🤖 AI Summary
Continuous-variable quantum programs have long lacked formal semantics and verification methods due to their operation on infinite-dimensional Hilbert spaces, unbounded measurements, and potentially divergent expectation values. This work presents the first formal semantic framework for such programs and introduces the first unary Hoare logic tailored to this setting. By integrating polynomial assertions based on regular observables with symbolic weakest precondition calculations, the approach overcomes the theoretical challenges posed by infinite dimensionality and unboundedness. The method successfully verifies standard continuous-variable quantum algorithms, establishes gate decomposition equivalences, quantifies the photon-number-state resources required for classical simulation, and provides precise analyses of approximation errors arising in physical implementations.
📝 Abstract
We provide a formal framework for Continuous-Variable Quantum Computing (CQC). While CQC is supported by photonic quantum hardware, we are not aware of a formal semantics for continuous-variable quantum programs nor of a unary Hoare logic for their verification. There are several technical obstacles to extending to CQC any of the formal frameworks available for Discrete-Variable Quantum Computing (DQC). Most importantly, continuous-variable quantum programs act on {\em infinite-dimensional} Hilbert spaces; their measurement outcomes are often {\em unbounded} and have expected values that are defined by an improper integral (or an infinite series), which may not converge. We overcome these challenges to give a formal semantics to a universal programming language for CQC and to provide the first Hoare logic for CQC. The assertions of our logic are built from polynomials over canonical observables. Besides proving relative completeness, we implement a symbolic weakest-precondition calculator for CQC based on our logic. Our tool has successfully verified CQC algorithms from textbooks and calculated their approximation errors for physically realizable implementations, proved the correctness (i.e., equivalence) of gate decompositions for CQC hardware, and computed the resource requirements (i.e., number of photon-number states) for achieving a desired accuracy in the classical simulation of continuous-variable quantum programs.