nLab Quipper

Contents

Context

Computation

Quantum systems

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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Idea
Related concepts
References

Precursors
Quipper as such
Dependent linear types and categorical semantics
Dynamic lifting
Reversing and control

Idea

Quipper is a functional quantum programming language, specifically a domain specific programming language for quantum circuits which is embedded into Haskell. As such it is similar to QWIRE.

Related concepts

References

Quipper landing page:

www.mathstat.dal.ca/~selinger/quipper

Precursors

Quipper has grown out of developments initiated in

Peter Selinger, Towards a quantum programming language, Mathematical Structures in Computer Science 14 4 (2004) 527–586 [doi:10.1017/S0960129504004256, pdf, web]

and specifically the quantum lambda-calculus of:

Benoît Valiron, A functional programming language for quantum computation with classical control, MSc thesis, University of Ottawa (2004) [doi:10.20381/ruor-18372, pdf]
Peter Selinger, Benoît Valiron, A lambda calculus for quantum computation with classical control, Proc. of TLCA 2005 [arXiv:cs/0404056, doi:10.1007/11417170_26]
Peter Selinger, Benoît Valiron, Quantum Lambda Calculus, in: Semantic Techniques in Quantum Computation, Cambridge University Press (2009) 135-172 [doi:10.1017/CBO9781139193313.005, pdf]

[Selinger (2016):] When the QPL workshop series was first founded, it was called “Quantum Programming Languages”. One year I wasn’t participating, and while I wasn’t looking they changed the name to “Quantum Physics and Logic” — same acronym!

Back in those days in the early 21st century we were actually trying to do programming languages for quantum computing $[$ Selinger & Valiron 2004 $]$ , but the sad thing is: In those days nobody really cared. $[...]$

Now it’s 15 years later and several of these parameters have changed: There has been a renewed interest, from government agencies and also from companies who are actually building quantum computers. $[...]$ .

So now people are working on quantum programming languages again.

On categorical semantics on von Neumann algebras:

Kenta Cho, Semantics for a Quantum Programming Language by Operator Algebras, EPTCS 172 (2014) 165-190 [arXiv:1412.8545, doi:10.4204/EPTCS.172.12]

Quipper as such

Exposition of the general idea of quantum programming languages for classically controlled quantum computation with an eye towards the Quipper-language:

Benoît Valiron, Neil J. Ross, Peter Selinger, D. Scott Alexander, Jonathan M. Smith, Programming the quantum future, Communications of the ACM 58 8 (2015) 52–61 [doi:10.1145/2699415, pdf, web, promo vid:YT]

Original articles on Quipper:

Alexander Green, Peter LeFanu Lumsdaine, Neil Ross, Peter Selinger, Benoît Valiron, Quipper: A Scalable Quantum Programming Language, ACM SIGPLAN Notices 48 6 (2013) 333-342 [arXiv:1304.3390, doi:10.1145/2499370.2462177]
Jonathan Smith, Neil Ross, Peter Selinger, Benoît Valiron, Quipper: Concrete Resource Estimation in Quantum Algorithms, QAPL 2014 (arXiv:1412.0625)
Neil J. Ross, Chapters 8-9 in: Algebraic and Logical Methods in Quantum Computation, Ph.D. thesis, Dalhousie University (2015) [arXiv:1510.02198]

(introducing Proto-Quipper)

Introduction and review:

Alexander Green, Peter LeFanu Lumsdaine, Neil Ross, Peter Selinger, Benoît Valiron, An Introduction to Quantum Programming in Quipper, Lecture Notes in Computer Science 7948 Springer (2013) 110-124 [arXiv:1304.5485, doi:10.1007/978-3-642-38986-3_10]
Peter Selinger, Introduction to Quipper, talk at QPL2016 (2016) [video]

Example algorithms:

Safat Siddiqui, Mohammed Jahirul Islam, Omar Shehab, Five Quantum Algorithms Using Quipper [arXiv:1406.4481]

On quantum software verification for/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)

Dependent linear types and categorical semantics

Discussion of some dependent linear type theory and categorical semantics for (proto-)Quipper:

Francisco Rios, Peter Selinger, A Categorical Model for a Quantum Circuit Description Language, EPTCS 266 (2018) 164-178 [arXiv:1706.02630, doi:10.4204/EPTCS.266.11]
Peng Fu, Kohei Kishida, Neil Ross, Peter Selinger, A Tutorial Introduction to Quantum Circuit Programming in Dependently Typed Proto-Quipper, in I. Lanese, M. Rawski (eds.) Reversible Computation RC 2020. Lecture Notes in Computer Science, vol 12227 [arXiv:2005.08396, doi:10.1007/978-3-030-52482-1_9]
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)
Francisco Rios, On a Categorically Sound Quantum Programming Language for Circuit Description, Dalhousie University (2021) [hdl:10222/80771]
Dongho Lee, Formal Methods for Quantum Programming Languages, Paris Saclay (Dec 2022) [hal:tel-03895847]

Semantics on quantum CPOs (via quantum relations on quantum sets) for interpreting Quipper with term recursion:

Andre Kornell, Bert Lindenhovius, Michael Mislove, Quantum CPOs, EPTCS 340 (2021) 174-187 [arXiv:2109.02196, doi:10.4204/EPTCS.340.9]

Dynamic lifting

The issue of “dynamic lifting” (of “bits” resulting from quantum measurement back into the “context”):

brief mentioning on GLRSV13, p. 5
Robert Rand, Formally Verified Quantum Programming, UPenn (2018) [ediss:3175]
Dongho Lee, Sebastien Bardin, Valentin Perrelle, Benoît Valiron, Formalization of a Programming Language for Quantum Circuits with Measurement and Classical Control, talk at Journées Informatique Quantique 2019 (Nov 2019) [pdf, pdf]
Dongho Lee, Valentin Perrelle, Benoît Valiron, Zhaowei Xu, Concrete Categorical Model of a Quantum Circuit Description Language with Measurement, Leibniz International Proceedings in Informatics 213 (2021) 51:1-51:20 [arXiv:2110.02691, doi:10.4230/LIPIcs.FSTTCS.2021.51]
Andrea Colledan, Ugo Dal Lago, On Dynamic Lifting and Effect Typing in Circuit Description Languages [arXiv:2202.07636]
Andrea Colledan, Ugo Dal Lago, On Dynamic Lifting and Effect Typing in Circuit Description Languages, talk at TYPES Workshop, Nantes (2022) [pdf, pdf]
Peng Fu, Kohei Kishida, Neil J. Ross, Peter Selinger, A biset-enriched categorical model for Proto-Quipper with dynamic lifting, in proceedings of Quantum Physics and Logic 2022 [arXiv:2204.13039]
Peng Fu, Kohei Kishida, Neil J. Ross, Peter Selinger, Proto-Quipper with dynamic lifting, in proceedings of Quantum Physics and Logic 2022 [arXiv:2204.13041]
Dongho Lee, Formal Methods for Quantum Programming Languages, Paris Saclay (Dec 2022) [hal:tel-03895847]
Frank (Peng) Fu, Proto-Quipper with Dynamic Lifting, talk at CQTS (23 Oct 2023) [video:YT]

Reversing and control

Formalization as Proto-Quipper-C of a semantics for the reversing, control, and with-computed operations is in:

Peng Fu, Kohei Kishida, Neil J. Ross, Peter Selinger, Proto-Quipper with Reversing and Control [arXiv:2410.22261]

Last revised on May 24, 2025 at 21:41:44. See the history of this page for a list of all contributions to it.