# "Higher-order Quantum Computations"

### Thursday, 04 Feb. 2016, 11:00 - 12:30

**Presenter:** Paolo Perinotti

**Host:** Caslav Brukner

**Where:** IQOQI Seminar Room, 2nd Floor, Boltzmanngasse 3, 1090 Vienna

Quantum computations are described by quantum circuits, that represent evolutions of states of systems from input to output. The property of quantum theory that allows us to interpret the input-to-output evolution as occurring in time is causality. In this talk we will consider computations that describe the transformation of an input circuit to an output circuit, namely computations that describe the transformation of an evolution. Differently from ordinary circuits, we will see that for maps on maps the input does not chronologically precede the output. Nevertheless, if these maps preserve complete positivity and normalisation they have a simple physical realisation in terms of causally ordered circuits. The construction of a hierarchy of transformations of transformations, however, proceeds arbitrarily far, and in the higher orders one encounters admissible functions that are a-temporal. Still, many of the maps in the hierarchy can be proved to have a sensible physical interpretation, and some of them have even been realised experimentally. Remarkably, higher—order maps were proved to provide computational advantages in achieving information processing tasks with respect to maps on states. We will discuss the problem of characterising mathematically admissible maps of every level, showing an important result in this direction, and then pose the open question of their physical achievability.