Typing Quantum Superpositions and Measurement

Abstract

Abstract. We propose a way to unify two approaches of non-cloning in quantum lambda-calculi. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as algebraiclinear functions. We illustrate this idea by defining a quantum extension of first-order simply-typed lambda-calculus, where the type is linear on superposition, while allows cloning base vectors. In addition, we provide an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis.

Two linearities for quantum computing in the lambda calculus

“…Assume there exists such an operator U , so given any |ϕ and |φ one has U |ψϕ = |ϕϕ and also U |ψφ = |φφ . Then U ϕψ|U ψφ = ϕϕ|φφ (1) where U ϕψ| is the conjugate transpose of U |ψϕ . However, notice that the left side of equation (1) can be rewritten as ϕψ| U † U |ψφ = ϕψ|ψφ = ϕ|φ While the right side of equation (1) can be rewritten as ϕ|φ ϕ|φ = ϕ|φ 2 So ϕ|φ = ϕ|φ 2 , which implies either ϕ|φ = 0 or ϕ|φ = 1, none of which can be assumed in the general case, since |ϕ and |φ were picked as random qubits.…”

“…However, interpreting all λ-terms as linear functions forbids to extend the calculus with non-linear operators, such as measurement. For instance, the term (λx.πx)(|0 +|1 ), where π represents a measurement in the computational basis, would reduce to ((λx.πx) |0 + (λx.πx) |1 ), while it should reduce to |0 with probability 1 2 and to |1 with probability 1 2 . In this paper, we propose a way to unify the two approaches, distinguishing duplicable and non-duplicable data by their type, like in the logic-linear calculi; and interpreting λ-terms as linear functions, like in the algebraic-linear calculi, when they expect duplicable data.…”

A Categorical Construction for the Computational Definition of Vector Spaces

“…This way, a functor from Set to Vec will allow to do the needed manipulation, while a forgetful functor from Vec to Set will return the result of the computation. The calculus Lambda-S [7,8] is a first-order typed fragment of Lineal, extended with measurements. The type system has been designed as a quantum lambda calculus, where the main goal was to study the non-cloning restrictions.…”

“…In [7,8] a first denotational semantics (in environment style) is given where the type B is interpreted as {|0 , |1 } while S(B) is interpreted as Span({|0 , |1 }) = C 2 , and, in general, a type A is interpreted as a basis while S(A) is the vector space generated by such a basis. In [10,11] we went further and gave a preliminary concrete categorical interpretation of Lambda-S where S is a functor of an adjunction between the category Set and the category Vec.…”

A Lambda Calculus for Density Matrices with Classical and Probabilistic Controls

“…A generalisation to any arbitrary measurement can be considered in a future, however, for the sake of simplicity in the classical control, we consider only measurements in the computational basis, which is a common practice in quantum lambda calculi[10,16,18,20,21,30].…”