 
Summary: Quantum Circuits with Mixed States
Dorit Aharonov \Lambda Alexei Kitaev y Noam Nisan z
Abstract
Current formal models for quantum computation
deal only with unitary gates operating on ``pure quan
tum states''. In these models it is difficult or impossible
to deal formally with several central issues: measure
ments in the middle of the computation; decoherence
and noise, using probabilistic subroutines, and more.
It turns out, that the restriction to unitary gates and
pure states is unnecessary. In this paper we general
ize the formal model of quantum circuits to a model in
which the state can be a general quantum state, namely
a mixed state, or a ``density matrix'', and the gates can
be general quantum operations, not necessarily unitary.
The new model is shown to be equivalent in compu
tational power to the standard one, and the problems
mentioned above essentially disappear.
The main result in this paper is a solution for the
subroutine problem. The general function that a quan
