Semantics of Higher-Order Quantum Computation via Geometry of Interaction

Particle-style token machines are a way to interpret proofs and programs, when the latter are written following the principles of linear logic. In this paper, we show that token machines also make sense when the programs at hand are those of a simple quantum λ -calculus with implicit qubits. This, however, requires generalising the concept of a token machine to one in which more than one particle travel around the term at the same time. The presence of multiple tokens is intimately related to entanglement and allows us to give a simple operational semantics to the calculus, coherently with the principles of quantum computation.

“…In [13], a geometry of interaction model for Selinger and Valiron's quantum λ -calculus [16] is defined. The model is formulated in particle-style.…”

Section: Related Workmentioning

confidence: 99%

“…On the one hand, the Hilbert space on top of which the first formulation of GoI is given is precisely the canonical state space of a quantum Turing machine [2]. On the other hand, the definition of a token machine provides a mathematically simpler setting, which has already found a role in this context [4,13].…”

Section: Introductionmentioning

confidence: 99%

Wave-Style Token Machines and Quantum Lambda Calculi

Lago

Zorzi

2015

“…Although the present paper has not yet been able to give a semantics for a higher-order quantum programming language, compared to such works [16,24,33], our semantics is quite simple: a type is interpreted just as a W * -algebra, and a program as a map of them (in opposite direction). However, the theory of operator algebras itself could be complicated enough.As is the case in previous works [16,24,33,39], quantum computation is usually modelled by using finite dimensional Hilbert spaces C n (or matrix algebras M n ∼ = B(C n )). It seems that the (explicit) use of operator algebras is not so common in the area of quantum computation.…”

mentioning

confidence: 94%

“…As is the case in previous works [16,24,33,39], quantum computation is usually modelled by using finite dimensional Hilbert spaces C n (or matrix algebras M n ∼ = B(C n )). It seems that the (explicit) use of operator algebras is not so common in the area of quantum computation.…”

mentioning

confidence: 99%

Semantics for a Quantum Programming Language by Operator Algebras

Cho

2014

This paper presents a novel semantics for a quantum programming language by operator algebras, which are known to give a formulation for quantum theory that is alternative to the one by Hilbert spaces. We show that the opposite category of the category of W*-algebras and normal completely positive subunital maps is an elementary quantum flow chart category in the sense of Selinger. As a consequence, it gives a denotational semantics for Selinger's first-order functional quantum programming language QPL. The use of operator algebras allows us to accommodate infinite structures and to handle classical and quantum computations in a unified way.Comment: In Proceedings QPL 2014, arXiv:1412.810

“…In recent years, following the endeavors of Abramsky and Coecke to express some of the basic quantummechanical concepts in an abstract axiomatic category theory setting, several models have been worked out to capture the semantics of quantum information protocols [1] and programming languages [12,16,24]. Concerning quantum hardware, an algebra of automata which include both classical and quantum entities has been studied in [13].…”

Section: Introductionmentioning

confidence: 99%

Quantum Turing automata

Bartha¹

2014

A denotational semantics of quantum Turing machines having a quantum control is defined in the dagger compact closed category of finite dimensional Hilbert spaces. Using the Moore-Penrose generalized inverse, a new additive trace is introduced on the restriction of this category to isometries, which trace is carried over to directed quantum Turing machines as monoidal automata. The Joyal-Street-Verity Int construction is then used to extend this structure to a reversible bidirectional one.Comment: In Proceedings DCM 2012, arXiv:1403.757