Continuous-variable quantum information

Short description: Continuous (non-quantized) quantities in quantum information science

Continuous-variable (CV) quantum information is the area of quantum information science that makes use of physical observables, like the strength of an electromagnetic field, whose numerical values belong to continuous intervals.[1][2][3] One primary application is quantum computing. In a sense, continuous-variable quantum computation is "analog", while quantum computation using qubits is "digital." In more technical terms, the former makes use of Hilbert spaces that are infinite-dimensional, while the Hilbert spaces for systems comprising collections of qubits are finite-dimensional.[4] One motivation for studying continuous-variable quantum computation is to understand what resources are necessary to make quantum computers more powerful than classical ones.[5]

## Contents

* 1 Implementation
* 2 Applications
* 3 Classical emulation
* 4 Computing continuous functions with discrete quantum systems
* 5 See also
* 6 References

## Implementation

One approach to implementing continuous-variable quantum information protocols in the laboratory is through the techniques of quantum optics.[6][7][8] By modeling each mode of the electromagnetic field as a quantum harmonic oscillator with its associated creation and annihilation operators, one defines a canonically conjugate pair of variables for each mode, the so-called "quadratures", which play the role of position and momentum observables. These observables establish a phase space on which Wigner quasiprobability distributions can be defined. Quantum measurements on such a system can be performed using homodyne and heterodyne detectors.

Quantum teleportation of continuous-variable quantum information was achieved by optical methods in 1998.[9][10] (Science deemed this experiment one of the "top 10" advances of the year.[11]) 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 entangled temporal modes, available two at a time.[12] In another implementation, 60 modes were simultaneously entangled in the frequency domain, in the optical frequency comb of an optical parametric oscillator.[13]

Another proposal is to modify the 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.[14]

## Applications

Continuous-variable quantum systems can be used for quantum cryptography, and in particular, quantum key distribution.[1] Quantum computing is another potential application, and a variety of approaches have been considered.[1] The first method, proposed by Seth Lloyd and Samuel L. Braunstein in 1999, was in the tradition of the circuit model: quantum logic gates are created by Hamiltonians that, in this case, are quadratic functions of the harmonic-oscillator quadratures.[5] Later, measurement-based quantum computation was adapted to the setting of infinite-dimensional Hilbert spaces.[15][16] Yet a third model of continuous-variable quantum computation encodes finite-dimensional systems (collections of qubits) into infinite-dimensional ones. This model is due to Daniel Gottesman, Alexei Kitaev and John Preskill.[17]

## 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.[18] When the 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.[15] This type of construction can be thought of as a continuum generalization of the Spekkens toy model.[19]

## 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 functions. A major motivation for investigating the quantum computation of continuous functions is that many scientific problems have mathematical formulations in terms of continuous quantities.[20] 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 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 qubits available to a computer and the number of 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 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.[21]

Other continuous problems for which quantum algorithms have been studied include finding matrix eigenvalues,[22] phase estimation,[23] the Sturm–Liouville eigenvalue problem,[24] solving differential equations with the Feynman–Kac formula,[25] initial value problems,[26] function approximation[27] high-dimensional integration.[28], and quantum cryptography [29]

## See also

* Quantum inequalities

## References

1. ↑ 1.0 1.1 1.2 Weedbrook, Christian; Pirandola, Stefano; García-Patrón, Raúl; Cerf, Nicolas J.; Ralph, Timothy C.; Shapiro, Jeffrey H.; Lloyd, Seth (2012-05-01). "Gaussian quantum information". Reviews of Modern Physics 84 (2): 621–669. doi:10.1103/RevModPhys.84.621. Bibcode: 2012RvMP...84..621W.
2. ↑ Braunstein, Samuel L.; van Loock, Peter (2005-06-29). "Quantum information with continuous variables". Reviews of Modern Physics 77 (2): 513–577. doi:10.1103/RevModPhys.77.513. Bibcode: 2005RvMP...77..513B.
3. ↑ Adesso, Gerardo; Ragy, Sammy; Lee, Antony R. (2014-03-12). "Continuous Variable Quantum Information: Gaussian States and Beyond". Open Systems & Information Dynamics 21 (1n02): 1440001. doi:10.1142/S1230161214400010. ISSN 1230-1612.
4. ↑ Braunstein, S. L.; Pati, A. K. (2012-12-06) (in en). Quantum Information with Continuous Variables. Springer Science & Business Media. doi:10.1007/978-94-015-1258-9. ISBN 9789401512589.
5. ↑ 5.0 5.1 Lloyd, Seth; Braunstein, Samuel L. (1999-01-01). "Quantum Computation over Continuous Variables". Physical Review Letters 82 (8): 1784–1787. doi:10.1103/PhysRevLett.82.1784. Bibcode: 1999PhRvL..82.1784L.
6. ↑ Bartlett, Stephen D.; Sanders, Barry C. (2002-01-01). "Universal continuous-variable quantum computation: Requirement of optical nonlinearity for photon counting". Physical Review A 65 (4): 042304. doi:10.1103/PhysRevA.65.042304. Bibcode: 2002PhRvA..65d2304B.
7. ↑ Menicucci, Nicolas C.; Flammia, Steven T.; Pfister, Olivier (2008-07-14). "One-way quantum computing in the optical frequency comb". Physical Review Letters 101 (13): 130501. doi:10.1103/PhysRevLett.101.130501. PMID 18851426. Bibcode: 2008PhRvL.101m0501M.
8. ↑ Tasca, D. S.; Gomes, R. M.; Toscano, F.; Souto Ribeiro, P. H.; Walborn, S. P. (2011-01-01). "Continuous-variable quantum computation with spatial degrees of freedom of photons". Physical Review A 83 (5): 052325. doi:10.1103/PhysRevA.83.052325. Bibcode: 2011PhRvA..83e2325T.
9. ↑ Furusawa, A.; Sørensen, J. L.; Braunstein, S. L.; Fuchs, C. A.; Kimble, H. J.; Polzik, E. S. (1998-10-23). "Unconditional Quantum Teleportation" (in en). Science 282 (5389): 706–709. doi:10.1126/science.282.5389.706. ISSN 0036-8075. PMID 9784123. Bibcode: 1998Sci...282..706F.
10. ↑ Braunstein, Samuel L.; Fuchs, Christopher A.; Kimble, H. J. (2000-02-01). "Criteria for continuous-variable quantum teleportation". Journal of Modern Optics 47 (2–3): 267–278. doi:10.1080/09500340008244041. ISSN 0950-0340. Bibcode: 2000JMOp...47..267B.
11. ↑ "The Runners-Up: The News and Editorial Staffs" (in en). Science 282 (5397): 2157–2161. 1998-12-18. doi:10.1126/science.282.5397.2157. ISSN 0036-8075. Bibcode: 1998Sci...282.2157..
12. ↑ Yokoyama, Shota; Ukai, Ryuji; Armstrong, Seiji C.; Sornphiphatphong, Chanond; Kaji, Toshiyuki; Suzuki, Shigenari; Yoshikawa, Jun-ichi; Yonezawa, Hidehiro et al. (2013). "Ultra-large-scale continuous-variable cluster states multiplexed in the time domain". Nature Photonics 7 (12): 982–986. doi:10.1038/nphoton.2013.287. Bibcode: 2013NaPho...7..982Y.
13. ↑ Chen, Moran; Menicucci, Nicolas C.; Pfister, Olivier (2014-03-28). "Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb". Physical Review Letters 112 (12): 120505. doi:10.1103/PhysRevLett.112.120505. PMID 24724640. Bibcode: 2014PhRvL.112l0505C.
14. ↑ Ortiz-Gutiérrez, Luis; Gabrielly, Bruna; Muñoz, Luis F.; Pereira, Kainã T.; Filgueiras, Jefferson G.; Villar, Alessandro S. (2017-08-15). "Continuous variables quantum computation over the vibrational modes of a single trapped ion". Optics Communications 397: 166–174. doi:10.1016/j.optcom.2017.04.011. Bibcode: 2017OptCo.397..166O.
15. ↑ 15.0 15.1 Menicucci, Nicolas C.; van Loock, Peter; Gu, Mile; Weedbrook, Christian; Ralph, Timothy C.; Nielsen, Michael A. (2006-09-13). "Universal Quantum Computation with Continuous-Variable Cluster States". Physical Review Letters 97 (11): 110501. doi:10.1103/PhysRevLett.97.110501. PMID 17025869. Bibcode: 2006PhRvL..97k0501M.
16. ↑ Zhang, Jing; Braunstein, Samuel L. (2006-03-16). "Continuous-variable Gaussian analog of cluster states". Physical Review A 73 (3): 032318. doi:10.1103/PhysRevA.73.032318. Bibcode: 2006PhRvA..73c2318Z.
17. ↑ Gottesman, Daniel; Kitaev, Alexei; Preskill, John (2001-06-11). "Encoding a qubit in an oscillator". Physical Review A 64 (1): 012310. doi:10.1103/PhysRevA.64.012310. Bibcode: 2001PhRvA..64a2310G.
18. ↑ Bartlett, Stephen D.; Sanders, Barry C.; Braunstein, Samuel L.; Nemoto, Kae (2002-02-14). "Efficient Classical Simulation of Continuous Variable Quantum Information Processes". Physical Review Letters 88 (9): 097904. doi:10.1103/PhysRevLett.88.097904. PMID 11864057. Bibcode: 2002PhRvL..88i7904B.
19. ↑ Bartlett, Stephen D.; Rudolph, Terry; Spekkens, Robert W. (2012-07-10). "Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction". Physical Review A 86 (1): 012103. doi:10.1103/PhysRevA.86.012103. Bibcode: 2012PhRvA..86a2103B.
20. ↑ Papageorgiou, A.. "Continuous Quantum Computation: Project Description". http://quantum.cs.columbia.edu/html/project.html.
21. ↑ Traub, J. F.; Woźniakowski, H. (2002-10-01). "Path Integration on a Quantum Computer" (in en). Quantum Information Processing 1 (5): 365–388. doi:10.1023/A:1023417813916. ISSN 1570-0755.
22. ↑ Jaksch, Peter; Papageorgiou, Anargyros (2003-12-19). "Eigenvector Approximation Leading to Exponential Speedup of Quantum Eigenvalue Calculation". Physical Review Letters 91 (25): 257902. doi:10.1103/PhysRevLett.91.257902. PMID 14754158. Bibcode: 2003PhRvL..91y7902J.
23. ↑ Bessen, Arvid J. (2005-04-08). "Lower bound for quantum phase estimation". Physical Review A 71 (4): 042313. doi:10.1103/PhysRevA.71.042313. Bibcode: 2005PhRvA..71d2313B.
24. ↑ Papageorgiou, A.; Woźniakowski, H (2005). "Classical and Quantum Complexity of the Sturm–Liouville Eigenvalue Problem" (in en). Quantum Information Processing 4 (2): 87–127. doi:10.1007/s11128-005-4481-x. Bibcode: 2005quant.ph..2054P.
Papageorgiou, A.; Woźniakowski, H. (2007-04-01). "The Sturm-Liouville Eigenvalue Problem and NP-Complete Problems in the Quantum Setting with Queries" (in en). Quantum Information Processing 6 (2): 101–120. doi:10.1007/s11128-006-0043-0. ISSN 1570-0755.
25. ↑ Kwas, Marek (2004-10-18). "Complexity of multivariate Feynman-Kac path integration in randomized and quantum settings". arXiv:quant-ph/0410134.
26. ↑ Kacewicz, Bolesław (2004). "Randomized and quantum algorithms yield a speed-up for initial-value problems" (in en). Journal of Complexity 20 (6): 821–834. doi:10.1016/j.jco.2004.05.002.
Szczesny, Marek (2006-12-12). "Randomized and Quantum Solution of Initial-Value Problems for Ordinary Differential Equations of Order k". arXiv:quant-ph/0612085.
Kacewicz, Bolesław (2005). "Improved bounds on the randomized and quantum complexity of initial-value problems" (in en). Journal of Complexity 21 (5): 740–756. doi:10.1016/j.jco.2005.05.003.
27. ↑ Novak, Erich; Sloan, Ian H.; Woźniakowski, Henryk (2004-04-01). "Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers" (in en). Foundations of Computational Mathematics 4 (2): 121–156. doi:10.1007/s10208-002-0074-6. ISSN 1615-3375.
Heinrich, Stefan (2004). "Quantum approximation I. Embeddings of finite-dimensional Lp spaces" (in en). Journal of Complexity 20 (1): 5–26. doi:10.1016/j.jco.2003.08.002.
Heinrich, Stefan (2004). "Quantum approximation II. Sobolev embeddings" (in en). Journal of Complexity 20 (1): 27–45. doi:10.1016/j.jco.2003.08.003.
28. ↑ Heinrich, Stefan (2002). "Quantum Summation with an Application to Integration" (in en). Journal of Complexity 18 (1): 1–50. doi:10.1006/jcom.2001.0629.
Heinrich, Stefan (2003-02-01). "Quantum integration in Sobolev classes". Journal of Complexity 19 (1): 19–42. doi:10.1016/S0885-064X(02)00008-0.
Novak, Erich (2001). "Quantum Complexity of Integration" (in en). Journal of Complexity 17 (1): 2–16. doi:10.1006/jcom.2000.0566.
29. ↑ Mu, Yi (1996). "Shared cryptographic bits via quantized quadrature phase amplitudes of light" (in en). Journal of Optics Communication 123 (1–3): 334–352. doi:10.1016/0030-4018(95)00688-5. Bibcode: 1996OptCo.123..344M.

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