The Zeta Calculus
Nicklas Bot\"o, Fabian Forslund

TL;DR
The paper introduces the Zeta Calculus, a quantum programming language extending lambda calculus with entanglement semantics, basis-dependent abstractions, and ZX-calculus semantics, enabling higher-order quantum functions.
Contribution
It presents the Zeta Calculus with basis-specific abstractions and formal semantics, advancing quantum programming language design and higher-order quantum function representation.
Findings
Language semantics are grounded in ZX-calculus.
Equational theory is proven sound.
Supports higher-order quantum functions.
Abstract
We propose a quantum programming language that generalizes the -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, and , 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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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Computability, Logic, AI Algorithms · Quantum Mechanics and Applications
