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quantum computation

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Contents

Context

Computation

constructive mathematics, realizability, computability

intuitionistic mathematics

propositions as types, proofs as programs, computational trinitarianism

Constructive mathematics

Realizability

Computability

Quantum systems

quantum logic


quantum probability theory – observables and states


quantum information


quantum computation

quantum algorithms:


quantum physics

Contents

Idea

General

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.

Classical control, quantum data

Any practical quantum computer will be classically controlled (Knill 96, Ömer 03, Nagarajan, Papanikolaou & Williams 07, Devitt 14):

From Miszczak 11


From Nagarajan, Papanikolaou Williams 07

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).

Quantum languages and quantum circuits

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 controlquantum data
intuitionistic typesdependent 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.

References

General

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

Textbook accounts:

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:

Discussion in terms of monoidal category-theory and finite quantum mechanics in terms of dagger-compact categories:

Experimental demonstration of “quantum supremacy” (“quantum advantage”):

Review:

Quantum programming languages

On quantum programming languages (programming languages for quantum computation):

General:

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:

The corresponding string diagrams are known in quantum computation as quantum circuit diagrams:

functional programming languages for quantum computation:

QPL:

Quipper:

QML:

QWIRE:

On classically controlled quantum computation:

Quantum programming via dependent linear type theory/indexed monoidal (∞,1)-categories:

specifically with Quipper:

On quantum software verification:

with Quipper:

  • Linda Anticoli, Carla Piazza, Leonardo Taglialegne, Paolo Zuliani, Towards Quantum Programs Verification: From Quipper Circuits to QPMC, In: Devitt S., Lanese I. (eds) Reversible Computation. RC 2016. Lecture Notes in Computer Science, vol 9720. Springer, Cham (doi:10.1007/978-3-319-40578-0_16)

with QWIRE:

Classically controlled quantum computing

Theory of classically controlled quantum computing and parameterized quantum circuits:

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)

  • TensorFlow.org (Google), Hybrid quantum classical models

Noisy intermediate-scale quantum computing

  • 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)

Quantum programming via monads

Discussion of aspects of quantum programming in terms of monads in functional programming are in

As linear logic

Discussion of quantum computation as the internal linear logic/linear type theory of compact closed categories is in

An exposition along these lines is in

  • John Baez, Mike Stay, Physics, topology, logic and computation: a rosetta stone, arxiv/0903.0340; in “New Structures for Physics”, ed. Bob Coecke, Lecture Notes in Physics 813, Springer, Berlin, 2011, pp. 95-174

In terms of dagger-compact categories

Discussion in terms of finite quantum mechanics in terms of dagger-compact categories:

  • Jamie Vicary, Section 3 of: The Topology of Quantum Algorithms, (LICS 2013) Proceedings of 28th Annual ACM/IEEE Symposium on Logic in Computer Science, pages 93-102 (arXiv:1209.3917)

Topological quantum computing

topological quantum computation is discussed in

Relation to tensor networks

Relation to tensor networks, specifically matrix product states:

  • Yiqing Zhou, E. Miles Stoudenmire, Xavier Waintal, What limits the simulation of quantum computers?, arXiv:2002.07730

Experimental realization

  • Han-Shen Zhong et al. Quantum computational advantage using photons, Science 370, n. 6523 (2020) 1460-1463 doi
category: computer science, physics

Last revised on December 23, 2021 at 10:20:15. See the history of this page for a list of all contributions to it.

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