nLab
quantum computation

Contents

Context

Constructivism, Realizability, Computability

Quantum systems

quantum logic


quantum probability theoryobservables 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

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)

It became a plausible practical possibility with the understanding of quantum error correction in

Introduction and survey:

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

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

Last revised on May 13, 2021 at 14:50:37. See the history of this page for a list of all contributions to it.