We develop a new framework of algebraic theories with linear parameters, and use it to analyze the equational reasoning principles of quantum computing and quantum programming languages. We use the framework as follows:

- we present a new elementary algebraic theory of quantum computation, built from unitary gates and measurement;
- we provide a completeness theorem or the elementary algebraic theory by relating it with a model from operator algebra;
- we extract an equational theory for a quantum programming language from the algebraic theory;
- we compare quantum computation with other local notions of computation by investigating variations on the algebraic theory.