A programming language combining quantum and classical control

TL;DR

This paper introduces a novel quantum programming language that seamlessly integrates quantum and classical control paradigms, enabling more versatile quantum computations within a unified framework.

Contribution

It presents a new language design combining pure quantum and mixed state control using syntactic, operational, and denotational approaches.

Findings

Successfully integrates quantum and classical control within a single language.

Defines a type system accommodating pure and mixed quantum data.

Provides categorical semantics for the combined control paradigms.

Abstract

The two main notions of control in quantum programming languages are often referred to as "quantum" control and "classical" control. With the latter, the control flow is based on classical information, potentially resulting from a quantum measurement, and this paradigm is well-suited to mixed state quantum computation. Whereas with quantum control, we are primarily focused on pure quantum computation and there the "control" is based on superposition. The two paradigms have not mixed well traditionally and they are almost always treated separately. In this work, we show that the paradigms may be combined within the same system. The key ingredients for achieving this are: (1) syntactically: a modality for incorporating pure quantum types into a mixed state quantum type system; (2) operationally: an adaptation of the notion of "quantum configuration" from quantum lambda-calculi, where the quantum data is replaced with pure quantum primitives; (3) denotationally: suitable (sub)categories of Hilbert spaces, for pure computation and von Neumann algebras, for mixed state computation in the Heisenberg picture of quantum mechanics.