation by Uniform Spectral Amplification, July 2017. URL http://arxiv.org/abs/1707.05391. arXiv:1707.05391 [quant-ph].
arXiv:1707.05391
[46] Yulong Dong, Xiang Meng, K. Birgitta Whaley, and Lin Lin. Efficient phase-factor evaluation in quantum signal processing. arXiv:2002.11649 [physics, physics:quant-ph], July 2021. 10.1103/PhysRevA.103.042419. URL http://arxiv.org/abs/2002.11649. arXiv: 2002.11649.
https://doi.org/10.1103/PhysRevA.103.042419
arXiv:2002.11649
Cited by
[1] J.-T. Bu, Lei Zhang, Zhan Yu, Jing-Bo Wang, W.-Q. Ding, W.-F. Yuan, B. Wang, H.-J. Du, W.-J. Chen, L. Chen, J.-W. Zhang, J.-C. Li, F. Zhou, Xin Wang, and M. Feng, "Exploring the experimental limit of deep quantum signal processing using a trapped-ion simulator", Physical Review Applied 23 3, 034073 (2025).
[2] 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.
[3] Dong An, Andrew M. Childs, and Lin Lin, "Quantum Algorithm for Linear Non-unitary Dynamics with Near-Optimal Dependence on All Parameters", Communications in Mathematical Physics 407 1, 19 (2026).
[4] Anna Bernasconi, Alessandro Berti, Gianna Maria del Corso, and Alessandro Poggiali, "Quantum Subroutine for Efficient Matrix Multiplication", IEEE Access 12, 116274 (2024).
[5] Almudena Carrera Vazquez and Aleksandros Sobczyk, "Spectral Gaps with Quantum Counting Queries and Oblivious State Preparation", Quantum 10, 2067 (2026).
[6] David Jennings, Matteo Lostaglio, Sam Pallister, Andrew T. Sornborger, and Yiğit Subaşı, "Randomized Adiabatic Quantum Linear Solver Algorithm with Optimal Complexity Scaling and Detailed Running Costs", PRX Quantum 6 4, 040373 (2025).
[7] Matteo Scandi and Álvaro M. Alhambra, "Thermalization in Open Many-Body Systems and KMS Detailed Balance", Physical Review X 16 1, 011040 (2026).
[8] Dong An, Akwum Onwunta, and Gengzhi Yang, "Fast-forwarding quantum algorithms for linear dissipative differential equations", Quantum 10, 1986 (2026).
[9] Cyrill Bösch, Malte Schade, Giacomo Aloisi, Scott D. Keating, and Andreas Fichtner, "Quantum wave simulation with sources and loss functions", Physical Review Research 7 3, 033225 (2025).
[10] Charles Yuan, "Cobble: Compiling Block Encodings for Quantum Computational Linear Algebra", Proceedings of the ACM on Programming Languages 10 PLDI, 200 (2026).
[11] Eky Febrianto, Yiren Wang, Burigede Liu, Michael Ortiz, and Fehmi Cirak, "A quantum spectral method for non-periodic boundary value problems", Computer Methods in Applied Mechanics and Engineering 457, 118934 (2026).
[12] Zexian Li, Xiao-Ming Zhang, Chunlin Yang, and Guofeng Zhang, "Binary Tree Block Encoding of Classical Matrix", IEEE Transactions on Quantum Engineering 7, 1 (2026).
[13] Austin Pechan, John Golden, and Daniel O’Malley, "Block encoding of the three-dimensional heterogeneous Poisson equation with application to fracture flow", Physical Review Applied 25 4, 044038 (2026).
[14] Cezary Pilaszewicz, 2026 IEEE Conference on Artificial Intelligence (CAI) 1968 (2026) ISBN:979-8-3315-6039-3.
[15] Arnab Roy, Joseph L. Pachuau, and Anish Kumar Saha, "Quantum Algorithm for Numerical Computation Using Post-selection Measurement", Iranian Journal of Science and Technology, Transactions of Electrical Engineering 50 2, 1309 (2026).
[16] Andreas Sturm and Niclas Schillo, "Efficient and Explicit Block Encoding of Finite Difference Discretizations of the Laplacian", ACM Transactions on Quantum Computing 3837860 (2026).
[17] Maria Tudorovskaya and David Muñoz Ramo, "Quantum computing simulation of a mixed spin-boson Hamiltonian and its performance for a cavity quantum electrodynamics problem", Physical Review A 109 3, 032612 (2024).
[18] Allan Tosta, Thais de Lima Silva, Giancarlo Camilo, and Leandro Aolita, "Randomized semi-quantum matrix processing", npj Quantum Information 10 1, 93 (2024).
[19] Yasunori Lee and Keita Kanno, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 2170 (2025) ISBN:979-8-3315-5736-2.
[20] Abeynaya Gnanasekaran and Amit Surana, "Efficient quantum access model for sparse structured matrices using linear combination of “things”", Physical Review A 113 2, 022437 (2026).
[21] Andrew Patterson and Leigh Lapworth, "Measurement schemes for quantum linear equation solvers", Quantum Science and Technology 10 2, 025037 (2025).
[22] Grzegorz Rajchel-Mieldzioć, Szymon Pliś, and Emil Zak, "Quantum algorithm for solving generalized eigenvalue problems with application to the Schrödinger equation", Physical Review Research 8 2, 023192 (2026).
[23] Leigh Lapworth, "The probability of success in solving linear systems of equations using quantum singular value transformation", Acta Mechanica Sinica 42 6, 725776 (2026).
[24] Di Fang, Diyi Liu, and Rahul Sarkar, "Time-Dependent Hamiltonian Simulation via Magnus Expansion: Algorithm and Superconvergence", Communications in Mathematical Physics 406 6, 128 (2025).
[25] Siyi Wang, Xiufan Li, Wei Jie Bryan Lee, Suman Deb, Eugene Lim, and Anupam Chattopadhyay, "A comprehensive study of quantum arithmetic circuits", Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 383 2288, 20230392 (2025).
[26] Xiao-Ming Zhang and Xiao Yuan, "Circuit complexity of quantum access models for encoding classical data", npj Quantum Information 10 1, 42 (2024).
[27] Rihan Hai, Shih-Han Hung, Tim Coopmans, Tim Littau, and Floris Geerts, "Quantum Data Management in the NISQ Era", Proceedings of the VLDB Endowment 18 6, 1720 (2025).
[28] Mauro E. S. Morales, Lirandë Pira, Philipp Schleich, Kelvin Koor, Pedro C. S. Costa, Dong An, Alán Aspuru-Guzik, Lin Lin, Patrick Rebentrost, and Dominic W. Berry, "Quantum linear system solvers: A survey of algorithms and applications", Reviews of Modern Physics 98 2, 025005 (2026).
[29] I. Novikau, I.Y. Dodin, and E.A. Startsev, "Encoding of linear kinetic plasma problems in quantum circuits via data compression", Journal of Plasma Physics 90 4, 805900401 (2024).
[30] Zhiyan Ding, Bowen Li, and Lin Lin, "Efficient Quantum Gibbs Samplers with Kubo–Martin–Schwinger Detailed Balance Condition", Communications in Mathematical Physics 406 3, 67 (2025).
[31] Oliver O'Brien and Christoph Sünderhauf, "Quantum state preparation via piecewise QSVT", Quantum 9, 1786 (2025).
[32] Leigh Lapworth and Christoph Sünderhauf, "Preconditioned block encodings for quantum linear systems", Quantum Science and Technology 10 4, 045064 (2025).
[33] Tae-Won Kim and Byung-Soo Choi, "Analysis of quantum primitives for quantum utility", Quantum Information Processing 25 6, 181 (2026).
[34] Ashish Joshi and Takahiko Koyama, "Quantum Algorithm for Metabolic Network Analysis", (2025).
[35] I. Novikau and I. Joseph, "Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems", Computer Physics Communications 309, 109498 (2025).
[36] Chunlin Yang, Zexian Li, Hongmei Yao, Zhaobing Fan, Guofeng Zhang, and Jianshe Liu, "Dictionary-based Block Encoding of Sparse Matrices with Low Subnormalization and Circuit Depth", Quantum 9, 1805 (2025).
[37] Nahid Binandeh Dehaghani, Rafal Wisniewski, and A. Pedro Aguiar, 2025 IEEE International Conference on Quantum Computing and Engineering (QCE) 282 (2025) ISBN:979-8-3315-5736-2.
[38] Nishitha Reddy Venkannagari, Varun Puram, and Johnson P Thomas, 2025 5th International Conference on AI-ML-Systems (AIMLSystems) 436 (2025) ISBN:979-8-3315-7318-8.
[39] Leonhard Hölscher, Oliver Ahrend, Lukas Karch, Carlotta L’Estocq, Marc Marfany Andreu, Tobias Stollenwerk, Frank K Wilhelm, and Julia Kowalski, "End-to-end quantum algorithm for topology optimization in structural mechanics", Quantum Science and Technology 11 2, 025029 (2026).
[40] Sasan Moradi, "Hybrid classical-quantum computation of heat diffusion in multilayer materials", (2025).
[41] Chelsea A. Williams, Annie E. Paine, Antonio A. Gentile, Daniel Berger, and Oleksandr Kyriienko, "Vortex detection from quantum data", Physical Review A 112 6, 062409 (2025).
[42] Xinchi Huang, Hirofumi Nishi, Yoshifumi Kawada, Tomofumi Zushi, and Yu-ichiro Matsushita, "Fourier space readout method for efficiently recovering functions encoded in quantum states", Quantum Science and Technology 11 3, 035050 (2026).
[43] Siyi Wang and Anupam Chattopadhyay, Computer Architecture and Design Methodologies 9 (2026) ISBN:978-981-95-6038-7.
[44] David Jennings, Matteo Lostaglio, Robert B. Lowrie, Sam Pallister, and Andrew T. Sornborger, "The cost of solving linear differential equations on a quantum computer: fast-forwarding to explicit resource counts", Quantum 8, 1553 (2024).
[45] Hiroaki Murakami, Kenzo Makino, Yasunori Lee, Keita Kanno, and Tomonori Fukuta, Communications in Computer and Information Science 2743, 329 (2026) ISBN:978-3-032-13851-4.
[46] Jezer Jojo, Ankit Khandelwal, and M Girish Chandra, 2024 16th International Conference on COMmunication Systems & NETworkS (COMSNETS) 1018 (2024) ISBN:979-8-3503-8311-9.
[47] Jiaqi Leng, Joseph Li, Yuxiang Peng, and Xiaodi Wu, "Expanding Hardware-Efficiently Manipulable Hilbert Space via Hamiltonian Embedding", Quantum 9, 1857 (2025).
[48] Leonhard Hölscher, Lukas Karch, Or Samimi, and Tamuz Danzig, "Quantum simulation-based optimization for cooling system design", Journal of Physics A: Mathematical and Theoretical 59 12, 125301 (2026).
[49] Ashish Joshi and Takahiko Koyama, "Quantum algorithm for metabolic network analysis", Machine Learning with Applications 24, 100913 (2026).
[50] R Au-Yeung, B Camino, O Rathore, and V Kendon, "Quantum algorithms for scientific computing", Reports on Progress in Physics 87 11, 116001 (2024).
[51] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).
[52] Jessica Lemieux, Matteo Lostaglio, Sam Pallister, William Pol, Karthik Seetharam, Sukin Sim, and Burak Şahinoğlu, "Quantum sampling algorithms for quantum state preparation and matrix block-encoding", arXiv:2405.11436, (2024).
[53] Tomer Goldfriend, Israel Reichental, Amir Naveh, Lior Gazit, Nadav Yoran, Ravid Alon, Shmuel Ur, Shahak Lahav, Eyal Cornfeld, Avi Elazari, Peleg Emanuel, Dor Harpaz, Tal Michaeli, Nati Erez, Lior Preminger, Roman Shapira, Erik Michael Garcell, Or Samimi, Sara Kisch, Gil Hallel, Gilad Kishony, Vincent van Wingerden, Nathaniel A. Rosenbloom, Ori Opher, Matan Vax, Ariel Smoler, Tamuz Danzig, Eden Schirman, Guy Sella, Ron Cohen, Roi Garfunkel, Tali Cohn, Hanan Rosemarin, Ron Hass, Klem Jankiewicz, Karam Gharra, Ori Roth, Barak Azar, Shahaf Asban, Natalia Linkov, Dror Segman, Ohad Sahar, Niv Davidson, Nir Minerbi, and Yehuda Naveh, "Design and synthesis of scalable quantum programs", arXiv:2412.07372, (2024).
[54] Quynh T. Nguyen, "The mixed Schur transform: efficient quantum circuit and applications", arXiv:2310.01613, (2023).
[55] Parker Kuklinski and Benjamin Rempfer, "S-FABLE and LS-FABLE: Fast approximate block-encoding algorithms for unstructured sparse matrices", arXiv:2401.04234, (2024).
[56] Rihan Hai, Shih-Han Hung, Tim Coopmans, Tim Littau, and Floris Geerts, "Quantum Data Management in the NISQ Era: Extended Version", arXiv:2409.14111, (2024).
[57] Chelsea A. Williams, Antonio A. Gentile, Vincent E. Elfving, Daniel Berger, and Oleksandr Kyriienko, "Quantum Iterative Methods for Solving Differential Equations with Application to Computational Fluid Dynamics", arXiv:2404.08605, (2024).
[58] Oscar Watts, Yuta Kikuchi, and Luuk Coopmans, "Quantum Semidefinite Programming with Thermal Pure Quantum States", arXiv:2310.07774, (2023).
[59] Abtin Ameri, Erika Ye, Paola Cappellaro, Hari Krovi, and Nuno F. Loureiro, "Quantum algorithm for the linear Vlasov equation with collisions", Physical Review A 107 6, 062412 (2023).
[60] Ryotaro Niwa, Zane Marius Rossi, Philip Taranto, and Mio Murao, "Singular value transformation for unknown quantum channels", arXiv:2506.24112, (2025).
[61] Xinchi Huang, Hirofumi Nishi, Taichi Kosugi, Yoshifumi Kawada, and Yu-ichiro Matsushita, "A probabilistic imaginary-time evolution quantum algorithm for advection-diffusion equation: Explicit gate-level implementation and comparisons to quantum linear system algorithms", arXiv:2409.18559, (2024).
[62] Leigh Lapworth, "Evaluation of block encoding for sparse matrix inversion using QSVT", arXiv:2402.17529, (2024).
[63] Ivan Novikau and Ilon Joseph, "Quantum algorithm for the advection-diffusion equation and the Koopman-von Neumann approach to nonlinear dynamical systems", arXiv:2410.03985, (2024).
[64] Diyi Liu, Weijie Du, Lin Lin, James P. Vary, and Chao Yang, "An Efficient Quantum Circuit for Block Encoding a Pairing Hamiltonian", arXiv:2402.11205, (2024).
[65] Jielun Chen and Garnet Kin-Lic Chan, "A framework for robust quantum speedups in practical correlated electronic structure and dynamics", arXiv:2508.15765, (2025).
[66] Vasileios Kalantzis, Mark S. Squillante, and Shashanka Ubaru, "On Efficient Solutions of General Structured Markov Processes in Quantum Computational Environments", arXiv:2404.17959, (2024).
[67] Charles Yuan, "Cobble: Compiling Block Encodings for Quantum Computational Linear Algebra", arXiv:2511.01736, (2025).
[68] Tae-Won Kim and Byung-Soo Choi, "Dequantizability from inputs", arXiv:2405.13273, (2024).
[69] Siddharth Hariprakash, Roel Van Beeumen, Katherine Klymko, and Daan Camps, "The Practicality of Randomized Quantum Linear Systems Solvers", arXiv:2510.13766, (2025).
[70] V. Anurag K. S., Ashish Kumar Patra, Manas Mukherjee, Alok Shukla, Sai Shankar P., Ruchika Bhat, L. Radhika T. S., and Jaiganesh G, "Towards Chemically Accurate and Scalable Quantum Simulations on IQM Quantum Hardware: A Quantum-HPC Hybrid Approach", arXiv:2604.01983, (2026).
[71] Zexian Li, Xiao-Ming Zhang, Chunlin Yang, and Guofeng Zhang, "Binary Tree Block Encoding of Classical Matrix", arXiv:2504.05624, (2025).
[72] Xinchi Huang, Hirofumi Nishi, Yoshifumi Kawada, Tomofumi Zushi, and Yu-ichiro Matsushita, "Real and Fourier space readout methods: Comparison of complexity and applications to CFD problems", arXiv:2511.20017, (2025).
[73] Niclas Schillo, Andreas Sturm, and Rüdiger Quay, "TARE: Block Encoding Linear Combinations of Pauli Strings Without Ancilla State Preparation", arXiv:2601.05740, (2026).
[74] Rakshit M. Gharat, Gopikrishnan Muraleedharan, Dominic W. Berry, and Gavin K. Brennen, "Quantum algorithm for solving differential equations using SLAC derivatives", arXiv:2605.04861, (2026).
[75] Krishnan Suresh, "Spectrally Corrected Polynomial Approximation for Quantum Singular Value Transformation", arXiv:2603.03998, (2026).
[76] Dekuan Dong, Yingzhou Li, and Jungong Xue, "Products between block-encodings", arXiv:2509.15779, (2025).
The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 05:16:48) and SAO/NASA ADS (last updated successfully 2026-08-11 16:43:22). The list may be incomplete as not all publishers provide suitable and complete citation data.
Could not fetch ADS cited-by data during last attempt 2026-08-12 05:16:48: Cannot retrieve data from ADS due to rate limitations.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.