constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
quantum algorithms:
Quantum computation is computation in terms of quantum information theory, possibly implemented on quantum computers, hence on physical systems for which phenomena of quantum mechanics are not negligible. In terms of computational trinitarianism quantum computation is the computation corresponding to (some kind of) quantum logic.
Specifically, topological quantum computation is (or is meant to be) quantum computation implemented on physical systems governed by topological quantum field theory, such as Chern-Simons theory. A prominent example of this is the (fractional) quantum Hall effect in solid state physics.
Any practical quantum computer will be classically controlled (Knill 96, Ömer 03, Nagarajan, Papanikolaou & Williams 07, Devitt 14):
See the list of references below.
The paradigm of classically controlled quantum computation applies in particular (Kim & Swingle 17) to the currently and near-term available noisy intermediate-scale quantum (NISQ) computers (Preskill 18, see the references below), which are useful for highly specialized tasks (only) and need to be emdedded in and called from a more comprehensive classical computing environment.
This, in turn, applies particularly to applications like quantum machine learning (Benedetti, Lloyd, Sack & Fiorentini 19, TensorFlow Quantum, for more see the references there).
A natural way (via computational trinitarianism) to understand quantum programming languages is as linear logic/linear type theory (Pratt 92, for more see at quantum logic) with categorical semantics in non-cartesian symmetric monoidal categories (Abramsky & Coecke 04, Abramsky & Duncan 05, Duncan 06, Lago-Faffian 12). .
The corresponding string diagrams are known as quantum circuit diagrams.
In fact, languages for classically controlled quantum computation should be based on dependent linear type theory (Vakar 14, Vakar 15, Vakar 17, Sec. 3, Lundfall 17, Lundfall 18, following Schreiber 14) with categorical semantics in indexed monoidal categories:
classical control | quantum data |
---|---|
intuitionistic types | dependent linear types |
This idea of classically controlled quantum programming via dependent linear type theory has been implemented for the Quipper language in FKS 20, FKRS 20.
The idea of quantum computation was first expressed in:
Paul Benioff, The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines, J Stat Phys 22, 563–591 (1980) (doi:10.1007/BF01011339)
Richard Feynman, Simulating physics with computers, Int J Theor Phys 21, 467–488 (1982) (doi:10.1007/BF02650179)
Quantum computation became a plausible practical possibility with the understanding of quantum error correction in
Peter W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493(R) 1995 (doi:10.1103/PhysRevA.52.R2493)
Andrew M. Steane, Error Correcting Codes in Quantum Theory, Phys. Rev. Lett. 77, 793 1996 (doi:10.1103/PhysRevLett.77.793)
Robert Calderbank, Peter W. Shor, Good Quantum Error-Correcting Codes Exist, Phys. Rev. A, Vol. 54, No. 2, pp. 1098-1106, 1996 (doi:10.1103/PhysRevA.54.1098)
Textbook accounts:
Ranee K. Brylinski, Goong Chen (eds.), Mathematics of Quantum Computation, Chapman and Hall/CRC 2002 (doi:10.1201/9781420035377)
Goong Chen, Louis Kauffman, Samuel J. Lomonaco (eds.), Mathematics of Quantum Computation and Quantum Technology, Chapman and Hall/CRC (2007) (ISBN:9780367388614, doi:10.1201/9781584889007)
Further introduction and survey:
Michael A. Nielsen, Isaac L. Chuang, Quantum computation and quantum information, Cambridge University Press 2000 (pdf)
Giuliano Benenti, Giulio Casati, Davide Rossini, Principles of Quantum Computation and Information, World Scientific (2004, 2018) (doi:10.1142/10909, 2004 pdf)
John Preskill, Quantum Computation lecture notes, since 2004 (web)
Jens Eisert, M. M. Wolf, Quantum computing, In: Handbook of Nature-Inspired and Innovative Computing, Springer 2006 (arXiv:quant-ph/0401019)
Greg Kuperberg, A concise introduction to quantum probability, quantum mechanics, and quantum computation, 2005 (pdf)
Scott Aaronson, Lecture notes Quantum Computing Since Democritus 2006 (web)
Michael Loceff, A course in quantum computing, 2013 (pdf)
National Academies of Sciences, Engineering, and Medicine, Quantum Computing: Progress and Prospects, The National Academies Press 2019 (doi:10.17226/25196)
Qiang Zhang, Feihu Xu, Li Li, Nai-Le Liu, Jian-Wei Pan, Quantum information research in China, Quantum Sci. Technol. 4 040503 (doi:10.1088/2058-9565/ab4bea)
Farzan Jazaeri, Arnout Beckers, Armin Tajalli, Jean-Michel Sallese, A Review on Quantum Computing: Qubits, Cryogenic Electronics and Cryogenic MOSFET Physics, 2019 MIXDES - 26th International Conference “Mixed Design of Integrated Circuits and Systems”, 2019, pp. 15-25, (arXiv:1908.02656, doi:10.23919/MIXDES.2019.8787164)
Quantum Computing Review Q4 2020, IDQ January 2021
Melanie Swan, Renato P dos Santos, Frank Witte, Between Science and Economics, Volume 2: Quantum Computing Physics, Blockchains, and Deep Learning Smart Networks, World Scientific 2020 (doi:10.1142/q0243)
Jiajun Chen, Review on Quantum Communication and Quantum Computation, Journal of Physics: Conference Series, Volume 1865, 2021 International Conference on Advances in Optics and Computational Sciences (ICAOCS) 2021 21-23 January 2021, Ottawa, Canada (doi:10.1088/1742-6596/1865/2/022008)
David Matthews, How to get started in quantum computing, Nature 591 March 2021 (nature:d41586-021-00533-x, pdf)
Christine Middleton, What’s under the hood of a quantum computer?, Physics Today, March 2021 (doi:10.1063/PT.6.1.20210305a)
See also:
Wikipedia, Quantum computation
Quantiki – Quantum Information Portal and Wiki
Discussion in terms of monoidal category-theory and finite quantum mechanics in terms of dagger-compact categories:
Experimental demonstration of “quantum supremacy” (“quantum advantage”):
F. Arute et al. Quantum supremacy using a programmable superconducting processor, Nature 574 (2019) 505–510 (doi:10.1038/s41586-019-1666-5)
Han-Sen Zhong et al., Quantum computational advantage using photons, Science 370 6523 (2020) 1460-1463 (doi:10.1126/science.abe8770)
Review:
Adrian Cho, Google claims quantum computing milestone, Science 365 6460 (2019) 1364 (doi:10.1126/science.365.6460.1364)
Philip Ball, Physicists in China challenge Google’s “quantum advantage”, Nature 588 380 (2020) (doi:10.1038/d41586-020-03434-7)
On quantum programming languages (programming languages for quantum computation):
General:
Jarosław Adam Miszczak, Models of quantum computation and quantum programming languages, Bull. Pol. Acad. Sci.-Tech. Sci., Vol. 59, No. 3 (2011), pp. 305-324 (arXiv:1012.6035)
Jarosław Adam Miszczak, High-level Structures for Quantum Computing, Morgan&Claypool 2012 (doi:10.2200/S00422ED1V01Y201205QMC006, pdf)
See also:
Surveys of existing languages:
Simon Gay, Quantum programming languages: Survey and bibliography, Mathematical Structures in Computer Science16(2006) (doi:10.1017/S0960129506005378, pdf)
Sunita Garhwal, Maryam Ghorani , Amir Ahmad, Quantum Programming Language: A Systematic Review of Research Topic and Top Cited Languages, Arch Computat Methods Eng 28, 289–310 (2021) (doi:10.1007/s11831-019-09372-6)
Quantum programming via quantum logic understood as linear type theory interpreted in symmetric monoidal categories:
Vaughan Pratt, Linear logic for generalized quantum mechanics, in Proc. of Workshop on Physics and Computation (PhysComp’92) (pdf, web)
Samson Abramsky, Bob Coecke, A categorical semantics of quantum protocols , Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004) (arXiv:quant-ph/0402130)
Samson Abramsky, Ross Duncan, A Categorical Quantum Logic, Mathematical Structures in Computer Science, Volume 16, Issue 3 (2006) pp. 469 - 489 (arXiv:quant-ph/0512114, doi:10.1017/S0960129506005275)
Ross Duncan, Types for Quantum Computing, 2006 (pdf)
Ugo Dal Lago, Claudia Faggian, On Multiplicative Linear Logic, Modality and Quantum Circuits, EPTCS 95, 2012, pp. 55-66 (arXiv:1210.0613)
The corresponding string diagrams are known in quantum computation as quantum circuit diagrams:
Giuliano Benenti, Giulio Casati, Davide Rossini, Chapter 3 of: Principles of Quantum Computation and Information, World Scientific 2018 (doi:10.1142/10909, 2004 pdf)
functional programming languages for quantum computation:
QPL:
Peter Selinger, The Quipper Language (web)
Alexander Green, Peter LeFanu Lumsdaine, Neil Ross, Peter Selinger, Benoît Valiron, Quipper: A Scalable Quantum Programming Language, ACM SIGPLAN Notices 48(6):333-342, 2013 (arXiv:1304.3390)
Alexander Green, Peter LeFanu Lumsdaine, Neil Ross, Peter Selinger, Benoît Valiron, An Introduction to Quantum Programming in Quipper, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 (arXiv:1304.5485)
Jonathan Smith, Neil Ross, Peter Selinger, Benoît Valiron, Quipper: Concrete Resource Estimation in Quantum Algorithms, QAPL 2014 (arXiv:1412.0625)
Francisco Rios, Peter Selinger, A Categorical Model for a Quantum Circuit Description Language, EPTCS 266, 2018, pp. 164-178 (arXiv:1706.02630)
QML:
Thorsten Altenkirch, Jonathan Grattage, A functional quantum programming language, Logic in Computer Science, 2005. Proceedings. 20th Annual IEEE Symposium on, 26-29 June 2005 Page(s):249 - 258 (arXiv:quant-ph/0409065, doi:10.1109/LICS.2005.1, pdf)
Jonathan Grattage, An overview of QML with a concrete implementation in Haskell, talk at Quantum Physics and Logic 2008, ENTCS: Proceedings of QPL V - DCV IV, 157-165, Reykjavik, Iceland, 2008 (arXiv:0806.2735)
Jennifer Paykin, Robert Rand, Steve Zdancewic, QWIRE: a core language for quantum circuits, POPL 2017: Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming LanguagesJanuary 2017 Pages 846–858 (doi:10.1145/3009837.3009894)
(using the exponential modality between linear type theory and intuitionistic type theory)
Jennifer Paykin, Linear/non-Linear Types For Embedded Domain-Specific Languages, 2018 (upenn:2752)
Jennifer Paykin, Steve Zdancewic, A HoTT Quantum Equational Theory, talk at QPL2019 (arXiv:1904.04371)
(using ambient homotopy type theory)
On classically controlled quantum computation:
Quantum programming via dependent linear type theory/indexed monoidal (∞,1)-categories:
Urs Schreiber, Quantization via Linear Homotopy Types (arXiv:1402.7041)
Peng Fu, Kohei Kishida, Peter Selinger, Linear Dependent Type Theory for Quantum Programming Languages, LICS ‘20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer ScienceJuly 2020 Pages 440–453 (arXiv:2004.13472, doi:10.1145/3373718.3394765, pdf, video)
specifically with Quipper:
On quantum software verification:
with Quipper:
with QWIRE:
Theory of classically controlled quantum computing and parameterized quantum circuits:
Emanuel Knill, Conventions for quantum pseudocode, 1996 (LA-UR-96-2724, pdf)
Bernhard Ömer, Structured Quantum Programming, 2003/2009 (pdf)
Simon Perdrix, Philippe Jorrand, Classically-Controlled Quantum Computation, Math. Struct. in Comp. Science, 16:601-620, 2006 (arXiv:quant-ph/0407008)
Rajagopal Nagarajan, Nikolaos Papanikolaou, David Williams, Simulating and Compiling Code for the Sequential Quantum Random Access Machine, Electronic Notes in Theoretical Computer Science Volume 170, 6 March 2007, Pages 101-124 (doi:10.1016/j.entcs.2006.12.014)
Jarosław Adam Miszczak, Models of quantum computation and quantum programming languages, Bull. Pol. Acad. Sci.-Tech. Sci., Vol. 59, No. 3 (2011), pp. 305-324 (arXiv:1012.6035)
Simon J. Devitt, Classical Control of Large-Scale Quantum Computers, In: S. Yamashita, S. Minato (eds.) Reversible Computation Lecture Notes in Computer Science, vol 8507. Springer 2014 (arXiv:1405.4943, doi:10.1007/978-3-319-08494-7_3)
Jarrod R McClean, Jonathan Romero, Ryan Babbush and Alán Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms, New Journal of Physics, Volume 18, February 2016 (doi:10.1088/1367-2630/18/2/023023)
Murray Thom (D-Wave), Three Truths and the Advent of Hybrid Quantum Computing, Jun 25, 2019
Application of classically controlled quantum computation:
Isaac H. Kim, Brian Swingle, Robust entanglement renormalization on a noisy quantum computer (arXiv:1711.07500)
(in terms of holographic tensor network states)
Sukin Sim, Peter D. Johnson, Alan Aspuru-Guzik, Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms, Adv. Quantum Technol. 2 (2019) 1900070 (arXiv:1905.10876)
Mateusz Ostaszewski, Edward Grant, Marcello Benedetti, Structure optimization for parameterized quantum circuits, Quantum 5, 391 (2021) (arXiv:1905.09692)
Ruslan Shaydulin et al., A Hybrid Approach for Solving Optimization Problems on Small Quantum Computers (doi:10.1109/MC.2019.2908942)
Eneko Osaba, Esther Villar-Rodriguez, Izaskun Oregi, Aitor Moreno-Fernandez-de-Leceta, Focusing on the Hybrid Quantum Computing – Tabu Search Algorithm: new results on the Asymmetric Salesman Problem (arXiv:2102.05919)
in particular in quantum machine learning:
Marcello Benedetti, Erika Lloyd, Stefan Sack, Mattia Fiorentini, Parameterized quantum circuits as machine learning models, Quantum Science and Technology 4, 043001 (2019) (arXiv:1906.07682)
D. Zhu et al. Training of quantum circuits on a hybrid quantum computer, Science Advances, 18 Oct 2019: Vol. 5, no. 10, eaaw9918 (doi: 10.1126/sciadv.aaw9918)
Andrea Mari, Thomas R. Bromley, Josh Izaac, Maria Schuld, Nathan Killoran, Transfer learning in hybrid classical-quantum neural networks, Quantum 4, 340 (2020) (arXiv:1912.08278)
Thomas Hubregtsen, Josef Pichlmeier, Patrick Stecher, Koen Bertels, Evaluation of Parameterized Quantum Circuits: on the relation between classification accuracy, expressibility and entangling capability, Quantum Machine Intelligence volume 3, Article number: 9 (2021) (arXiv:2003.09887, doi:10.1007/s42484-021-00038-w)
Nikolaj Moll et al., Quantum optimization using variational algorithms on near-term quantum devices, Quantum Science and Technology, Volume 3, Number 3 2018 (arXiv:1710.01022)
John Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2018-08-06, volume 2, page 79 (arXiv:1801.00862, doi:10.22331/q-2018-08-06-79)
Daniel Koch, Brett Martin, Saahil Patel, Laura Wessing, and Paul M. Alsing, Demonstrating NISQ era challenges in algorithm design on IBM’s 20 qubit quantum computer, AIP Advances 10, 095101 (2020) (doi:10.1063/5.0015526)
Frank Leymann, Johanna Barzen, The bitter truth about gate-based quantum algorithms in the NISQ era, Quantum Sci. Technol. 5 044007 (2020) (doi:10.1088/2058-9565/abae7d)
Discussion of aspects of quantum programming in terms of monads in functional programming are in
Thorsten Altenkirch, Alexander Green, The quantum IO monad, in Semantic Techniques in Quantum Computation, January 2009, appeared in 2010 (pdf, talk slides)
J. K. Vizzotto, Thorsten Altenkirch, A. Sabry, Structuring quantum effects: superoperators as arrows (arXiv:quant-ph/0501151)
Discussion of quantum computation as the internal linear logic/linear type theory of compact closed categories is in
Samson Abramsky, Ross Duncan, A Categorical Quantum Logic, Mathematical Structures in Computer Science, 2006 (arXiv:quant-ph/0512114)
Ross Duncan, Types for quantum mechanics, 2006 (pdf, slides)
Ugo Dal Lago, Claudia Faggian, On Multiplicative Linear Logic, Modality and Quantum Circuits, EPTCS 95, 2012, pp. 55-66 (arXiv:1210.0613)
An exposition along these lines is in
Discussion in terms of finite quantum mechanics in terms of dagger-compact categories:
topological quantum computation is discussed in
Michael Freedman, Alexei Kitaev, Michael J. Larsen, Zhenghan Wang, Topological Quantum Computation (arXiv:quant-ph/0101025)
Zhenghan Wang, Topologization of electron liquids with Chern-Simons theory and quantum computation (arXiv:cond-mat/0601285)
Michael Freedman, Michael Larsen, Zhenghan Wang, A modular functor which is universal for quantum computation (arXiv:quant-ph/0001108)
Alexei Kitaev, Fault-tolerant quantum computation by anyons, Annals Phys. 303 (2003) 2-30 (arXiv:quant-ph/9707021)
Relation to tensor networks, specifically matrix product states:
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