Exact dynamics of single qubit gate fidelities under the measurement-based quantum computation scheme

TL;DR

This paper analyzes the exact dynamics of single qubit gate fidelities in measurement-based quantum computation under dephasing noise, revealing conditions for fidelity oscillations and their dependence on initial resource state geometry.

Contribution

It provides a necessary condition for fidelity oscillations in resource states and highlights the importance of initial geometry in measurement-based quantum computation.

Findings

Fidelity oscillates for initial canonical cluster states under dephasing.

Oscillations depend solely on the initial geometry of the resource state.

Timing of measurements critically affects gate fidelity in noisy environments.

Abstract

Measurement-based quantum computation is an efficient model to perform universal computation. Nevertheless, theoretical questions have been raised, mainly with respect to realistic noise conditions. In order to shed some light on this issue, we evaluate the exact dynamics of some single qubit gate fidelities using the measurement-based quantum computation scheme when the qubits which are used as resource interact with a common dephasing environment. We report a necessary condition for the fidelity dynamics of a general pure N-qubit state, interacting with this type of error channel, to present an oscillatory behavior and we show that for the initial canonical cluster state the fidelity oscillates as a function of time. This state fidelity oscillatory behavior brings significant variations to the values of the computational results of a generic gate acting on that state depending on the instants we choose to apply our set of projective measurements. As we shall see, considering some specific gates that are frequently found in the literature, neither fast application of the set of projective measurements necessarily implies high gate fidelity, nor slow application thereof necessarily implies low gate fidelity. Our condition for the occurrence of the fidelity oscillatory behavior shows that the oscillation presented by the cluster state is due exclusively to its initial geometry. Other states that can be used as resources for measurement-based quantum computation can present the same initial geometrical condition. Therefore, it is very important for the present scheme to know when the fidelity of a particular resource state will oscillate in time and, if this is the case, what are the best times to perform the measurements.