Continuous quantum computation: Difference between revisions

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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.

Two major motivations for studying '''continuous quantum computation''' are:

== Motivations ==

* Many scientific problems have continuous mathematical formulations. Examples of such formulations are

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 of such formulations are

** [[Path Integral Formulation|Path integration]]

** [[Feynman–Kac]] path integration

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== 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. In the case of continuous-variable quantum computation, oneOne also specifies a degree of uncertainty, typically by setting the maximum acceptable error. Thus, the goal of a continuous-variable 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>

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*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 WozniakowskiWoźniakowski, 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 Wo´zniakowskiWoźniakowski, 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 Wo´zniakowskiWoźniakowski, 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].

*Woźniakowski, H. (2006), The Quantum Setting with Randomized Queries for Continuous Problems, Quantum Information Processing, 5(2), 83–130. Also [https://arXiv.org/abs/quant-ph/0601196 arXiv:quant-ph/0601196].