The Zeta Calculus

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 $\lambda$-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, $\zeta$ and $\xi$, 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.