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.