A universal, large-scale quantum computer would be a powerful tool with applications of high value to mankind. For example, such a computer could significantly speed up the search for new medications or materials. However, the error rates of current qubit designs are simply too large to enable interesting computations. Therefore, both error correction and improved designs of qubits are needed. In 2001, Gottesman, Kitaev and Preskill proposed an encoding (GKP code) where a qubit is stored in a harmonic oscillator — a system that can be controlled and manufactured with high precision, and therefore have comparatively high coherence times. Moreover, the code offers good protection against losses, a simple gate set, and error correction circuits that are comparatively easy to implement. The drawback is that encoding a qubit into a GKP code state is a challenging task. In this thesis, we develop efficient schemes to encode a GKP qubit. Bosonic codes, where a qubit is stored in an oscillator, and in particular the GKP code are still relatively unknown. Therefore, we will start the thesis with an overview of the field, and provide the reader with the tools to analyze a GKP code, as these are quite different from standard error correcting codes. A tool which is important to understand, and that describes a protocol that encodes a GKP qubit is the so-called phase estimation algorithm. This algorithm allows to measure the eigenvalue of any unitary operation, and is one of the cornerstones of quantum information. We will show how phase estimation can be applied to encode a GKP qubit, and what the requirements for an experiment attempting to do so are. A major advantage of the GKP code over other encodings is that it can tolerate significant photon loss before the encoded information is lost. In addition, states that are closely related to the GKP qubit can be used to violate Bell’s inequalities (i. e. prove the presence of entanglement), even in the presence of large noise. Both these applications make the code particularly interesting in the optical regime, where error correction usually cannot be done while the signal is travelling. In this thesis, we will analyze an encoding protocol originally proposed by H.M. Vasconcelos, L. Sanz, and S. Glancy, Optics Letters 35, 3261 (2010) that relied on post-selection, and show that any output state can be used as a GKP code state with a simple change of frame, providing an exponential speedup. In 2019, two separate experiments generated a GKP code state for the first time: C.Flühmann et al., Nature 566, 513 (2019) realized a GKP qubit in the motional mode of a trapped ion, while P. Campagne-Ibarcq et al., Nature 584, 368 (2020) realized it with a transmon qubit coupled to a microwave cavity. However, both these experiments employ phase estimation, which is slow because it requires many measurements in sequence. We propose a circuit that allows a single-shot measurement of the GKP stabilizers, and analyze the performance of such a measurement as well as the impact of noise.@en

One of the most direct preparations of a Gottesman-Kitaev-Preskill (GKP) qubit in an oscillator uses a tunable photon-pressure (also called optomechanical) coupling of the form qˆbˆ†bˆ, enabling us to imprint the modular value of the position qˆ of one oscillator onto the state of an ancilla oscillator. We analyze the practical feasibility of executing such modular quadrature measurements in a parametric circuit-QED realization of this coupling. We provide estimates for the expected GKP squeezing induced by the protocol and discuss the effect of photon loss and other errors on the resulting squeezing.@en

Grid (or comb) states are an interesting class of bosonic states introduced by Gottesman, Kitaev, and Preskill [D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001)PLRAAN1050-294710.1103/PhysRevA.64.012310] to encode a qubit into an oscillator. A method to generate or "breed" a grid state from Schrödinger cat states using beam splitters and homodyne measurements is known [H. M. Vasconcelos, L. Sanz, and S. Glancy, Opt. Lett. 35, 3261 (2010)OPLEDP0146-959210.1364/OL.35.003261], but this method requires postselection. In this paper we show how postprocessing of the measurement data can be used to entirely remove the need for postselection, making the scheme much more viable. We bound the asymptotic behavior of the breeding procedure and demonstrate the efficacy of the method numerically.@en

3D-polarized light imaging (3D-PLI) reconstructs nerve fibers in histological brain sections by measuring their birefringence. This study investigates another effect caused by the optical anisotropy of brain tissue – diattenuation. Based on numerical and experimental studies and a complete analytical description of the optical system, the diattenuation was determined to be below 4 % in rat brain tissue. It was demonstrated that the diattenuation effect has negligible impact on the fiber orientations derived by 3D-PLI. The diattenuation signal, however, was found to highlight different anatomical structures that cannot be distinguished with current imaging techniques, which makes Diattenuation Imaging a promising extension to 3D-PLI.@en

3D-polarized light imaging (3D-PLI) reconstructs nerve fibers in histological brain sections by measuring their birefringence. This study investigates another effect caused by the optical anisotropy of brain tissue – diattenuation. Based on numerical and experimental studies and a complete analytical description of the optical system, the diattenuation was determined to be below 4 % in rat brain tissue. It was demonstrated that the diattenuation effect has negligible impact on the fiber orientations derived by 3D-PLI. The diattenuation signal, however, was found to highlight different anatomical structures that cannot be distinguished with current imaging techniques, which makes Diattenuation Imaging a promising extension to 3D-PLI.@en

3D-polarized light imaging (3D-PLI) reconstructs nerve fibers in histological brain sections by measuring their birefringence. This study investigates another effect caused by the optical anisotropy of brain tissue – diattenuation. Based on numerical and experimental studies and a complete analytical description of the optical system, the diattenuation was determined to be below 4 % in rat brain tissue. It was demonstrated that the diattenuation effect has negligible impact on the fiber orientations derived by 3D-PLI. The diattenuation signal, however, was found to highlight different anatomical structures that cannot be distinguished with current imaging techniques, which makes Diattenuation Imaging a promising extension to 3D-PLI.@en

3D-polarized light imaging (3D-PLI) reconstructs nerve fibers in histological brain sections by measuring their birefringence. This study investigates another effect caused by the optical anisotropy of brain tissue – diattenuation. Based on numerical and experimental studies and a complete analytical description of the optical system, the diattenuation was determined to be below 4 % in rat brain tissue. It was demonstrated that the diattenuation effect has negligible impact on the fiber orientations derived by 3D-PLI. The diattenuation signal, however, was found to highlight different anatomical structures that cannot be distinguished with current imaging techniques, which makes Diattenuation Imaging a promising extension to 3D-PLI.@en

Grid states can encode a qubit into a bosonic mode or be used for sensing. We propose protocols that prepare grid states using Schro¨dinger cat states, linear optics and homodyne detection without post-selection.@en

We show that one can determine both parameters of a displacement acting on an oscillator with an accuracy which scales inversely with the square root of the number of photons in the oscillator. Our results are obtained by using a grid state as a sensor state for detecting small translations in phase space (displacements). Grid states were first proposed [D. Gottesman, Phys. Rev. A 64, 012310 (2001)PLRAAN1050-294710.1103/PhysRevA.64.012310] for encoding a qubit into an oscillator: an efficient preparation protocol of such states, using a coupling to a qubit, was later developed [B. M. Terhal and D. Weigand, Phys. Rev. A 93, 012315 (2016)1050-294710.1103/PhysRevA.93.012315]. We compare the performance of the grid state with the quantum compass or cat code state and place our results in the context of the two-parameter quantum Cramér-Rao lower bound on the variances of the displacement parameters. We show that the accessible information about the displacement for a grid state increases with the number of photons in the state when we measure and prepare the state using a phase estimation protocol. This is in contrast with the accessible information in the quantum compass state which we show is always upper bounded by a constant, independent of the number of photons. We present numerical simulations of a phase estimation based preparation protocol of a grid state in the presence of photon loss, nonlinearities, and qubit measurement, using no post-selection, showing how the two effective squeezing parameters which characterize the grid state change during the preparation. The idea behind the phase estimation protocol is a simple maximal-information gain strategy.@en

We show that one can determine both parameters of a displacement acting on an oscillator with an accuracy which scales inversely with the square root of the number of photons in the oscillator. Our results are obtained by using a grid state as a sensor state for detecting small translations in phase space (displacements). Grid states were first proposed [D. Gottesman, Phys. Rev. A 64, 012310 (2001)PLRAAN1050-294710.1103/PhysRevA.64.012310] for encoding a qubit into an oscillator: an efficient preparation protocol of such states, using a coupling to a qubit, was later developed [B. M. Terhal and D. Weigand, Phys. Rev. A 93, 012315 (2016)1050-294710.1103/PhysRevA.93.012315]. We compare the performance of the grid state with the quantum compass or cat code state and place our results in the context of the two-parameter quantum Cramér-Rao lower bound on the variances of the displacement parameters. We show that the accessible information about the displacement for a grid state increases with the number of photons in the state when we measure and prepare the state using a phase estimation protocol. This is in contrast with the accessible information in the quantum compass state which we show is always upper bounded by a constant, independent of the number of photons. We present numerical simulations of a phase estimation based preparation protocol of a grid state in the presence of photon loss, nonlinearities, and qubit measurement, using no post-selection, showing how the two effective squeezing parameters which characterize the grid state change during the preparation. The idea behind the phase estimation protocol is a simple maximal-information gain strategy.@en

Gottesman, Kitaev, and Preskill have formulated a way of encoding a qubit into an oscillator such that the qubit is protected against small shifts (translations) in phase space. The idea underlying this encoding is that error processes of low rate can be expanded into small shift errors. The qubit space is defined as an eigenspace of two mutually commuting displacement operators Sp and Sq which act as large shifts or translations in phase space. We propose and analyze the approximate creation of these qubit states by coupling the oscillator to a sequence of ancilla qubits. This preparation of the states uses the idea of phase estimation where the phase of the displacement operator, say Sp, is approximately determined. We consider several possible forms of phase estimation. We analyze the performance of repeated and adaptive phase estimation as the simplest and experimentally most viable schemes given a realistic upper limit on the number of photons in the oscillator. We propose a detailed physical implementation of this protocol using the dispersive coupling between a transmon ancilla qubit and a cavity mode in circuit QED. We provide an estimate that in a current experimental setup one can prepare a good code state from a squeezed vacuum state using eight rounds of adaptive phase estimation, lasting in total about 4μs, with 94% (heralded) chance of success.@en

Gottesman, Kitaev, and Preskill have formulated a way of encoding a qubit into an oscillator such that the qubit is protected against small shifts (translations) in phase space. The idea underlying this encoding is that error processes of low rate can be expanded into small shift errors. The qubit space is defined as an eigenspace of two mutually commuting displacement operators Sp and Sq which act as large shifts or translations in phase space. We propose and analyze the approximate creation of these qubit states by coupling the oscillator to a sequence of ancilla qubits. This preparation of the states uses the idea of phase estimation where the phase of the displacement operator, say Sp, is approximately determined. We consider several possible forms of phase estimation. We analyze the performance of repeated and adaptive phase estimation as the simplest and experimentally most viable schemes given a realistic upper limit on the number of photons in the oscillator. We propose a detailed physical implementation of this protocol using the dispersive coupling between a transmon ancilla qubit and a cavity mode in circuit QED. We provide an estimate that in a current experimental setup one can prepare a good code state from a squeezed vacuum state using eight rounds of adaptive phase estimation, lasting in total about 4μs, with 94% (heralded) chance of success.@en