Logic and Quantum Computation

The goal of these lectures is to describe a “structural” theory of quantum computation. Quantum programs can be reasoned about at many different levels. They can be reasoned about numerically and algebraically, which involves specific calculations with complex numbers and vector space notation. Programs can also be reasoned about symbolically, by expressing them in a formal language and formulating axioms and reasoning principles which apply to all programs. We term this latter approach the “structural” approach, and it is the main topic of these lectures. To allow symbolic reasoning, programs must be expressed in a formal language. This language should be structured: large programs are built up from smaller ones by means of syntactic operations, such as sequential or parallel composition or the introduction of loops. A basic principle which should be enjoyed by a structured programming language is the principle of compositionality, according to which the behavior of a composite program should be uniquely determined by the behaviors of its parts. The set of programs, under these structural operations, is then subject to some axioms and equations. In many cases, these axioms and equations will be sufficient to reason about a given quantum algorithm in a purely symbolic way. There are several potential benefits to studying quantum computation in the context of formal languages. First, the use of formal languages allows the creation of a metatheory: it allows one to quantify over all possible programs, rather than reasoning about one program at a time. Also, the uniform construction of large programs from smaller parts gives rise to a principle of reasoning about programs by induction. Another potential benefit of this approach is the extension of known links between logic and complexity theory. In the classical (i.e., non-quantum) world, there are many such connections. For example, for many complexity classes (such as P, NP, etc.) there exist logics which characterize these complexity classes precisely, in the sense that a function is provably total in the logic if and only if it belongs to the given complexity class. By studying quantum computing in terms of formal languages, one opens the door to extending such results to quantum complexity classes. The structural approach to quantum computing also offers the possibility to investigate and employ new high-level language features and abstractions. Currently, it seems that each newly discovered quantum algorithm relies on some particular and unique