Quantum Gate and Quantum State Preparation through Neighboring Optimal Control

Title of dissertation: QUANTUM GATE AND QUANTUM STATE PREPARATION THROUGH NEIGHBORING OPTIMAL CONTROL Yuchen Peng, Doctor of Philosophy, 2016 Dissertation directed by: Dr. Frank Gaitan Laboratory for Physical Sciences Professor Victor Yakovenko Department of Physics Successful implementation of fault-tolerant quantum computation on a system of qubits places severe demands on the hardware used to control the many-qubit state. It is known that an accuracy threshold Pa exists for any quantum gate that is to be used for such a computation to be able to continue for an unlimited number of steps. Specifically, the error probability Pe for such a gate must fall below the accuracy threshold: Pe < Pa. Estimates of Pa vary widely, though Pa ∼ 10−4 has emerged as a challenging target for hardware designers. I present a theoretical framework based on neighboring optimal control that takes as input a good quantum gate and returns a new gate with better performance. I illustrate this approach by applying it to a universal set of quantum gates produced using non-adiabatic rapid passage. Performance improvements are substantial comparing to the original (unimproved) gates, both for ideal and non-ideal controls. Under suitable conditions detailed below, all gate error probabilities fall by 1 to 4 orders of magnitude below the target threshold of 10−4. After applying the neighboring optimal control theory to improve the performance of quantum gates in a universal set, I further apply the general control theory in a two-step procedure for fault-tolerant logical state preparation, and I illustrate this procedure by preparing a logical Bell state fault-tolerantly. The twostep preparation procedure is as follow: Step 1 provides a one-shot procedure using neighboring optimal control theory to prepare a physical qubit state which is a highfidelity approximation to the Bell state |β01⟩ = 1/ √ 2 (|01⟩+ |10⟩). I show that for ideal (non-ideal) control, an approximate |β01⟩ state could be prepared with error probability ε ∼ 10−6 (10−5) with one-shot local operations. Step 2 then takes a block of p pairs of physical qubits, each prepared in |β01⟩ state using Step 1, and faulttolerantly prepares the logical Bell state ∣∣β01⟩ for the C4 quantum error detection code. QUANTUM GATE AND QUANTUM STATE PREPARATION THROUGH NEIGHBORING OPTIMAL CONTROL