**Title**: A Complete Classification of the Toffoli Gate with Ancillary bits

**Abstract**:

The Toffoli gate is a universal gate for classical reversible computation. This means that if we are allowed to fix the values of certain inputs and outputs (called ancillary bits), we can simulate any Boolean function from with a circuit from wires consisting only of Toffoli gates (with extra ignored outputs).

Iwama found a complete set of identities for circuits solely consisting of Toffoli gates. I present a complete set of identities for the symmetric monoidal category generated by the Toffoli gate \emph{and ancillary bits}. I also provide a normal form for these circuits and prove an equivalence of categories into a subcategory of

**Title: **Structures for decoherence

** **

**Abstract:**

This talk will introduce and develop the structure required for studying

decoherence in certain monoidal categories. Our driving example is a

decoherence structure in

Our goal is to move towards understanding the following:

Theorem:

Let C be a dagger compact closed category and

be a subcategory of C that inherits the dagger and compact closed structure. Suppose

has a decoherence structure with purification, then there exists an invertible dagger functor

such that

.