Title: Performing Non-Local Phase Estimation with a Rydberg-Superconducting Qubit Hybrid

Abstract: Distributed quantum computation is the key to high volume computation in the NISQ era. This investigation explores the key aspects necessary for the construction of a quantum network by numerically simulating the execution of the distributed phase estimation algorithm in a proposed novel superconducting-resonator-atom hybrid system. The phase estimation algorithm is used to estimate the phase or eigenvalue of a given unitary operator and is a sub-process of many other known quantum algorithms such as Shor's algorithm and the quantum counting algorithm . An entangling gate between two qubits is utilised in the distributed phase estimation algorithm, called an E2 gate which provides the possibility to transfer quantum information from one quantum computer to another, which was numerically shown to have a construction time of 17ns at a fidelity of 93%. This investigation analytically derives the Hamiltonian dynamics as well as the noise sources of each system and utilizes quantum optimal control (QOC), namely the gradient ascent pulse engineering (GRAPE) algorithm, to minimize fidelity error in the corresponding systems gate construction. The GRAPE algorithm showed very accurate engineering of Rydberg atom single and multi-qubit gates with fidelities higher than 90% while the flux qubit suffered greatly from noise with multi-qubit gate fidelities lower than 90%. The C-shunt factor was shown to decrease the noise of the flux qubit which in turn increased the probability of accurately estimating the phase using 4 counting qubits. A trade off was observed between the number of time steps in the descent/ascent and the number of GRAPE iterations ran on the optimisation for low C-shunt factors $0<\zeta<1000$. For $\zeta = 1000$, the GRAPE algorithm showed effectiveness for a large number of time steps and large amount of GRAPE iterations by reaching estimation accuracies greater than 90%.