This Master's thesis explores some of the limitations of the rare-earth quantum computing (REQC) scheme by investigating how the design of the computing scheme and experimental errors will affect gate fidelity.The quantum bits that constitute a quantum logic gate were modeled and their interaction with the laser pulses that control the gate operation was simulated in MATLAB.Emphasis was put on exploring and finding pulse shapes and developing the gate implementation scheme to assure robust quantum operations.Expected sources of errors were introduced and modeled and their impact on the fidelity of the operation was studied.The results showed that it was possible to find pulse shapes that yielded high fidelity gate operations also in the presence of experimental complications.Detta examensarbete utforskar nagra av begransningarna i en kvantdatormodell baserad pa sallsynta jordartsjoner genom att undersoka hur designen av beriikningsschemat och experimentella fel paverkar tillfOrlitligheten hos de logiska grindarna.Kvantbitarna, som kvantgrindarna utfcir operationer pa, modellerades och deras viixelverkan med laserpulserna som anviinds for att kontrollera de logiska operationerna simulerades i MATLAB.Arbetet fokuserades darefter pa att utforska och designa pulsformer samt pa att utveckla metoderna fcir implementation av kvantgrindar med avsikt att siikerstalla robusta kvantoperationer.Formodade felkiillor introducerades och modellerades och deras inverkan pa operationens fidelitet studerades.De erhallna resultaten visar att designen av laserpulserna har mycket stor inverkan pa fideliteten hos kvantgrindsoperationerna men att det anda gar att hitta pulsformer som ger mycket god fidelitet.
This Master's thesis explores some of the limitations of the rare-earth quantum computing (REQC) scheme by investigating how the design of the computing scheme and experimental errors will affect gate fidelity.The quantum bits that constitute a quantum logic gate were modeled and their interaction with the laser pulses that control the gate operation was simulated in MATLAB.Emphasis was put on exploring and finding pulse shapes and developing the gate implementation scheme to assure robust quantum operations.Expected sources of errors were introduced and modeled and their impact on the fidelity of the operation was studied.The results showed that it was possible to find pulse shapes that yielded high fidelity gate operations also in the presence of experimental complications.Detta examensarbete utforskar nagra av begransningarna i en kvantdatormodell baserad pa sallsynta jordartsjoner genom att undersoka hur designen av beriikningsschemat och experimentella fel paverkar tillfOrlitligheten hos de logiska grindarna.Kvantbitarna, som kvantgrindarna utfcir operationer pa, modellerades och deras viixelverkan med laserpulserna som anviinds for att kontrollera de logiska operationerna simulerades i MATLAB.Arbetet fokuserades darefter pa att utforska och designa pulsformer samt pa att utveckla metoderna fcir implementation av kvantgrindar med avsikt att siikerstalla robusta kvantoperationer.Formodade felkiillor introducerades och modellerades och deras inverkan pa operationens fidelitet studerades.De erhallna resultaten visar att designen av laserpulserna har mycket stor inverkan pa fideliteten hos kvantgrindsoperationerna men att det anda gar att hitta pulsformer som ger mycket god fidelitet.
We present a composite pulse controlled phase gate which, together with a bus architecture, improves the feasibility of a recent quantum computing proposal based on rare-earth-ion-doped crystals. The proposed gate operation is tolerant to variations between ions of coupling strengths, pulse lengths, and frequency shifts. In the absence of decoherence effects, it achieves worst case fidelities above 0.999 with relative variations in coupling strength as high as 10 and frequency shifts up to several percent of the resonant Rabi frequency of the laser used to implement the gate. We outline an experiment to demonstrate the creation and detection of maximally entangled states in the system.
Quantum computers based on rare-earth-ion-doped crystals show promising properties in terms of scalability and connectivity if single ions can be used as qubits. Through simulations, we investigate gate operations on such qubits. We discuss how gate and system parameters affect gate errors, the required frequency bandwidth per qubit, and the risk for instantaneous spectral diffusion (ISD). Furthermore, we examine how uncertainties in the system parameters affect the gate errors, and how precisely the system needs to be known. We find gate errors for arbitrary single-qubit gates of $2.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ when ISD is not considered and $3.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ when we take heed to minimize it. Additionally, we construct two-qubit gates with errors ranging from $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\ensuremath{\rightarrow}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ over a broad range of dipole-dipole interaction strengths.
We propose and analyze a high-fidelity readout scheme for a single-instance approach to quantum computing in rare-earth-ion-doped crystals. The scheme is based on using different elements as qubit and readout ions, where the readout ions are doped into the material at a much lower concentration than the qubit ions. It is shown that by allowing the qubit ion sitting closest to a readout ion to act as a readout buffer, the readout error can be reduced by more than an order of magnitude. The scheme is shown to be robust against certain experimental variations, such as varying detection efficiencies, and we use the scheme to predict the attainable quantum fidelity of a controlled not (cnot) gate in these solid-state systems. In addition, we discuss the potential scalability of the protocol to larger qubit systems. The results are based on parameters which we believe are experimentally feasible with current technology and which can be simultaneously realized.
Experimental single qubit gate fidelities in the rare-earth quantum computing (REQC) scheme was found to be lower than predicted by simulation of the optical Bloch equations for the system. The cause was believed to be excitation-induced frequency shifts (EFS), so this BSc thesis investigates a model for EFS in Eu:YSO where stochastic frequency shifts are imposed on the ions. This was implemented as a convolution of the solution to the Bloch equations in each time step with a convolution kernel, where the width of the kernel was dependent on the change in excitation. The simulation was run for a 7 µs complex hyperbolic secant pulse, one of the pulses used in previous experiments. The result was decreased fidelity, but not enough to explain the experimental result. This indicates that EFS may not be the main cause of the experimental fidelity losses, but more thorough investigation is required to draw more certain conclusions.
This thesis describes a number of coherent processes, such as quantum information processing, superradiance and electromagnetically induced transparency, which have been experimentally implemented using rare-earth-ion-doped crystals. The rare-earths are a class of elements that have in common, an atomic structure that allows for very long lifetimes as well as coherence times, both on optical transitions and on spin transitions. These ions can be naturally trapped inside host crystals, and with the phonon vibrations removed by cooling down to cryogenic temperatures (< 4 K), they can be used for a number of different quantum information processes. Most experiments were carried out using PrYSO, but also other crystal types were investigated, such as Nd:YVO4, and La2(WO4)3. The characterization of the ions were done with a variety of methods, including among others, photon echo techniques and electromagnetically induced transparency. Quantum computing is a rapidly growing field and there are still many potential candidates for its implementation. In our work we have utilized the spin states of the Pr rare-earth ion as a qubit, and demonstrated arbitrary single qubit gates, which are important pieces towards a quantum computing realization in these systems. Most of the experimental work done for the thesis was carried out using an ensemble approach, that has the advantage of giving a strong readout signal, but for future scaling to multiple qubits, single instances are more promising, which is discussed. Quantum memories are crucial components in applications such as quantum networks and long distance quantum communication. In this thesis, work has been done to investigate how quantum memories can be implemented in rare-earth crystals. In particular, there are two protocols, CRIB and AFC, which were suggested with rare-earth crystals directly in mind, and both of these require high optical depth, for maximum recall efficiency. In this thesis, implications of being in the high optical depth regime, with phenomena such as superradiance, a collective effect that could cause the stored light to be immediately reemitted, and slow light effects that come from performing the storage inside spectrally structured materials, were investigated. In order to carry out the phase-sensitive experiments, a laser system with a very narrow linewidth of ~1 kHz at 606 nm was constructed, by locking the laser to a semi-persistent spectral hole. In addition, many experiments required advanced pulse shapes, such as complex sechyp pulses or pulses obtained from optimal control theory. In order to be able to accurately create such shapes, an elaborate system using an arbitrary waveform generator and two well calibrated AOMs, controlled from a computer, was also built.
This thesis work contributes to the effort of detecting single rare earth ions doped into crystals, an operation crucial for the development of rare earth quantum computer hardware. Doping crystals with rare earth elements, their trivalent ions substitute some of the crystal’s original bonds. These ions have a partially filled 4f shell, protected by the full 5s and 5p shell from the environment which makes them good candidates for qubits. Because not every ion experiences the same electric field from the imperfect crystal, their total linewidth is inhomogeneously broadened. If the doping is sufficiently low and the crystal sufficiently small, each ion undergoes a different optical shift and can thus be addressed separately. Communication between ions happens due to their permanent dipole moments which change according to their energy level. So far no single ions could be detected in Lund because highly doped bulk crystals were used. It was nevertheless tried to construct quantum logic gates using qubits that consisted of an ensemble of ions. Constructing logic gates with these means that every single ion of one qubit needs to be able to communicate with every single ion of another qubit, a method that becomes increasingly difficult when more and more qubits are to be entangled. Connecting as many qubits as possible is a crucial step in building a quantum computer that could outperform classical computers and this could be facilitated if every qubit consisted of only one ion which is the reason for the recent explorations towards nano crystals. This development also makes it possible to address another problem of rare earth ions, their low probability of spontaneous emission. Placing nano crystals into an optical cavity promises the enhancement of spontaneous emission by the Purcell effect. Ideal cavities are Fiber Fabry-Pérot resonators of which one side consists of a mirror coated fiber tip and the other of a plane mirror functioning as the substrate for the nano crystals. It was the aim of this thesis to build a cryostat holder that would bring these two mirrors together with high stability and would allow for vertical and horizontal scanning.