Qubit recycling and the path counting problem

TL;DR

This paper analyzes the fidelity of a protocol for resetting qudits in quantum circuits, establishing a connection with path counting on graphs, and provides exact results for convolutional circuits, including effects of noise.

Contribution

It introduces an exact expression for the fidelity of qudit reset protocols in a family of quantum circuits, linking it to graph path counting and analyzing convergence rates.

Findings

Fidelity converges to 1 at rate q^2/(q^2+1) for constant window size.

Correlation between reset qudits decays exponentially with distance.

Derived fidelity expression in the presence of noise based on channel properties.

Abstract

Recently, it was shown that the qudits used in circuits of a convolutional form (e.g., Matrix Product State sand Multi-scale Entanglement Renormalization Ansatz) can be reset unitarily \href{https://doi.org/10.1103/PhysRevA.103.042613}{[Phys. Rev. A 103, 042613 (2021)]}, even without measurement. We analyze the fidelity of this protocol for a family of quantum circuits that interpolates between such circuits and local quantum circuits, averaged over Haar-random gates. We establish a connection between this problem and a counting of directed paths on a graph, which is determined by the shape of the quantum circuit. This connection leads to an exact expression for the fidelity of the protocol for the entire family that interpolates between convolutional circuit and random quantum circuit. For convolutional circuits of constant window size, the rate of convergence to unit fidelity is shown to be $\frac{q^2}{q^2+1}$, independent of the window size, where $q$ is the local qudit dimension. Since most applications of convolutional circuits use constant-sized windows, our result suggests that the unitary reset protocol will likely work well in such a regime. We also derive two extra results in the convolutional limit, which may be of an independent interest. First, we derive exact expressions for the correlations between reset qudits and show that it decays exponentially in the distance. Second, we derive an expression for the the fidelity in the presence of noise, expressed in terms of the quantities that define the property of the channel, such as the entanglement fidelity.