Continuous-variable quantum information: Difference between revisions

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== Computing continuous functions with discrete quantum systems ==

Occasionally, and somewhat confusingly, the term "continuous quantum computation" is used to refer to a different area of quantum computing: the study of how to use quantum systems having ''finite''-dimensional Hilbert spaces to calculate or approximate the answers to mathematical questions involving [[continuous function]]s. A major motivation for investigating the quantum computation of continuous functions is that many scientific problems have mathematical formulations in terms of continuous quantities.<ref>{{Cite web|url=http://quantum.cs.columbia.edu/html/project.html|title=Continuous Quantum Computation: Project Description|last=Papageorgiou|first=A.|date=|website=quantum.cs.columbia.edu|archive-url=|archive-date=|dead-url=|access-date=2017-05-15}}</ref> A second motivation is to explore and understand the ways in which quantum computers can be more capable or powerful than classical ones. The [[Computational complexity theory|computational complexity]] of a problem can be quantified in terms of the minimal computational resources necessary to solve it. In quantum computing, resources include the number of [[qubit]]s available to a computer and the number of [[Quantum complexity theory|queries]] that can be made to that computer. 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. For example, the classical complexity of [[integer factorization]] is unknown.

One example of a scientific problem that is naturally expressed in continuous terms is [[Functional integration|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 that quantum algorithms can outperform their classical counterparts, and the computational complexity of path integration, 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.<ref>{{Cite journal|last=Traub|first=J. F.|last2=Woźniakowski|first2=H.|date=2002-10-01|title=Path Integration on a Quantum Computer|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>