## Fermionic Quantum Information in Surface Acoustic Waves

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## Abstract

Quantum computers are on the verge of revolutionising modern technology by providing scientists with unparalleled computational resources. With quantum-mechanical phenomena such as the superposition principle and entanglement, these computers could solve certain computational problems that are otherwise impossible for even the most powerful classical supercomputers. One of the major challenges standing in the way of this computing revolution is the accurate control of quantum bits. Quantum systems are extremely fragile and, by their nature, cannot be measured without destroying their quantum state. I wrote a numerical program to solve the time-dependent Schrödinger equation, the differential equation that describes the evolution of wave functions. The advantage of my code over other solvers is its speed. I used graphics processing units (GPUs), a technology that has only recently matured, to accelerate high-performance computing. Hardware- acceleration allows me to solve complex time-evolution problems within days rather than years. Such an exceptional speedup has enabled me to calculate the behaviour of single electrons in semiconductor devices. Electrons are particularly interesting because they are ubiquitous in modern technology, as well as being fundamental quantum particles. Using the simulations produced by my code, I track the time evolution of an electron wave function as it propagates along quantum circuits. By animating the evolution of the wave function, I am able to visualise the wave function of electrons propagating in space and time. This is a remarkable tool for studying the behaviour of quantum particles in nanodevices. I focused my thesis on the realistic modelling of devices that are readily available in a laboratory or on designs that could be fabricated in the near future. I began by modelling single electrons as quantum bits. I provide a definition for an optimal qubit and lay out the set of operations required to manipulate the quantum information carried by the electron. In all my simulations, I aim to model experimentally realistic devices. I calculated the electrostatic potential of a real nanodevice and simulated the time-evolution of a single electron. I show that it is possible to create a single-electron beam splitter by tuning the voltages applied to various parts of the device and I calculate the range of voltages in which quantum information is preserved and manipulated accurately. These results were verified experimentally by collaborators at the Institut Néel and were published in Nature Communications 10, 4557 (2019). Using my code, I developed a framework for general measurements of electron qubits and provided a design for a semiconductor device capable of performing positive-operator valued measures (POVMs). A POVM is a powerful measurement technique in quantum mechanics that allows quantum information to be manipulated in interesting ways. The proposed setup is suggested as an implementation of entanglement distillation, which is a useful error correction tool that transforms an arbitrary entangled state into a pure Bell pair. Entanglement is one of the most fascinating aspects of quantum mechanics and it remains a challenge to generate perfectly entangled particle pairs. An experimentally viable method for distilling – or perfecting – entanglement is crucial for the design of quantum computers or quantum communication systems. Using this design, I introduced a protocol to use electrons, rather than photons, in quantum-optics-like systems. These results were published in Phys. Rev. A 96, 052305 (2017). Going beyond single-particle behaviour, I compare different methods for generating entanglement between electron-spin qubits using the power-of-SWAP operation. By using realistic experimental parameters in my simulations, I demonstrate that generating entan- glement via electron-electron collisions in a harmonic channel cannot be implemented for multidimensional systems. These findings go against what researchers thought was possible and put forward the need for new solutions to particle entanglement. I provide an alternative by demonstrating that a method based on the exchange energy is more viable than previously thought. I present a semiconductor device structure and an electrostatic potential that experi- mental groups can use in order to obtain the most efficient entangling quantum logic gates. These findings were published in Phys. Rev. A 101, 022329 (2020). The results presented in this thesis provide a comprehensive description of the control of single electrons in a surface-acoustic-wave-based quantum circuit. However, work in this field is far from over. I present various research paths for future projects. These include going beyond the time-dependent Schrödinger equation to capture more complicated dynamics, using different hardware solutions to further accelerate numerical problem solving, and studying new systems of interest to extend this project beyond semiconductor physics.In all my simulations, I aim to model experimentally realistic devices. I calculated the electrostatic potential of a real nanodevice and simulated the time-evolution of a single electron. I show that it is possible to create a single-electron beam splitter by tuning the voltages applied to various parts of the device and I calculate the range of voltages in which quantum information is preserved and manipulated accurately. These results were verified experimentally by collaborators at the Institut Néel and were published in Nature Communications 10, 4557 (2019). Using my code, I developed a framework for general measurements of electron qubits and provided a design for a semiconductor device capable of performing positive-operator valued measures (POVMs). A POVM is a powerful measurement technique in quantum mechanics that allows quantum information to be manipulated in interesting ways. The proposed setup is suggested as an implementation of entanglement distillation, which is a useful error correction tool that transforms an arbitrary entangled state into a pure Bell pair. Entanglement is one of the most fascinating aspects of quantum mechanics and it remains a challenge to generate perfectly entangled particle pairs. An experimentally viable method for distilling – or perfecting – entanglement is crucial for the design of quantum computers or quantum communication systems. Using this design, I introduced a protocol to use electrons, rather than photons, in quantum-optics-like systems. These results were published in Phys. Rev. A 96, 052305 (2017). Going beyond single-particle behaviour, I compare different methods for generating entanglement between electron-spin qubits using the power-of-SWAP operation. By using realistic experimental parameters in my simulations, I demonstrate that generating entan- glement via electron-electron collisions in a harmonic channel cannot be implemented for multidimensional systems. These findings go against what researchers thought was possible and put forward the need for new solutions to particle entanglement. I provide an alternative by demonstrating that a method based on the exchange energy is more viable than previously thought. I present a semiconductor device structure and an electrostatic potential that experi- mental groups can use in order to obtain the most efficient entangling quantum logic gates. These findings were published in Phys. Rev. A 101, 022329 (2020). The results presented in this thesis provide a comprehensive description of the control of single electrons in a surface-acoustic-wave-based quantum circuit. However, work in this field is far from over. I present various research paths for future projects. These include going beyond the time-dependent Schrödinger equation to capture more complicated dynamics, using different hardware solutions to further accelerate numerical problem solving, and studying new systems of interest to extend this project beyond semiconductor physics.