The Zeta Calculus
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We propose a quantum programming language that generalizes the λ-calculus. The language is non-linear; duplicated variables denote, not cloning of quantum data, but sharing a qubit's state; that is, producing an entangled pair of qubits whose amplitudes are identical with respect to a chosen basis. The language has two abstraction operators, ζ and ξ, corresponding to the Z- and X-bases; each abstraction operator is also parameterised by a phase, indicating a rotation that is applied to the input before it is shared. We give semantics for the language in the ZX-calculus and prove its equational theory sound. We show how this language can provide a good representation of higher-order functions in the quantum world.
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