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#REDIRECT[[Continuous-variable quantum information]]
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'''Continuous quantum computation''' is the study of how to use the techniques of [[quantum computation]] to compute or approximate the answers to mathematical questions involving [[continuous function]]s.
== Motivations ==
One major motivation for studying the quantum computation of continuous functions is that many scientific problems have mathematical formulations in terms of continuous quantities. Examples include evaluating [[Path integral formulation|path integrals]], solving [[differential equation]]s using the [[Feynman–Kac formula]], and [[numerical optimization]] of continuous functions. A second motivation is to explore and understand the ways in which quantum computers can be more capable or powerful than classical ones. By computational complexity (complexity for brevity) is meant the '''minimal''' computational resources needed to solve a problem. Two of the most important resources for quantum computing are [[qubit]]s and queries. Classical complexity has been extensively studied in [[information-based complexity]]. The classical complexity of many continuous problems is known. Therefore, when the quantum complexity of these problems is obtained, the question as to whether quantum computers are more powerful than classical can be answered. Furthermore, the degree of the improvement can be quantified. In contrast, the complexity of discrete problems is typically unknown; one has to settle for the complexity hierarchy. For example, the classical complexity of integer factorization is unknown.
== Applications ==
One example of a scientific problem that is naturally expressed in continuous terms is [[path integration]]. The general technique of path integration has numerous applications including [[quantum mechanics]], [[quantum chemistry]], [[statistical mechanics]], and [[computational finance]]. Because randomness is present throughout quantum theory, one typically requires that a quantum computational procedure yield the correct answer, not with certainty, but with high probability. For example, one might aim for a procedure that computes the correct answer with probability at least 3/4. One also specifies a degree of uncertainty, typically by setting the maximum acceptable error. Thus, the goal of a quantum computation could be to compute the numerical result of a path-integration problem to within an error of at most ε with probability 3/4 or more. In this context, it is known<ref>{{Cite journal|last=Traub|first=J. F.|last2=Woźniakowski|first2=H.|date=2002-10-01|title=Path Integration on a Quantum Computer|url=https://link.springer.com/article/10.1023/A:1023417813916|journal=Quantum Information Processing|language=en|volume=1|issue=5|pages=365–388|arxiv=quant-ph/0109113|doi=10.1023/A:1023417813916|issn=1570-0755}}</ref> that quantum algorithms can outperform their classical counterparts, and the computational complexity of the problem, as measured by the number of times one would expect to have to query a quantum computer to get a good answer, grows as the inverse of ε.
In the standard model of quantum computation the probabilistic nature of quantum computation enters only through measurement; the queries are deterministic. In analogy with classical [[Monte Carlo method|Monte Carlo methods]], Woźniakowski introduced the idea of a quantum setting with randomized queries. He showed that in this setting the qubit complexity is of order <math>\scriptstyle \log\varepsilon^{-1}</math>, thus achieving an exponential improvement over the qubit complexity in the standard quantum computing setting.<ref>{{Cite journal|last=Woźniakowski|first=H.|date=2006-04-01|title=The Quantum Setting with Randomized Queries for Continuous Problems|url=https://link.springer.com/article/10.1007/s11128-006-0013-6|journal=Quantum Information Processing|language=en|volume=5|issue=2|pages=83–130|arxiv=quant-ph/0601196|doi=10.1007/s11128-006-0013-6|issn=1570-0755}}</ref>
Besides path integration there have been numerous recent papers studying algorithms and quantum speedups for continuous problems. These include finding matrix [[Eigenvalues and eigenvectors|eigenvalues]], phase estimation, the Sturm–Liouville eigenvalue problem, solving [[Differential equation|differential equations]] with the [[Feynman–Kac formula]], initial value problems, function approximation and high-dimensional integration.
==External links==
*http://quantum.cs.columbia.edu – Continuous quantum computing web page at [[Columbia University]]
==References==
*Bessen, A. J. (2005), A lower bound for phase estimation, Physical Review A, 71(4), 042313. Also [https://arXiv.org/abs/quant-ph/0412008 arXiv:quant-ph/0412008].
*Heinrich, S. (2002), Quantum Summation with an Application to Integration, J. Complexity, 18(1), 1–50. Also [https://arXiv.org/abs/quant-ph/0105116 arXiv:quant-ph/0105116].
*Heinrich, S. (2003), Quantum integration in Sobolev spaces, J. Complexity, 19, 19–42.
*Heinrich, S. (2004), Quantum Approximation I. Embeddings of Finite Dimensional <math>L_p</math> Spaces, J. Complexity, 20, 5–26. Also [https://arXiv.org/abs/quant-ph/0305030 arXiv:quant-ph/0305030].
*Heinrich, S. (2004), Quantum Approximation II. Sobolev Embeddings, J. Complexity, 20, 27–45. Also [https://arXiv.org/abs/quant-ph/0305031 arXiv:quant-ph/0305031].
*Jaksch, P. and Papageorgiou, A. (2003), Eigenvector approximation leading to exponential speedup of quantum eigenvalue calculation, Phys. Rev. Lett., 91, 257902. Also [https://arXiv.org/abs/quant-ph/0308016 arXiv:quant-ph/0308016].
*Kacewicz, B. Z. (2005), Randomized and quantum solution of initial value problems, J. Complexity, 21, 740–756.
*Kwas, M., Complexity of multivariate Feynman–Kac Path Integration in Randomized and Quantum settings, 2004. Also [https://arXiv.org/abs/quant-ph/0410134 arXiv:quant-ph/0410134].
*Novak, E. (2001), Quantum complexity of integration, J. Complexity, 17, 2–16. Also [https://arXiv.org/abs/quant-ph/0008124 arXiv:quant-ph/0008124].
*Novak, E., Sloan, I. H., and Woźniakowski, H., Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers, J. Foundations of Computational Mathematics, 4, 121-156, 2004. Also [https://arXiv.org/abs/quant-ph/0206023 arXiv:quant-ph/0206023]
*Papageorgiou, A. and Woźniakowski, H. (2005), Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem, Quantum Information Processing, 4(2), 87–127. Also [https://arXiv.org/abs/quant-ph/0502054 arXiv:quant-ph/0502054].
*Papageorgiou, A. and Woźniakowski, H. (2007), The Sturm–Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries, Quantum Information Processing, 6(2), 101–120. Also [https://arXiv.org/abs/quant-ph/0504194 arXiv:quant-ph/0504194].
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[[Category:Quantum information science]]
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