Continuous-variable quantum information: Difference between revisions

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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=":40">{{Cite journal|lastlast1=LloydWeedbrook|firstfirst1=SethChristian|authorlast2=Pirandola|first2=Stefano|last3=García-linkPatrón|first3=Seth LloydRaúl|last2last4=BraunsteinCerf|first2first4=SamuelNicolas LJ.|author-link2last5=SamuelRalph|first5=Timothy LC.|last6=Shapiro|first6=Jeffrey BraunsteinH.|last7=Lloyd|first7=Seth|date=19992012-0105-01|title=QuantumGaussian Computationquantum overinformation|journal=Reviews Continuousof VariablesModern Physics|urlvolume=https://link.aps84|issue=2|pages=621–669|arxiv=1110.org/3234|doi/=10.1103/PhysRevLettRevModPhys.8284.1784621|bibcode=2012RvMP...84..621W|s2cid=119250535}}</ref><ref>{{Cite journal|last1=Braunstein|first1=Samuel L.|last2=van Loock|first2=Peter|date=2005-06-29|title=Quantum information with continuous variables|journal=[[PhysicalReviews Reviewof LettersModern Physics]]|volume=8277|issue=82|pages=1784–1787513–577|arxiv=quant-ph/98100820410100|doi=10.1103/PhysRevLettRevModPhys.8277.1784513|bibcode=1999PhRvL2005RvMP..82.1784L77..513B|s2cid=118990906}}</ref><ref>{{Cite journal|last1=Adesso|first1=Gerardo|last2=Ragy|first2=Sammy|last3=Lee|first3=Antony R.|date=2014-03-12|title=Continuous Variable Quantum Information: Gaussian States and Beyond|journal=[[Open Systems & Information Dynamics]]|volume=21|issue=1n02|pages=1440001|arxiv=1401.4679|doi=10.1142/S1230161214400010|s2cid=15318256|issn=1230-1612}}</ref> One primary application is [[quantum computing]]. In a sense, continuous-variable quantum computation is "analog", while quantum computation using [[qubit]]s is "digital." In more technical terms, the former makes use of [[Hilbert space]]s that are [[Dimension|infinite-dimensional]], while the Hilbert spaces for systems comprising collections of qubits are finite-dimensional.<ref>{{Cite book|url=|title=Quantum Information with Continuous Variables|lastlast1=Braunstein|firstfirst1=S. L.|last2=Pati|first2=A. K.|date=2012-12-06|publisher=Springer Science & Business Media|year=|isbn=9789401512589|___location=|pages=|language=en|doi=10.1007/978-94-015-1258-9}}</ref><ref>{{Cite journal|lastciteseerx=Braunstein|first=Samuel L.|last2=van Loock|first2=Peter|date=2005-06-29|title=Quantum information with continuous variables|url=https://link.aps.org/doi/10.1103/RevModPhys1.771.513|journal=[[Reviews of Modern Physics]]|volume=77|issue=2|pages=513–577|arxiv=quant-ph/0410100|doi=10762.1103/RevModPhys.77.513|bibcode=2005RvMP...77..513B4959}}</ref> One motivation for studying continuous-variable quantum computation is to understand what resources are necessary to make quantum computers more powerful than classical ones.<ref name=":4">{{Cite journal|lastlast1=AdessoLloyd|firstfirst1=GerardoSeth|author-link=Seth Lloyd|last2=RagyBraunstein|first2=SammySamuel L.|last3author-link2=Lee|first3=AntonySamuel RL. Braunstein|date=20141999-0301-1201|title=Continuous Variable Quantum Information:Computation Gaussianover StatesContinuous and Beyond|url=http://www.worldscientific.com/doi/abs/10.1142/S1230161214400010Variables|journal=[[OpenPhysical SystemsReview & Information DynamicsLetters]]|volume=2182|issue=1n028|pages=14400011784–1787|arxiv=1401.4679quant-ph/9810082|doi=10.11421103/S1230161214400010PhysRevLett.82.1784|issnbibcode=1230-16121999PhRvL..82.1784L|s2cid=119018466}}</ref>

== 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|lastlast1=Bartlett|firstfirst1=Stephen D.|last2=Sanders|first2=Barry C.|date=2002-01-01|title=Universal continuous-variable quantum computation: Requirement of optical nonlinearity for photon counting|url=https://link.aps.org/doi/10.1103/PhysRevA.65.042304|journal=[[Physical Review A]]|volume=65|issue=4|pages=042304|arxiv=quant-ph/0110039|doi=10.1103/PhysRevA.65.042304|bibcode=2002PhRvA..65d2304B|s2cid=118896298}}</ref><ref name=":1b">{{Cite journal|lastlast1= Menicucci |firstfirst1=Nicolas C.|last2= Flammia |first2=Steven T.|last3= Pfister |first3=Olivier|date=2008-07-14|title=One-way quantum computing in the optical frequency comb|url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.130501|journal=[[Physical Review Letters]]|volume= 101 |issue=13|pages=130501|doi=10.1103/PhysRevLett.101.130501|pmid=18851426|arxiv=0804.4468|bibcode=2008PhRvL.101m0501M|s2cid=1307950}}</ref><ref name=":2">{{Cite journal|lastlast1=Tasca|firstfirst1=D. S.|last2=Gomes|first2=R. M.|last3=Toscano|first3=F.|last4=Souto Ribeiro|first4=P. H.|last5=Walborn|first5=S. P.|date=2011-01-01|title=Continuous-variable quantum computation with spatial degrees of freedom of photons|url=https://link.aps.org/doi/10.1103/PhysRevA.83.052325|journal=[[Physical Review A]]|volume=83|issue=5|pages=052325|arxiv=1106.3049|doi=10.1103/PhysRevA.83.052325|bibcode=2011PhRvA..83e2325T|s2cid=118688635}}</ref> By modeling each mode of the electromagnetic field as a [[quantum harmonic oscillator]] with its associated creation and annihilation operators, one defines a [[Conjugate variables|canonically conjugate]] pair of variables for each mode, the so-called "quadratures", which play the role of [[Position and momentum space|position and momentum]] observables. These observables establish a [[phase space]] on which [[Wigner quasiprobability distribution]]s can be defined. [[Measurement in quantum mechanics|Quantum measurements]] on such a system can be performed using [[Homodyne detection|homodyne]] and [[Heterodyne detection|heterodyne detectors]].

[[Quantum teleportation]] of continuous-variable quantum information was achieved by optical methods in 1998.<ref>{{Cite journal|lastlast1=Furusawa|firstfirst1=A.|last2=Sørensen|first2=J. L.|last3=Braunstein|first3=S. L.|last4=Fuchs|first4=C. A.|last5=Kimble|first5=H. J.|last6=Polzik|first6=E. S.|date=1998-10-23|title=Unconditional Quantum Teleportation|url=http://science.sciencemag.org/content/282/5389/706|journal=Science|language=en|volume=282|issue=5389|pages=706–709|doi=10.1126/science.282.5389.706|issn=0036-8075|pmid=9784123|bibcode=1998Sci...282..706F}}</ref><ref>{{Cite journal|lastlast1=Braunstein|firstfirst1=Samuel L.|last2=Fuchs|first2=Christopher A.|last3=Kimble|first3=H. J.|date=2000-02-01|title=Criteria for continuous-variable quantum teleportation|url=http://www.tandfonline.com/doi/abs/10.1080/09500340008244041|journal=Journal of Modern Optics|volume=47|issue=2–3|pages=267–278|arxiv=quant-ph/9910030|doi=10.1080/09500340008244041|issn=0950-0340|via=|bibcode=2000JMOp...47..267B|s2cid=16713029}}</ref> ([[Science (journal)|''Science'']] deemed this experiment one of the "top 10" advances of the year.<ref>{{Cite journal|last=|first=|date=1998-12-18|title=The Runners-Up: The News and Editorial Staffs|url=http://science.sciencemag.org/content/282/5397/2157|journal=Science|language=en|volume=282|issue=5397|pages=2157–2161|doi=10.1126/science.282.5397.2157|issn=0036-8075|via=|bibcode=1998Sci...282.2157.|s2cid=220101560}}</ref>) In 2013, quantum-optics techniques were used to create a "[[cluster state]]", a type of preparation essential to one-way (measurement-based) quantum computation, involving over 10,000 [[Quantum entanglement|entangled]] temporal modes, available two at a time.<ref>{{Cite journal|lastlast1=Yokoyama|firstfirst1=Shota|last2=Ukai|first2=Ryuji|last3=Armstrong|first3=Seiji C.|last4=Sornphiphatphong|first4=Chanond|last5=Kaji|first5=Toshiyuki|last6=Suzuki|first6=Shigenari|last7=Yoshikawa|first7=Jun-ichi|last8=Yonezawa|first8=Hidehiro|last9=Menicucci|first9=Nicolas C.|date=|title=Ultra-large-scale continuous-variable cluster states multiplexed in the time ___domain|url=http://www.nature.com/doifinder/10.1038/nphoton.2013.287|journal=Nature Photonics|volume=7|issue=12|pages=982–986|arxiv=1306.3366|doi=10.1038/nphoton.2013.287|via=|bibcode=2013NaPho...7..982Y|year=2013|s2cid=53575929}}</ref> In another implementation, 60 modes were simultaneously entangled in the frequency ___domain, in the optical frequency comb of an optical parametric oscillator.<ref>{{Cite journal|lastlast1= Chen |firstfirst1=Moran|last2= Menicucci |first2=Nicolas C.|last3= Pfister |first3=Olivier|date=2014-03-28|title=Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb|url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.120505|journal=[[Physical Review Letters]]|volume= 112 |issue=12|pages= 120505 |doi= 10.1103/PhysRevLett.112.120505|pmid=24724640|arxiv=1311.2957|bibcode=2014PhRvL.112l0505C|s2cid=18093254}}</ref>

Another proposal is to modify the [[Trapped ion quantum computer|ion-trap quantum computer]]: instead of storing a single [[qubit]] in the internal energy levels of an ion, one could in principle use the position and momentum of the ion as continuous quantum variables.<ref>{{Cite journal|lastlast1=Ortiz-Gutiérrez|firstfirst1=Luis|last2=Gabrielly|first2=Bruna|last3=Muñoz|first3=Luis F.|last4=Pereira|first4=Kainã T.|last5=Filgueiras|first5=Jefferson G.|last6=Villar|first6=Alessandro S.|date=2017-08-15|title=Continuous variables quantum computation over the vibrational modes of a single trapped ion|url=http://www.sciencedirect.com/science/article/pii/S0030401817302821|journal=Optics Communications|volume=397|pages=166–174|arxiv=1603.00065|doi=10.1016/j.optcom.2017.04.011|via=|bibcode=2017OptCo.397..166O|s2cid=118617424}}</ref>

== Applications ==

Continuous-variable quantum systems can be used for [[quantum cryptography]], and in particular, [[quantum key distribution]].<ref name=":0">{{Cite journal|last=Weedbrook|first=Christian|last2=Pirandola|first2=Stefano|last3=García-Patrón|first3=Raúl|last4=Cerf|first4=Nicolas J.|last5=Ralph|first5=Timothy C.|last6=Shapiro|first6=Jeffrey H.|last7=Lloyd|first7=Seth|date=2012-05-01|title=Gaussian quantum information|url=https://link.aps.org/doi/10.1103/RevModPhys.84.621|journal=Reviews of Modern Physics|volume=84|issue=2|pages=621–669|arxiv=1110.3234|doi=10.1103/RevModPhys.84.621|via=|bibcode=2012RvMP...84..621W}}</ref> [[Quantum computing]] is another potential application, and a variety of approaches have been considered.<ref name=":0" /> The first method, proposed by [[Seth Lloyd]] and [[Samuel L. Braunstein]] in 1999, was in the tradition of the [[Quantum circuit|circuit model]]: quantum [[logic gate]]s are created by [[Hamiltonian (quantum mechanics)|Hamiltonians]] that, in this case, are quadratic functions of the harmonic-oscillator quadratures.<ref name=":4" /> Later, [[One-way quantum computer|measurement-based quantum computation]] was adapted to the setting of infinite-dimensional Hilbert spaces.<ref name=":3">{{Cite journal|lastlast1=Menicucci|firstfirst1=Nicolas C.|last2=van Loock|first2=Peter|last3=Gu|first3=Mile|last4=Weedbrook|first4=Christian|last5=Ralph|first5=Timothy C.|last6=Nielsen|first6=Michael A.|author-link6=Michael Nielsen|date=2006-09-13|title=Universal Quantum Computation with Continuous-Variable Cluster States|url=https://link.aps.org/doi/10.1103/PhysRevLett.97.110501|journal=[[Physical Review Letters]]|volume=97|issue=11|pages=110501|arxiv=quant-ph/0605198|doi=10.1103/PhysRevLett.97.110501|pmid=17025869|bibcode=2006PhRvL..97k0501M|s2cid=14715751}}</ref><ref>{{Cite journal|lastlast1=Zhang|firstfirst1=Jing|last2=Braunstein|first2=Samuel L.|date=2006-03-16|title=Continuous-variable Gaussian analog of cluster states|url=https://link.aps.org/doi/10.1103/PhysRevA.73.032318|journal=Physical Review A|volume=73|issue=3|pages=032318|doi=10.1103/PhysRevA.73.032318|bibcode=2006PhRvA..73c2318Z|arxiv=quant-ph/0501112|s2cid=119511825 }}</ref> Yet a third model of continuous-variable quantum computation encodes finite-dimensional systems (collections of [[qubit]]squbits) into infinite-dimensional ones. This model is due to [[Daniel Gottesman]], [[Alexei Kitaev]] and [[John Preskill]] who introduced the [[Gottesman–Kitaev–Preskill code]].<ref>{{Cite journal|lastlast1=Gottesman|firstfirst1=Daniel|last2=Kitaev|first2=Alexei|last3=Preskill|first3=John|date=2001-06-11|title=Encoding a qubit in an oscillator|url=https://link.aps.org/doi/10.1103/PhysRevA.64.012310|journal=Physical Review A|volume=64|issue=1|pages=012310|arxiv=quant-ph/0008040|doi=10.1103/PhysRevA.64.012310|via=|bibcode=2001PhRvA..64a2310G|s2cid=18995200}}</ref>

== 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 analog computations. This class includes, in fact, some computational tasks that use [[quantum entanglement]].<ref>{{Cite journal|lastlast1=Bartlett|firstfirst1=Stephen D.|last2=Sanders|first2=Barry C.|last3=Braunstein|first3=Samuel L.|last4=Nemoto|first4=Kae|author4-link= Kae Nemoto |date=2002-02-14|title=Efficient Classical Simulation of Continuous Variable Quantum Information Processes|url=https://link.aps.org/doi/10.1103/PhysRevLett.88.097904|journal=[[Physical Review Letters]]|volume=88|issue=9|pages=097904|arxiv=quant-ph/0109047|doi=10.1103/PhysRevLett.88.097904|pmid=11864057|bibcode=2002PhRvL..88i7904B|s2cid=2161585}}</ref> When the [[Wigner quasiprobability distribution|Wigner quasiprobability representations]] of all the quantities—states, time evolutions ''and'' measurements—involved in a computation are nonnegative, then they can be interpreted as ordinary probability distributions, indicating that the computation can be modeled as an essentially classical one.<ref name=":3" /> This type of construction can be thought of as a continuum generalization of the [[Spekkens Toytoy Modelmodel]].<ref>{{Cite journal|lastlast1=Bartlett|firstfirst1=Stephen D.|last2=Rudolph|first2=Terry|last3=Spekkens|first3=Robert W.|date=2012-07-10|title=Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction|url=https://link.aps.org/doi/10.1103/PhysRevA.86.012103|journal=[[Physical Review A]]|volume=86|issue=1|pages=012103|arxiv=1111.5057|doi=10.1103/PhysRevA.86.012103|bibcode=2012PhRvA..86a2103B|s2cid=119235025}}</ref>

== 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 [[lineFunctional integralintegration|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|lastlast1=Traub|firstfirst1=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|s2cid=5821196|issn=1570-0755}}</ref>

Other continuous problems for which quantum algorithms have been studied include finding matrix [[Eigenvalues and eigenvectors|eigenvalues]],<ref>{{Cite journal|lastlast1=Jaksch|firstfirst1=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|pmid=14754158|bibcode=2003PhRvL..91y7902J|s2cid=1855075}}</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|bibcode=2005PhRvA..71d2313B|s2cid=118887469}}</ref> the Sturm–Liouville eigenvalue problem,<ref>{{Cite journal|lastlast1=Papageorgiou|firstfirst1=A.|last2=Woźniakowski|first2=H|date=|title=Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem|url=https://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|year=2005|bibcode=2005QuIP....4...87P|s2cid=11089349}}<br/>{{Cite journal|lastlast1=Papageorgiou|firstfirst1=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|s2cid=7604869|issn=1570-0755}}</ref> solving [[differential equation]]s with the [[Feynman–Kac formula]],<ref>{{cite arxivarXiv|last=Kwas|first=Marek|date=2004-10-18|title=Complexity of multivariate Feynman-Kac path integration in randomized and quantum settings|eprint=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|year=2004|arxiv=quant-ph/0311148|s2cid=9949704}}<br/>{{cite arxivarXiv|last=Szczesny|first=Marek|date=2006-12-12|title=Randomized and Quantum Solution of Initial-Value Problems for Ordinary Differential Equations of Order k|eprint=quant-ph/0612085}}<br/>{{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|year=2005|arxiv=quant-ph/0405018|s2cid=5934254}}</ref> function approximation,<ref>{{Cite journal|lastlast1=Novak|firstfirst1=Erich|last2=Sloan|first2=Ian H.|last3=Woźniakowski|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|s2cid=10519614|issn=1615-3375|via=}}<br>

{{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|year=2004|s2cid=6044488}}<br>

{{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|year=2004|s2cid=6061625}}</ref> and high-dimensional integration.<ref>{{Cite journal|last=Heinrich|first=Stefan|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|year=2002|s2cid=14365504}}<br/>{{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|s2cid=5471897}}<br/>{{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|year=2001|s2cid=2271590}}</ref> and [[quantum cryptography]].<ref>{{Cite journal|last=Mu|first=Yi|title=Shared cryptographic bits via quantized quadrature phase amplitudes of light|journal=Journal of Optics Communication|language=en|volume=123|pages=334–352|doi=10.1016/0030-4018(95)00688-5|year=1996|issue=1–3 |bibcode=1996OptCo.123..344M |s2cid=18374270 }}<br/></ref>

== See also ==

* [[Quantum inequalities]]

==References==

{{Reflist}}

{{Quantum information}}

[[Category:Quantum information science]]