Quantum Circuits with Mixed States
Dorit Aharonov, Alexei Kitaev, Noam Nisan
TL;DR
The paper introduces a mixed-state quantum circuit model that permits non-unitary gates and mid-computation measurements, thereby naturally incorporating noise and decoherence while remaining computationally equivalent to standard quantum circuits. It establishes unitary-dilation techniques to simulate any general gate, proves that probabilistic subroutines do not extend computational power (FQP^{FQP}=FQP), and develops robust metrics (trace and diamond norms) to bound errors. A causality-based lower bound and correlation-graph analysis yield depth constraints for circuits generating probabilistic outputs. Overall, the work provides a solid framework for analyzing quantum computations with density matrices, including error propagation and subroutine usage, with implications for algorithm design and fault tolerance.
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.