Visualizing Kraus operators for dephasing noise during application of the $\sqrt{\mathrm{\mathrm{SWAP}}}$ quantum gate

TL;DR

This paper visualizes the evolution of Kraus operators for a dephasing-noise-affected $\,\sqrt{\mathrm{SWAP}}$ quantum gate, highlighting optimized control strategies and their implications for quantum operations.

Contribution

It introduces a method to visualize Kraus operators' dynamics during a noisy quantum gate, linking formalism with measurement procedures and emphasizing optimized gate design.

Findings

Optimized Kraus operators are derived for the $\,\sqrt{\mathrm{SWAP}}$ gate under dephasing noise.

Kraus operators' evolution is visualized as curves in three-dimensional space.

The importance of optimized gates in quantum operational theory is discussed.

Abstract

We consider the case of a $\sqrt{\mathrm{SWAP}}$ quantum gate and its optimized entangling action, via continuous dynamical decoupling, in the presence of dephasing noise. We illustrate the procedure in the specific case where only the two-qubit operation is controlled and no single-qubit operations are included in the description. To compare the optimized dynamics in the presence of noise with the ideal case, we use the standard fidelity measure. Then we discuss the importance of using optimized gates in the quantum operational-probabilistic theory. Because of their importance for the explicit construction of the completely positive maps representing the operations, we derive optimized Kraus operators in this specific case, focusing on the entanglement operation. We then show how to visualize the time evolution of each Kraus operator as a curve in a three-dimensional Euclidean space. Finally, we connect this formalism with the operational framework of quantum mechanics by describing a possible set of measurements that could be performed to obtain the Kraus operators.