Continuous quantum computation: Difference between revisions

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We discuss the second motivation. 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, it can be established how much more powerful. 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 ==

==An example: path integration==

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, one 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.

Path integration has numerous applications including [[quantum mechanics]], [[quantum chemistry]], [[statistical mechanics]], and [[computational finance]]. We want to compute an approximation to within error at most <math>\scriptstyle\varepsilon</math> with probability, say, at least 3/4. Then the following was [http://arXiv.org/quant-ph/0109113 shown] by Traub and Woźniakowski:

* A quantum computer enjoys exponential speedup over the classical worst case and quadratic speedup over the classical randomized case.

* The query complexity is of order <math>\scriptstyle\varepsilon^{-1}</math>.

* The qubit complexity is of order <math>\scriptstyle\varepsilon^{-2}\log\varepsilon^{-1}</math>.

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 [http://arXiv.org/quant-ph/0601196 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>

Thus the qubit complexity of path integration is a second degree polynomial in <math>\scriptstyle\varepsilon^{-1}</math>. That seems pretty good but we probably won't have enough qubits for a long time to do new science especially with error correction. Since this is a complexity result we can't do better by inventing a clever new algorithm. But perhaps we can do better by slightly modifying the queries.

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, Feynman–Kacsolving [[Differential equation|differential equations]] pathwith integrationthe [[Feynman–Kac formula]], initial value problems, function approximation and high-dimensional integration. See the papers listed below and the references therein.

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 Woźniakowski introduced the idea of a quantum setting with [http://arXiv.org/quant-ph/0601196 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.

==ApplicationsExternal links==

*http://quantum.cs.columbia.edu – Continuous quantum computing web page at [[Columbia University http://quantum.cs.columbia.edu]]

Besides path integration there have been numerous recent papers studying algorithms and quantum speedups for continuous problems. These include matrix eigenvalues, phase estimation, the Sturm–Liouville eigenvalue problem, Feynman–Kac path integration, initial value problems, function approximation and high-dimensional integration. See the papers listed below and the references therein.

==References==

*Bessen, A. J. (2005), A lower bound for phase estimation, Physical Review A, 71(4), 042313. Also http://arXiv.org/quant-ph/0412008.

*Heinrich, S. (2002), Quantum Summation with an Application to Integration, J. Complexity, 18(1), 1–50. Also http://arXiv.org/quant-ph/0105116.

*Bessen, A. J. (2005), A lower bound for phase estimation, Physical Review A, 71(4), 042313. Also http[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 http[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 http[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 http[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 http[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 http[https://arXiv.org/abs/quant-ph/0410134 arXiv:quant-ph/0410134].

*Novak, E. (2001), Quantum complexity of integration, J. Complexity, 17, 2–16. Also http[https://arXiv.org/abs/quant-ph/0008124 arXiv:quant-ph/0008124].

*Novak, E., Sloan, I. H., and Wozniakowski, H., Tractability of Approximation for Weighted orobovKorobov Spaces on Classical and Quantum Computers, J. Foundations of Computational Mathematics, 4, 121-156, 2004. Also http[https://arXiv.org/abs/quant-ph/0206023 arXiv:quant-ph/0206023]

*Papageorgiou, A. and Wo´zniakowski, H. (2005), Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem, Quantum Information Processing, 4(2), 87–127. Also http[https://arXiv.org/abs/quant-ph/0502054 arXiv:quant-ph/0502054].

*Papageorgiou, A. and Wo´zniakowski, 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 http[https://arXiv.org/abs/quant-ph/0504194 arXiv:quant-ph/0504194].

*Traub, J. F. and Wo´zniakowskiWoźniakowski, H. (20022006), PathThe integrationQuantum onSetting awith quantumRandomized computerQueries for Continuous Problems, Quantum Information Processing, 1(5(2), 365–388, 200283–130. Also http[https://arXiv.org/abs/quant-ph/0601196 arXiv:quant-ph/01091130601196].

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

*Adesso, G., Ragy, S. and Lee, A. R. (2014), Continuous variable quantum information: Gaussian states and beyond. [http://arxiv.org/abs/1401.4679 arXiv:1401.4679]

==External links==

*Continuous quantum computing web page at Columbia University http://quantum.cs.columbia.edu

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