Quantum Circuits with Mixed States

TL;DR

This paper introduces a model of quantum circuits with mixed states allowing non-unitary operations, analyzes their computational power, and discusses metrics for error measurement, extending the standard quantum circuit framework.

Contribution

It defines quantum circuits with density matrices, including non-unitary gates and measurements, and provides a framework for analyzing their computational power and error metrics.

Findings

Quantum circuits with mixed states are computationally equivalent to standard quantum circuits.

Introduces trace and diamond metrics for error analysis in mixed state quantum computing.

Provides a lower bound for probabilistic functions in quantum circuits.

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.