  Measurement-based quantum computation has emerged from the physics community
as a new approach to quantum computation where the notion of measurement is the
main driving force of computation. This is in contrast with the more
traditional circuit model which is based on unitary operations. Among
measurement-based quantum computation methods, the recently introduced one-way
quantum computer stands out as fundamental.
  We develop a rigorous mathematical model underlying the one-way quantum
computer and present a concrete syntax and operational semantics for programs,
which we call patterns, and an algebra of these patterns derived from a
denotational semantics. More importantly, we present a calculus for reasoning
locally and compositionally about these patterns.
  We present a rewrite theory and prove a general standardization theorem which
allows all patterns to be put in a semantically equivalent standard form.
Standardization has far-reaching consequences: a new physical architecture
based on performing all the entanglement in the beginning, parallelization by
exposing the dependency structure of measurements and expressiveness theorems.
  Furthermore we formalize several other measurement-based models:
Teleportation, Phase and Pauli models and present compositional embeddings of
them into and from the one-way model. This allows us to transfer all the theory
we develop for the one-way model to these models. This shows that the framework
we have developed has a general impact on measurement-based computation and is
not just particular to the one-way quantum computer.
