In order to perform simulations of fermionic systems on a quantum computer, there is an encoding needed which maps the fermionic Hamiltonian onto a qubit Hamiltonian. In this thesis, a fermionic encoding is constructed which is less constraining in terms of the required operations that realize this mapping compared to other known mappings.
In the constructed mapping, the Bravyi-Kitaev superfast encoding is used to place qubits at the edges of a graph corresponding to a physical system of n fermionic modes. From this encoding certain stabilizer constraints arise which ensure that the correct fermionic operators are being encoded throughout the entire simulation, and which are used to detect the occurrence of any errors during the simulation.
In combination with the superfast encoding, a randomized algorithm developed by Freedman-Hastings is combined with a stacking- and sewing procedure of the graph and a Vertex Coloring algorithm to obtain a cycle basis which governs the properties of these stabilizer constraints.
This constructed mapping obtains a locality and sparsity of the qubit Hamiltonian terms and stabilizer constraints of O(1), while keeping the total number of qubits O(npoly(log(n))). Here poly(log(n)) means some polynomial in log(n). It is also numerically shown that the number of qubits can be further reduced, while keeping the locality of the stabilizer constraints O(1), and the locality and sparsity of the qubit Hamiltonian terms and the sparsity of the stabilizer constraints O(poly(log(n))).
It is furthermore shown how to use this encoding to perform simulations of two fermionic systems, namely the Fermi-Hubbard model on sparse hopping graphs and the sparse SYK model.

The recent progress in quantum technologies shines a bright light on the future of quantum computation. However, resource estimation for quantum computations remains a key challenge. The resource I study in this thesis is known as Magic or Non-stabilizerness and it represents the key requirement for quantum computational advantage in computation. Recent studies in quantum information suggested wide classes of quantifiers for non-stabilizerness.

In this thesis, I develop novel techniques for quantifying non-stabilizerness with the tools fromquantuminformation theory. I am specifically interested inmaking quantum resource estimation computationally as efficient as possible so that quantum resource estimation can become a routine step in both numerical and experimental exploration of quantumcomputing.

I show that by measuring the spreading of the local information in the quantumsystem, we can quantify non-stabilizerness. The measures of such information-spreading can be classified into two categories, entropic-based measures, such as mutual information, and correlator-based measures such as Out-of-Time Ordered Correlators. I investigated both classes of measures and I related them both numerically and analytically to the estimation of non-stabilizerness.

Finally, I relate non-stabilizerness quantification to classical variational methods. Classical methods designed to approximate quantum states are by construction not restricted by non-stabilizerness. The question therefore remained how well these techniques can capture quantumresources. I provide a systematic benchmark for both classical and quantum approximate methods of expressing quantum states and their tradeoff between non-stabilizerness expressivity and ground-state energy accuracy. We observed that having better energy accuracy is necessary but insufficient to have better accuracy in non-stabilizerness.

This Thesis forms a bridge between quantum information resource theory and condensed matter physics and offers a stepping stone towards further exchange between these two fields.

Scientific computing and applied mathematics enable the exploration of and, sometimes even, the simplification of complex systems through various optimized modeling and simulation methods. These fields create and utilize computational resources to do so. Many problems, however, are still unsolvable or very difficult to solve with current well-established methods – be it algorithms or computers. Quantum computers were theorized and are now being realized to extend computational abilities. Though commercially available and widely researched, quantum computers today are still more proof-of-concept than ”universally” applicable tools. A major part of the problem is these systems are riddled with noise, hence the phrase noisy intermediate scale quantum (NISQ) devices is commonly used to refer to current day devices. There are many approaches to mitigating the effects of noise in quantum systems from both hardware and software perspectives. This work focuses on optimizing the device hardware by combining aspects of two already well-explored systems – superconducting quantum integrated circuits and topological insulators.
The former are relatively easy to fabricate and have become one of the main focuses of today’s industry. The heart of superconducting quantum circuits is found in two circuit elements: the quantum bit (qubit) and the resonator. The qubit manages quantum information processing. The resonator is a simple inductor and capacitor (LC) in parallel and is a well understood classical circuit element. Resonators play a key role in this research. In contrast, topological quantum systems are incredibly difficult to fabricate. In theory, however, topological insulators promise a strong shield against noise across the device. By leveraging the simplicity and ease of simulating classical LC resonators and combining this with the noise-protection introduced in topological systems, this project lays the groundwork for developing a model supporting how such a hybrid hardware could process and protect quantum information. This thesis presents the characterization of how a two-dimensional array of LC resonators could behave similarly to a qubit when demonstrating edge modes, which are typically found in topological quantum systems. An approach for how to model such systems is proposed from the perspective of architecture, optimal control, and metrics.

For NP-hard optimisation problems no polynomial-time algorithms exist for finding a solution. Therefore, heuristic methods are often used, especially when approximate solutions can be satisfactory. One such method is quantum annealing, a method where some initial Hamiltonian is slowly perturbed to anneal towards a problem Hamiltonian. The annealing schedule should follow the adiabatic theorem of quantum mechanics and should therefore be slow to yield the most accurate results. Finding a schedule that is both fast and still accurate has been referred to as a 'black art' and is usually just guessed. In this thesis reinforcement learning (RL) is explored to see if it can help with finding the optimal quantum annealing (QA) schedules. The Hamiltonians used are of
the form of a transverse field Ising model. These are well studied Hamiltonians that can map to many NP-complete problems [1]. Finding a good balance between going both fast and being precise proved to be difficult and needs further research. Therefore this thesis mainly dealt with training an RL agent to find the most accurate QA schedule. The results show that the agent learns to find the most accurate (slowest) annealing schedule after 300,000 learning steps for a problem Hamiltonian with a single coupling constant J and no reward shaping. Annealing in an environment with a spin glass problem Hamiltonian proved to be difficult and has not shown good results in this thesis This needs further investigation. Nonetheless, the result on the single coupling constant Hamiltonian shows the potential of an RL agent for learning in a quantum annealing environment with very sparse rewards. This paves the way for further research into an RL agent that can find a schedule that is both fast and accurate on a wide range of problem Hamiltonians.