quantum computation



Quantum systems

quantum logic

quantum probability theoryobservables and states

quantum information

quantum computation

quantum algorithms:

quantum physics




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.



The idea of quantum computation was first expressed in:

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

Textbook accounts:

Further introduction and survey:

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”):


Quantum programming languages

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


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:





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:

in particular in quantum machine learning:

Noisy intermediate-scale 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

In terms of dagger-compact categories

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

Topological quantum computing

topological quantum computation is discussed in

Relation to tensor networks

Relation to tensor networks, specifically matrix product states:

Experimental realization