Quantum linear algebra algorithm offering exponential speedup under certain conditions
The Harrow–Hassidim–Lloyd algorithm or HHL algorithm is a quantum algorithm for numerically solving a system of linear equations, designed by Aram Harrow, Avinatan Hassidim, and Seth Lloyd. The algorithm estimates the result of a scalar measurement on the solution vector to a given linear system of equations.[1]
The algorithm is one of the main fundamental algorithms expected to provide a speedup over their classical counterparts, along with Shor's factoring algorithm, Grover's search algorithm, and the quantum Fourier transform. Provided the linear system is sparse[2] and has a low condition number , and that the user is interested in the result of a scalar measurement on the solution vector, instead of the values of the solution vector itself, then the algorithm has a runtime of , where is the number of variables in the linear system. This offers an exponential speedup over the fastest classical algorithm, which runs in (or for positive semidefinite matrices).
An implementation of the quantum algorithm for linear systems of equations was first demonstrated in 2013 by three independent publications.[3][4][5] The demonstrations consisted of simple linear equations on specially designed quantum devices.[3][4][5] The first demonstration of a general-purpose version of the algorithm appeared in 2018.[6]
Due to the prevalence of linear systems in virtually all areas of science and engineering, the quantum algorithm for linear systems of equations has the potential for widespread applicability.[7]
^Harrow, Aram W; Hassidim, Avinatan; Lloyd, Seth (2008). "Quantum algorithm for linear systems of equations". Physical Review Letters. 103 (15): 150502. arXiv:0811.3171. Bibcode:2009PhRvL.103o0502H. doi:10.1103/PhysRevLett.103.150502. PMID 19905613. S2CID 5187993.
^Johnston, Eric (2019-07-03). Programming Quantum Computers: Essential Algorithms and Code Samples. O'Reilly Media. p. 267. ISBN 9781492039655.
^ abCai, X.-D; Weedbrook, C; Su, Z.-E; Chen, M.-C; Gu, Mile; Zhu, M.-J; Li, Li; Liu, Nai-Le; Lu, Chao-Yang; Pan, Jian-Wei (2013). "Experimental Quantum Computing to Solve Systems of Linear Equations". Physical Review Letters. 110 (23): 230501. arXiv:1302.4310. Bibcode:2013PhRvL.110w0501C. doi:10.1103/PhysRevLett.110.230501. PMID 25167475. S2CID 20427454.
^ abBarz, Stefanie; Kassal, Ivan; Ringbauer, Martin; Lipp, Yannick Ole; Dakić, Borivoje; Aspuru-Guzik, Alán; Walther, Philip (2014). "A two-qubit photonic quantum processor and its application to solving systems of linear equations". Scientific Reports. 4: 6115. arXiv:1302.1210. Bibcode:2014NatSR...4E6115B. doi:10.1038/srep06115. ISSN 2045-2322. PMC 4137340. PMID 25135432.
^ abPan, Jian; Cao, Yudong; Yao, Xiwei; Li, Zhaokai; Ju, Chenyong; Peng, Xinhua; Kais, Sabre; Du, Jiangfeng; Du, Jiangfeng (2014). "Experimental realization of quantum algorithm for solving linear systems of equations". Physical Review A. 89 (2): 022313. arXiv:1302.1946. Bibcode:2014PhRvA..89b2313P. doi:10.1103/PhysRevA.89.022313. S2CID 14303240.
^Zhao, Zhikuan; Pozas-Kerstjens, Alejandro; Rebentrost, Patrick; Wittek, Peter (2019). "Bayesian Deep Learning on a Quantum Computer". Quantum Machine Intelligence. 1 (1–2): 41–51. arXiv:1806.11463. doi:10.1007/s42484-019-00004-7. S2CID 49554188.
^Quantum Computer Runs The Most Practically Useful Quantum Algorithm, by Lu and Pan.
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