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Quantum Algorithms: Equation Solving by Simulation news & views QUANTUM ALGORITHMS Equation solving by simulation Quantum computers can outperform their classical counterparts at some tasks, but the full scope of their power is unclear. A new quantum algorithm hints at the possibility of far-reaching applications. Andrew M. Childs uantum mechanical computers have state |b〉. Then, by a well-known technique can be encoded into an instance of solving the potential to quickly perform called phase estimation5, the ability to linear equations, even with the restrictions calculations that are infeasible with produce e–iAt|b〉 is leveraged to create a required for their quantum solver to be Q –1 present technology. There are quantum quantum state |x〉 proportional to A |b〉. efficient. Therefore, either ordinary classical algorithms to simulate efficiently the (A similar approach can be applied when computers can efficiently simulate quantum dynamics of quantum systems1 and to the matrix A is non-Hermitian, or even ones — a highly unlikely proposition — or decompose integers into their prime when A is non-square.) The result is a the quantum algorithm for solving linear factors2, problems thought to be intractable solution to the system of linear equations equations performs a task that is beyond the for classical computers. But quantum encoded as the quantum state |x〉. reach of classical computation. computation is not a magic bullet — some Producing a quantum state proportional Proving ‘hardness’ results of this kind problems cannot be solved dramatically to A–1|b〉 does not, by itself, solve the task is a widely used strategy for establishing faster by quantum computers than by at hand. To extract information from the non-triviality of quantum algorithms. classical ones. Is a quantum computer a quantum state, we must perform a Similar constructions are known for necessarily a special-purpose device measurement. Learning all N amplitudes problems such as finding ground states then, suitable only for number-theoretic of an N-dimensional quantum state by adiabatic evolution6, computing calculations or problems that are inherently requires a number of measurements at least invariants of knots7, estimating entries quantum? There might be hope for more proportional to N. Thus, if our goal is to of powers of matrices8 and contracting generality. Writing in Physical Review completely reconstruct a solution x, there tensor networks9, among other tasks. But Letters3, Aram Harrow, Avinatan Hassidim is no hope for a quantum algorithm to what sets linear equations apart is their and Seth Lloyd describe how quantum offer a significant advantage over classical ubiquity in scientific computing and computers can extract information methods. However, for some problems engineering applications. about the solutions to linear equations, a we might not be interested in the entire Will the quantum solution of linear fundamental task with broad applications. vector x, but rather in some special feature equations3 turn out to be a widely used The basic problem of finding a vector x of it, such as an expectation value. Then a tool, or are its limitations too great for the satisfying Ax = b for some given matrix A quantum computer may be able to solve the technique to be of practical significance? and vector b arises throughout science and problem rapidly. Unfortunately, no concrete task has yet been engineering. For example, signal processing, In addition to readout, the approach proposed for which the quantum algorithm convex optimization and finite-element suffers from other significant limitations. provides a clear advantage. But it will be analysis all rely on solving linear equations. The vector b must be given in a way that exciting to explore the impact of this and Owing to the importance of linear systems, allows quick preparation of the quantum related applications of quantum computing, considerable effort has been spent on state |b〉. Methods are available for doing in particular as the methods can be put finding fast algorithms for solving them. this if b has a suitable implicit description, into practice. ❐ One simple approach is known as Gaussian but the approach is useless if b is given elimination, a method that was already by explicit classical data. Furthermore, Andrew M. Childs is in the Department of used in ancient China, and that today is as in classical methods for solving linear Combinatorics and Optimization and at the standard fare in courses on linear algebra. equations, the performance depends Institute for Quantum Computing, University of Alternative methods offer improved speed crucially on the condition number κ, a Waterloo, 200 University Avenue West, Waterloo, and better numerical stability, and some can measure of how close A is to being singular. Ontario N2L 3G1, Canada. take advantage of special properties such The running time of the quantum algorithm e‑mail: [email protected] as sparsity of A. However, if A is an N × N is polynomial in log(N) and in κ, so the References matrix (or a rectangular matrix for which quantum advantage is only significant when 1. Feynman, R. P. Int. J. Theor. Phys. 21, 467–488 (1982). the larger dimension is N), all of these κ is not too large. 2. Shor, P. W. SIAM J. Comput. 26, 1484–1509 (1997). methods use a number of operations at least Having lowered the bar for the sense in 3. Harrow, A. W., Hassidim, A. & Lloyd, S. Phys. Rev. Lett. 103, 150502 (2009). proportional to N. which we hope to solve a system of linear 4. Aharonov, D. & Ta-Shma, A. in Proc. 35th ACM Symp. Theor. 3 The authors suggest approaching this equations, one might wonder whether Comput. 20–29 (ACM, 2003). problem using quantum simulation. Given this quantum algorithm3 offers a real 5. Cleve, R., Ekert, A., Macchiavello, C. & Mosca, M. an N × N sparse Hermitian matrix A, a advantage over classical computing at Proc. R. Soc. Lond. A 454, 339–354 (1998). 〉 6. Aharonov, D. et al. in Proc. 45th IEEE Found. Comput. Sci. 42–51 quantum state |b and a time t, quantum all. To address this concern, in perhaps (IEEE, 2004). simulation provides a method for preparing the most interesting aspect of their work, 7. Bordewich, M., Freedman, M., Lovász, L. & Welsh, D. the quantum state e–iAt|b〉 using a number of Harrow, Hassidim and Lloyd prove that the Comb. Probab. Comput. 14, 737–754 (2005). operations that is only polynomial in log(N) problem they solve is as hard as anything 8. Janzing, D. & Wocjan, P. Theor. Comput. 3, 61–79 (2007). (ref. 4). In the approach of Harrow et al., a quantum computer can do. In particular, 9. Arad, I. & Landau, Z. Preprint at <http://www.arxiv.org/ the vector b is first encoded into a quantum they show that any quantum computation abs/0805.0040> (2008). NATURE PHYSICS | VOL 5 | DECEMBER 2009 | www.nature.com/naturephysics 861 © 2009 Macmillan Publishers Limited. All rights reserved.
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