Quantum Term Rewrite Systems: Applications to Complexity Analysis

2026-07-22

AI Summary

This work addresses the verifiability of complexity analysis and termination guarantees in quantum computation. It introduces, for the first time, a physically realizable Quantum Term Rewriting System (QTRS) and establishes its equivalence with uniform families of quantum circuits, thereby precisely characterizing the class of functions FBQP. By incorporating termination proof techniques, the authors define a subclass of terminating QTRS whose reduction lengths directly bound the size of the corresponding quantum circuits. This framework not only enables static complexity analysis and certification but is also proven complete for functions computable in quantum polynomial time, providing a foundational theoretical basis for ensuring both reliability and efficiency in quantum programs.

Abstract

Term Rewrite Systems (TRS) is a computational model offering a level of abstraction well-suited towards static analysis, e.g., termination or complexity analyses. In this paper, we introduce Quantum Term Rewrite Systems (QTRS), an extension of TRS to quantum computing, thus allowing to benefit from quantum advantage while being able to certify the complexity. We ensure that QTRS correspond to physically realizable processes and adapt techniques to obtain termination certificates or generic bounds on the reduction length. We delineate a class of terminating QTRS that can be compiled to uniform families of quantum circuits of size bounded by the reduction length. Conversely, this class is universal for quantum circuits. In particular, we show a characterization of the class of functions computable in quantum polynomial time, known as $\mathtt{FBQP}$.

Problem

• Quantum Term Rewrite Systems
• Complexity Analysis
• Quantum Computing
• FBQP
• Termination

Innovation

• Quantum Term Rewrite Systems
• Complexity Analysis
• Termination Certification
• Uniform Quantum Circuits
• FBQP