Definition of compact models for the simulation of spin qubits in semiconductor quantum dots

Semiconductor quantum computation has been a rapidly growing field in the past years thanks to a global investigation on realizing fault-tolerant implementations. Experiments on physical devices are indeed yielding very good results in terms of gate fidelities and coherence times. Quantum dots are one of the most explored possibilities for semiconductor quantum implementation, thanks to their flexibility and technological readiness for such application. Yet, this is not the only option available for semiconductors: different technologies, namely dopants and optically addressable quantum defects, are accessible, but gate-defined quantum dots offer enough advantages over those alternatives to justify the extensive research carried out on them. Relevant benefits are the small qubit footprint (tens of nanometres) and the fast gate-controlled manipulation and read-out procedure, but the most notable advantage is the compatibility with the established CMOS integration techniques. This makes quantum dots a strong candidate for large-scale quantum computing integration. The route towards the optimization of these structures passes through the use of a reliable behavioral simulation of the quantum device in order to improve its performance based on the physical definition of the device, such as which materials are used and which type of manipulation is employed. The aim of this thesis is the implementation of this type of simulation on a Double Quantum Dot (DQD) structure in silicon. Unfortunately, a canonical simulation of a semiconductor quantum device needs the solution of a system of differential equations (Lindblad master equation), and its computational cost exponentially increases with the number of qubits in the system. This highlights the need for a compact model that efficiently describes a quantum system, and that can accurately take into account quantum noise sources. After briefly introducing the fundamental concepts needed to understand the device functioning, this work focuses on the theoretical derivation of the model, including the computation of the single-qubit control Hamiltonian and how to derive the unitary matrix evolution of the system using the Rotating Wave Approximation (RWA) to cancel any Hamiltonian time dependence. Also, the two-qubit system is analyzed, paying particular attention to the different possibilities for the two-qubit native gate choice: the sqrt(SWAP) and CPHASE gates are implemented and the mechanism that rules this native gate duality is thoroughly explored. The model is entirely developed in a MATLAB environment, and it also has a built-in QASM2.0 interpreter, which makes it compatible with quantum assembly, the standard language for quantum algorithms programming. The approximations employed in the compact model – most of them involving time dependences simplifications – are then verified using Quantum Toolbox in Python (QuTiP), a Python-based environment able to solve the Lindblad master equation. The error between QuTiP exact simulation and the MATLAB model is plotted with respect to some relevant system input parameters, in order to prove the validity of the model approximations. The model is finally validated in terms of state probability distribution and gate fidelity by extensively verifying its results when simulating different quantum algorithms.