Extending the Lambda Calculus to Express Randomized and Quantumized Algorithms

This paper introduces a formal metalanguage called the lambda-q calculus for the specification of quantum programming languages. This metalanguage is an extension of the lambda calculus, which provides a formal setting for the specification of classical programming languages. As an intermediary step, we introduce a formal metalanguage called the lambda-p calculus for the specification of programming languages that allow true random number generation. We demonstrate how selected randomized algorithms can be programmed directly in the lambda-p calculus. We also demonstrate how satisfiability can be solved in the lambda-q calculus.