Quantum Circuits with Mixed States
  We define the model of quantum circuits with density matrices, where non-unitary gates are allowed. Measurements in the middle of the computation, noise and decoherence are implemented in a natural way in this model, which is shown to be equivalent in computational power to standard quantum circuits.   The main result in this paper is a solution for the subroutine problem: The general function that a quantum circuit outputs is a probabilistic function, but using pure state language, such a function can not be used as a black box in other computations. We give a natural definition of using general subroutines, and analyze their computational power.   We suggest convenient metrics for quantum computing with mixed states. For density matrices we analyze the so called ``trace metric'', and using this metric, we define and discuss the ``diamond metric'' on superoperators. These metrics enable a formal discussion of errors in the computation.   Using a ``causality'' lemma for density matrices, we also prove a simple lower bound for probabilistic functions. 
Dorit Aharonov (Physics and Computer science, The Hebrew University),   Alexei Kitaev (L.D.Landau Institute for Theoretical ...