Quantum computing has many challenges that need to be overcome in order for quantum computers to become truly viable. One of these challenges is the tendency for most physical systems used as qubits to decohere. This means that their states do not remain stable in the presence of coupling to an environment, and this must somehow be countered. The main ways of addressing this problem are either by trying to construct physical systems with less or no decoherence, or to counter the decoherence with error correction schemes. This project will consider the Kitaev chain, which is a model that can host a so-called topological qubit, which in theory should have no decoherence of states in the limit of an infinitely long chain. The main goal of the project will be to simulate manipulations of the state of such qubits.
The Kitaev chain is an example of a topological superconductor, which generally falls under the category of topological materials. This is a field of physics that has seen active research for the past three decades, which offers the promise of materials with certain very stable properties. For creating a qubit, the ideal scenario is that there is some two-level system (potentially embedded in a larger system) where changing the state of the system from one to the other is somehow difficult. The Kitaev chain has a two-level degeneracy in the ground state and an energy gap between the ground states and the higher energy states. Moving from one of the ground states to the other requires both a non-local operation at the two edges of the chain This tends not to happen without willful manipulation of the system. If we use the two ground states as a two-level system and assume that the energy gap to the higher energy states is large enough that accidental excitations are unlikely, then we have a qubit. This qubit should have very little decoherence since very specific manipulations are necessary in order to change the state of the system, and this is very unlikely to happen due to random effects from the environment.
Such a qubit is therefore well suited to store information, but as changing the state is difficult, so is also computation. Several schemes are proposed in order to get around this problem, and one of the more common ones is by braiding operations and that is the one that this thesis will focus on. In theory, such operations should work, but generally they assume both an infinitely slow rate of change, and ignore some of the finer details of how such operations are performed. The main task of this project will be to simulate such braiding operations, and evaluate the effects of a finite rate of change. This includes identifying an efficient and proper method to find the states of the system, and then simulating time evolution between logical qubit states. In finding the states of the system, one of the main challenges will be that the Hilbert space is typically highly degenerate. This might cause issues with detecting all of the possible states (even beneath a potential energy cutoff), as there will be several global and local minima, which might also leave some methods wholly unable to determine a proper solution. Certain methods are therefore not well suited to this problem, and identifying which methods are well suited is considered a part of this project. As for simulating time evolution, the main challenge is that it can be very computationally expensive. The complexity of the problem might be reduced by only considering transitions to the lowest energy states of the system and eliminating most of the Hilbert space as a consideration. A proper scheme for time evolution with these limitations must also be identified.
For those interested, some further reading might be found in the following articles:
Kitaev's original article, https://arxiv.org/abs/cond-mat/0010440
Article explaining how computation may be performed using nanowires. The Kitaev chain may be considered a toy model of such a system. https://arxiv.org/pdf/1006.4395