Semantics of a purely quantum programming language

Quantum algorithm is the key to dig the potential power of quantum computing and make quantum computation more efficient than its classical analogue. However, present methods of designing quantum algorithms are too tricky and lack of systematic development. The aim of quantum programming languages is to deal with this problem and try to provide a high-level and unifying description and treatment of quantum algorithms. There are two equivalent ways to view classical programs: state transformers which map input states into output states, and predicate transformers which map postconditions into weakest preconditions. In this paper, we prove an analogous result for quantum programming. We show that quantum programs can be treated as either super-operators on density matrices or healthy transformers on quantum predicates. This result gives a complete characterization of physically realizable quantum programs in terms of healthy quantum predicate transformers. We propose a simple programming language which is a purely quantum fragment describing the quantum part of a future quantum computer in Knill’s architecture. The denotational semantics, weakest precondition semantics, and weakest liberal precondition semantics of this purely quantum language are introduced. Furthermore, the connections between these three semantics are also examined. To help reason about quantum programs involving quantum loops, we extend all existing proof rules for loops in classical probabilistic programs to the case of our purely quantum language.