While quantum devices have seen major advancements in recent years, there are still significant
challenges to scaling up their computational power. Geometry optimization techniques pose a
useful tool for tackling these challenges and improving the characteristics of quantum dot devices.
These devices consist of metal gate electrodes on a semiconductor heterostructure. Within the
semiconductor heterostructure, the electron wavefunctions used as the qubits are ‘trapped’ by the
potential induced by these metal gates.

    In this work, we have modeled the potential induced by the gates by discretizing the corresponding
Poisson equation using the finite-volume method. The discretized linear system is
solved with factorization-based solvers, of which we make repeated calls more efficient by applying
the Woodbury identity. The potential is used to solve the Schrodinger equation, of which
the eigenstates are transformed to a maximally localized basis to obtain the dot wavefunctions.
The gate voltages of the device are tuned so the effective Hamiltonian of the dots approaches a
target Hamiltonian. We have modelled the disorder sensitivity of the devices by inducing changes
to the boundary of the gate electrodes, for which the model is evaluated efficiently by utilizing
perturbation theory.

    Using this device model, we have implemented a discrete geometry optimization algorithm to
optimize for the gate electrode shapes. This algorithm generates a range of random changes to the
geometry shape and evaluates which one has the best characteristics.
We have demonstrated that this technique is effective for optimizing devices to be less sensitive
to gate shape disorder, to have higher level spacing, and to have more local gate-dot interactions.
We have applied it to double dot devices, triple dot devices, and double dot devices with wires.
The algorithm does not converge to the global minimum of the optimization problem, as different
initial conditions lead to marginally different results.

    We have implemented several strategies for the sake of computational efficiency. The use of
the Woodbury identity, perturbation theory for loss function gradients, and linear corrections for
disordered geometries lead to an estimated speedup of more than 62 times.
Since the aim of this project was to be a proof-of-concept for geometry optimization techniques
for quantum devices, we simplified some of the dynamics for computational efficiency or coding
efficiency. We have not modelled the Coulomb repulsion between electrons in different dots,
nor the effects of strain on the system. Additionally, the square-grid discretization of the gate
electrodes has an impact on the resulting geometries.

    Nonetheless, we have established that it is possible to apply discrete geometry optimization
techniques to improve the characteristics of modelled quantum dot devices. Moreover, we have
successfully introduced various strategies to improve the computational efficiency of the model.

Over the last decades, studies have shown that the gene regulation of a wide range of organisms can be described with networks [Jeong H., Tombor B., Albert R., Ottval Z.N., Barab´asi A.L., 2000] [Jeong H., Mason S.P., Barab´asi A.L., Oltvai Z.N., 2001]. The interactions between the mRNA strands and proteins form links in the network, while these molecules form the nodes of the network. Numerical models for the dynamics of such a network, through solving a minimally nonlinear stochastic differential equation show nontrivial dynamics. This behavior is caused by non linearity introduced by the positivity condition, this is due to the fact that molecular concentrations cannot cannot be negative. Whether these dynamics are stable, oscillatory or chaotic seems to depend on the average connectivity of the network. There appears to be a region of networks that can transition from chaotic behavior to stable behavior. For a simple differential equation this would not be a region, but just one value. This region, referred to as the “Edge-of-Chaos” region, is therefore rather interesting. In this report we analyse the Lyapunov spectrum in order to quantify these dynamics. This is a way to measure the stability of the solution of out differential equation. We look at how different parameters in the minimally nonlinear differential equation and the generation of the network affect these dynamics. The investigated parameters include different edge weights, different noise levels, different equilibrium values and different network types. After studying the dynamics of a single network, we study the dynamics of populations of networks using an evolutionary algorithm and a co-evolutionary algorithm. We found that an edge of chaos region exists universally, that is for all types of networks we looked at, for all feasible noise levels, for all equilibrium values and for all network sizes. Whether or not a network lies in this region is determined by how much self interaction its nodes have compared to the strength of the interactions between the nodes. Additionally, the size of the network determines how well this “Edge-of-Chaos” plateau forms. Larger networks in this region behave more stable than smaller ones. We also came to the conclusion that both the evolutionary and co-evolutionary algorithm, using the Kullback-Leibler divergence in accordance with a probability distribution defined by a Hamiltonian to compare the fitness of individuals, do not yield the expected results. That is, they do not show any correlation between the connectivity of a network and the development of the population over time.