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Quantum Circuits with Mixed States

Dorit Aharonov·Alexei Kitaev·Noam Nisan
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Abstract

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.

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Dorit Aharonov, Alexei Kitaev, Noam Nisan. 1998-06-08. Quantum Circuits with Mixed States. https://arxiv.org/abs/quant-ph/9806029

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