Unitary Expressions: A Necessary Abstraction for Extensible Quantum Programming Languages and Systems

Quantum gates are the fundamental instructions of digital quantum computers. Current programming languages, systems, and software development toolkits identify these operational gates by their titles, which requires a shared understanding of their meanings. However, in the continuously developing software ecosystem surrounding quantum computing -- spanning high-level programming systems to low-level control stacks -- this identification process is often error-prone, challenging to debug, maintenance-heavy, and resistant to change. In this paper, we propose replacing this nominal gate representation with a functional one. We introduce the OpenQudit system for describing, parsing, optimizing, analyzing, and utilizing programs comprising gates described as symbolic unitary expressions. As part of this effort, we design the Qudit Gate Language (QGL), a unitary-specific expression language, and implement a differentiating just-in-time compiler in OpenQudit towards embedding this language in quantum programming languages and systems. Additionally, we have precisely designed and implemented the Qudit Virtual Machine (QVM) to evaluate quantum programs and their gradients efficiently. This evaluation is performed millions of times during the compilation of quantum programs. Our QVM can compute gradients approximately ten times faster than current leading numerical quantum compilation frameworks in the most common use cases. Altogether, the OpenQudit system is envisioned to (1) support many-level or qudit-based quantum systems, (2) enable the safe composition of program transformation tools, (3) accelerate circuit optimizers and transpilers, (4) enable compiler extensibility, and (5) provide a productive, simple-to-use interface to quantum practitioners.