Approximate Relational Reasoning for Quantum Programs

Abstract Quantum computation is inevitably subject to imperfections in its implementation. These imperfections arise from various sources, including environmental noise at the hardware level and the introduction of approximate implementations by quantum algorithm designers, such as lower-depth computations. Given the significant advantage of relational logic in program reasoning and the importance of assessing the robustness of quantum programs between their ideal specifications and imperfect implementations, we design a proof system to verify the approximate relational properties of quantum programs. We demonstrate the effectiveness of our approach by providing the first formal verification of the renowned low-depth approximation of the quantum Fourier transform. Furthermore, we validate the approximate correctness of the repeat-until-success algorithm. From the technical point of view, we develop approximate quantum coupling as a fundamental tool to study approximate relational reasoning for quantum programs, a novel generalization of the widely used approximate probabilistic coupling in probabilistic programs, answering a previously posed open question for projective predicates.