SAQR-QC: A Logic for Scalable but Approximate Quantitative Reasoning about Quantum Circuits

Reasoning about quantum programs remains a fundamental challenge, regardless of the programming model or computational paradigm. Despite extensive research, existing verification techniques are insufficient -- even for quantum circuits, a deliberately restricted model that lacks classical control, but still underpins many current quantum algorithms. Many existing formal methods require exponential time and space to represent and manipulate (representations of) assertions and judgments, making them impractical for quantum circuits with many qubits. This paper presents a logic for reasoning in such settings, called SAQR-QC. The logic supports {S}calable but {A}pproximate {Q}uantitative {R}easoning about {Q}uantum {C}ircuits, whence the name. SAQR-QC has three characteristics: (i) some (deliberate) loss of precision is built into it; (ii) it has a mechanism to help the accumulated loss of precision during a sequence of reasoning steps remain small; and (iii) most importantly, to make reasoning scalable, every reasoning step is local -- i.e., it involves just a small number of qubits. We demonstrate the effectiveness of SAQR-QC via two case studies: the verification of GHZ circuits involving non-Clifford gates, and the analysis of quantum phase estimation -- a core subroutine in Shor's factoring algorithm.