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{{Short description|Continuous (non-quantized) quantities in quantum information science}}
{{Use American English|date=January 2019}}'''Continuous-variable''' ('''CV''') '''quantum information''' is the area of [[quantum information science]] that makes use of [[Observable|physical observables]], like the strength of an [[electromagnetic field]], whose numerical values belong to [[List of continuity-related mathematical topics|continuous]] [[Interval (mathematics)|intervals]].<ref name=":
== Implementation ==
One approach to implementing continuous-variable quantum information protocols in the laboratory is through the techniques of [[quantum optics]].<ref name=":1">{{Cite journal|
[[Quantum teleportation]] of continuous-variable quantum information was achieved by optical methods in 1998.<ref>{{Cite journal|
Another proposal is to modify the [[Trapped ion quantum computer|ion-trap quantum computer]]: instead of storing a single
== Applications ==
Continuous-variable quantum systems can be used for [[quantum cryptography]], and in particular, [[quantum key distribution]].<ref name=":0"
== Classical emulation ==
In all approaches to quantum computing, it is important to know whether a task under consideration can be carried out efficiently by a classical computer. An [[algorithm]] might be described in the language of quantum mechanics, but upon closer analysis, revealed to be implementable using only classical resources. Such an algorithm would not be taking full advantage of the extra possibilities made available by quantum physics. In the theory of quantum computation using finite-dimensional Hilbert spaces, the [[Gottesman–Knill theorem]] demonstrates that there exists a set of quantum processes that can be emulated efficiently on a classical computer. Generalizing this theorem to the continuous-variable case, it can be shown that, likewise, a class of continuous-variable quantum computations can be simulated using only classical
== 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.
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|
Other continuous problems for which quantum algorithms have been studied include finding matrix [[Eigenvalues and eigenvectors|eigenvalues]],<ref>{{Cite journal|
{{Cite journal|last=Heinrich|first=Stefan
{{Cite journal|last=Heinrich|first=Stefan
== See also ==
* [[Quantum inequalities]]
==References==
{{Reflist}}
{{Quantum information}}
[[Category:Quantum information science]]
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