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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]],<ref>{{Cite journal|last=Jaksch|first=Peter|last2=Papageorgiou|first2=Anargyros|date=2003-12-19|title=Eigenvector Approximation Leading to Exponential Speedup of Quantum Eigenvalue Calculation|url=https://link.aps.org/doi/10.1103/PhysRevLett.91.257902|journal=Physical Review Letters|volume=91|issue=25|pages=257902|arxiv=quant-ph/0308016|doi=10.1103/PhysRevLett.91.257902}}</ref> phase estimation,<ref>{{Cite journal|last=Bessen|first=Arvid J.|date=2005-04-08|title=Lower bound for quantum phase estimation|url=https://link.aps.org/doi/10.1103/PhysRevA.71.042313|journal=Physical Review A|volume=71|issue=4|pages=042313|arxiv=quant-ph/0412008|doi=10.1103/PhysRevA.71.042313}}</ref> the Sturm–Liouville eigenvalue problem,<ref>{{Cite journal|last=Papageorgiou|first=A.|last2=Woźniakowski|first2=H|date=|title=Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem|url=http://link.springer.com/article/10.1007/s11128-005-4481-x|journal=Quantum Information Processing|language=en|volume=4|issue=2|pages=87–127|arxiv=quant-ph/0502054|doi=10.1007/s11128-005-4481-x}}</ref><ref>{{Cite journal|last=Papageorgiou|first=A.|last2=Woźniakowski|first2=H.|date=2007-04-01|title=The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries|url=https://link.springer.com/article/10.1007/s11128-006-0043-0|journal=Quantum Information Processing|language=en|volume=6|issue=2|pages=101–120|arxiv=quant-ph/0504191|doi=10.1007/s11128-006-0043-0|issn=1570-0755}}</ref> solving [[Differential equation|differential equations]] with the [[Feynman–Kac formula]],<ref>{{Cite journal|last=Kwas|first=Marek|date=2004-10-18|title=Complexity of multivariate Feynman-Kac path integration in randomized and quantum settings|url=http://arxiv.org/abs/quant-ph/0410134|journal=arXiv:quant-ph/0410134}}</ref> initial value problems,<ref>{{Cite journal|last=Kacewicz|first=Bolesław|title=Randomized and quantum algorithms yield a speed-up for initial-value problems|url=https://doi.org/10.1016/j.jco.2004.05.002|journal=Journal of Complexity|language=en|volume=20|issue=6|pages=821–834|doi=10.1016/j.jco.2004.05.002}}</ref><ref>{{Cite journal|last=Szczesny|first=Marek|date=2006-12-12|title=Randomized and Quantum Solution of Initial-Value Problems for Ordinary Differential Equations of Order k|url=http://arxiv.org/abs/quant-ph/0612085|journal=arXiv:quant-ph/0612085}}</ref><ref>{{Cite journal|last=Kacewicz|first=Bolesław|title=Improved bounds on the randomized and quantum complexity of initial-value problems|url=https://doi.org/10.1016/j.jco.2005.05.003|journal=Journal of Complexity|language=en|volume=21|issue=5|pages=740–756|doi=10.1016/j.jco.2005.05.003}}</ref> function approximation<ref>{{Cite journal|last=Novak|first=Erich|last2=Sloan|first2=Ian H.|last3=Wo’zniakowski|first3=Henryk|date=2004-04-01|title=Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers|url=https://link.springer.com/article/10.1007/s10208-002-0074-6|journal=Foundations of Computational Mathematics|language=en|volume=4|issue=2|pages=121–156|arxiv=quant-ph/0206023|doi=10.1007/s10208-002-0074-6|issn=1615-3375}}</ref><ref>{{Cite journal|last=Heinrich|first=Stefan|date=|title=Quantum approximation I. Embeddings of finite-dimensional Lp spaces|url=https://doi.org/10.1016/j.jco.2003.08.002|journal=Journal of Complexity|language=en|volume=20|issue=1|pages=5–26|arxiv=quant-ph/0305030|doi=10.1016/j.jco.2003.08.002}}</ref><ref>{{Cite journal|last=Heinrich|first=Stefan|date=|title=Quantum approximation II. Sobolev embeddings|url=https://doi.org/10.1016/j.jco.2003.08.003|journal=Journal of Complexity|language=en|volume=20|issue=1|pages=27–45|arxiv=quant-ph/0305031|doi=10.1016/j.jco.2003.08.003}}</ref> and high-dimensional integration.<ref>{{Cite journal|last=Heinrich|first=S.|date=|title=Quantum Summation with an Application to Integration|url=https://doi.org/10.1006/jcom.2001.0629|journal=Journal of Complexity|language=en|volume=18|issue=1|pages=1–50|arxiv=quant-ph/0105116|doi=10.1006/jcom.2001.0629}}</ref><ref>{{Cite journal|last=Heinrich|first=Stefan|date=2003-02-01|title=Quantum integration in Sobolev classes|url=http://www.sciencedirect.com/science/article/pii/S0885064X02000080|journal=Journal of Complexity|volume=19|issue=1|pages=19–42|arxiv=quant-ph/0112153|doi=10.1016/S0885-064X(02)00008-0}}</ref><ref>{{Cite journal|last=Novak|first=Erich|date=|title=Quantum Complexity of Integration|url=https://doi.org/10.1006/jcom.2000.0566|journal=Journal of Complexity|language=en|volume=17|issue=1|pages=2–16|arxiv=quant-ph/0008124|doi=10.1006/jcom.2000.0566}}</ref>
==External links==
*http://quantum.cs.columbia.edu – Continuous quantum computing web page at [[Columbia University]]
==References==
{{Reflist}}
{{DEFAULTSORT:Continuous Quantum Computation}}
[[Category:Quantum information science]]
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