Quantum circuits with mixed states

Current formal models for quantum computation deal only with unitary gates operating on pure quantum states. In these models it is difficult or impossible to deal formally with several central issues: measurements 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 generalize 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 computational 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 quantum circuit outputs is a probabilistic function. However, the question of using probabilistic functions as subroutines was not previously dealt with, the reason being that in the language of pure states, this simply can not be done. We define a natural notion of using general subroutines, and show that using general subroutines does not strengthen the model. As an example of the advantages of analyzing quantum complexity using density matrices, we prove a simple lower bound on depth of circuits that compute probabilistic functions. Finally, we deal with the question of inaccurate quantum computation with mixed states. Using the so called trace metric on density matrices, we show how to keep track of errors in the new model. Institutes of Physics and Computer science, The Hebrew University, Jerusalem, Israel y L.D.Landau Institute for Theoretical Physics,Moscow, Russia z Institute of Compu...