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Quipper (Rev #7) in nLab

Context

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 = \linebreak 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

Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Idea

Quipper is a functional quantum programming language

References

General

Resources:

Peter Selinger, The Quipper Language (web)

Original articles:

Peter Selinger, Towards a quantum programming language, Mathematical Structures in Computer Science 14 4 (2004) 527–586 [[doi:10.1017/S0960129504004256](https://doi.org/10.1017/S0960129504004256), pdf, web]
Alexander Green, Peter LeFanu Lumsdaine, Neil Ross, Peter Selinger, Benoît Valiron, Quipper: A Scalable Quantum Programming Language, ACM SIGPLAN Notices 48(6):333-342, 2013 (arXiv:1304.3390)
Jonathan Smith, Neil Ross, Peter Selinger, Benoît Valiron, Quipper: Concrete Resource Estimation in Quantum Algorithms, QAPL 2014 (arXiv:1412.0625)

Review:

Alexander Green, Peter LeFanu Lumsdaine, Neil Ross, Peter Selinger, Benoît Valiron, An Introduction to Quantum Programming in Quipper, Lecture Notes in Computer Science 7948:110-124, Springer, 2013 (arXiv:1304.5485)

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 Quipper:

Francisco Rios, Peter Selinger, A Categorical Model for a Quantum Circuit Description Language, EPTCS 266 (2018) 164-178 [[arXiv:1706.02630](https://arxiv.org/abs/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](https://arxiv.org/abs/2005.08396), doi:10.1007/978-3-030-52482-1_9]

The issue of “dynamic lifting” (of “bits” back into the “context”):

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](https://arxiv.org/abs/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](https://arxiv.org/abs/2202.07636)]
Andrea Colledan, Ugo Dal Lago, On Dynamic Lifting and Effect Typing in Circuit Description Languages, talk at TYPES Workshop, Nantes (2022) [[pdf](https://types22.inria.fr/files/2022/06/TYPES_2022_slides_13.pdf), pdf]
Peng Fu, Kohei Kishida, Neil J. Ross, Peter Selinger, A biset-enriched categorical model for Proto-Quipper with dynamic lifting [[arXiv:2204.13039](https://arxiv.org/abs/2204.13039)]
Peng Fu, Kohei Kishida, Neil J. Ross, Peter Selinger, Proto-Quipper with dynamic lifting [[arXiv:2204.13041](https://arxiv.org/abs/2204.13041)]

Revision on October 22, 2022 at 18:03:41 by Urs Schreiber See the history of this page for a list of all contributions to it.