nLab Quipper




Quantum systems

quantum logic

quantum physics

quantum probability theoryobservables and states

quantum information

quantum computation


quantum algorithms:

quantum sensing

quantum communication

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

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

syntax object language

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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




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.


Quipper landing page:


Quipper has grown out of developments initiated in

and specifically the quantum lambda-calculus of:

[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.

Quipper as such

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

Original articles on Quipper:

Introduction and review:

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:

Dynamic lifting

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

Last revised on September 11, 2023 at 12:36:17. See the history of this page for a list of all contributions to it.