Qubit recycling and the path counting problem
- URL: http://arxiv.org/abs/2301.03725v2
- Date: Wed, 18 Oct 2023 23:25:34 GMT
- Title: Qubit recycling and the path counting problem
- Authors: Zijian Song, Isaac H. Kim
- Abstract summary: Recently, it was shown that the qudits used in circuits of a convolutional form (e.g., Matrix Product State sand Multi-scaleanglement Renormalization Ansatz) can be reset unitarily.
We analyze the fidelity of this protocol for a family of quantum circuits that interpolates between such circuits and local quantum circuits.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- 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.
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