In this thesis observations on the application of parametric drives to superconducting quantum circuits in disparate parameter regimes are presented. By the nonlinear inductance of the Josephson junction, a variety of interactions in circuit quantum electrodynamical systems comprised of strongly, moderately, and weakly nonlinear oscillators are realized.

Chapter 1 contains an introduction to classical and quantum information and introduces superconducting circuits as a platform for quantum information processing. An outline of the contents of the thesis is also provided.

In Chapter 2 a theoretical foundation for the later chapters is established, spanning from the classical harmonic oscillator to circuit quantum electrodynamical systems and parametric driving. The transmon qubit, junction-embedded coplanar waveguide, tunable coupler, and Josephson junction array resonator are introduced and some methods for realizing parametrically activated interactions in such systems are discussed.

Chapter 3 focuses on the steps necessary for constructing a superconducting quantum circuit. The design, simulation, and fabrication methods necessary for creating the experimental devices of later chapters are discussed.

In Chapter 4 results of the parametrically activated interactions between two tunably coupled transmon qubits by flux modulation of a SQUID are presented. When the coupling SQUID is modulated at the sum or difference frequencies of the transmons, level repulsion and attraction are observed spectroscopically. The viability of the platform for analog quantum simulations is discussed and the experimental results are compared to analytical models and numerical simulations of the quantum master equation.

In Chapter 5 spectroscopic signatures of a few-photon Kerr parametric oscillator are observed upon the application of an all-microwave bichromatic drive to a Josephson junction-embedded coplanar waveguide resonator. Semiclassical analytical, numerical, and quantum master equation simulations are performed and compared with the experimental results. An effective model based on semiclassical methods proves insufficient in modelling the behaviour of the system, indicating the presence of quantum effects.

In Chapter 6 a weakly nonlinear Josephson junction array resonator is bichromatically driven into a parametric phase state. Stochastic switching between the two non-equilibrium stationary states of the system is observed and the time between stochastic switching events is determined for a range of drive strengths. An additional microwave drive resonant with the frequency of parametric response is applied and the system is biased into one of the phase states. The biasing and change in switching time as a function of drive power and phase is shown. The contributions of classical and quantum effects to the occurrence of switching events is discussed.

In Chapter 7 measurements of a strongly parametrically driven Duffing oscillator are presented. As the system is strongly driven at a variety of large negative detunings, signatures of chaotic behaviour are observed in the output field spectrum and quadrature histograms. The observed features are discussed and compared to known markers of chaotic behaviour in classical parametrically driven Duffing oscillators.

Chapter 8 concludes the thesis, providing a review of the contents and findings of the previous chapters. The thesis ends with an outlook and suggestions for potential future topics of study.
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Quantum computers hold great potential to solve complex physics and chemistry problems that are beyond the capabilities of classical computers. Unlike classical systems that use bits represented as deterministic 0s and 1s, quantum computers operate with quantum bits or qubits, which can exist in a superposition of states. In addition to superposition, quantum computers harness the power of entanglement and quantum interference that enable quantum computers to encode and process information in ways that classical systems fundamentally cannot. These properties allow quantum computers to explore vast solution spaces in parallel, solving certain computational problems far more efficiently than classical systems. This has numerous potential applications ranging from the factoring of large numbers, optimizing complex systems, computational chemistry, machine learning, cryptography, and artificial intelligence.

Yet, current quantum processors are fragile, noisy and fairly limited in both quantity and quality with tens of qubits and physical error rates of around 10-³. To realize practical quantum applications, however, error rates need to be below 10-¹⁵ across millions of qubits. To bridge this gap and fully harness the potential of quantum computers, quantum error correction (QEC) is essential. QEC codes are designed to protect quantum information by redundantly encoding it onto multiple physical qubits. This encoding allows for the detection and correction of local errors affecting individual qubits, e.g., through stabilizer measurements. Importantly, if the physical error rates are below a specific threshold, QEC codes can exponentially suppress logical error rates by increasing the number of physical qubits involved. This is essential for achieving fault-tolerant computations, which are key to unlocking the full potential of quantum computers…
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Quantum computers hold the potential to revolutionize computation by harnessing the unique properties of qubits. However, qubits are highly susceptible to errors, posing a major challenge in building reliable quantum systems. To address this, the development of quantum error correction codes is essential. The surface code has become a well-known code, using a 2D square grid to encode qubits. However, its high qubit overhead may be improved by alternative codes with more efficient encoding schemes. This thesis focuses on the small stellated dodecahedron code, previously outperformed by the surface code, and proposes a new approach to optimize its performance by developing an algorithm that finds interleaved measurement schedules. The developed algorithm is tested on both the surface code and a code based on the Tetrahemi hexahedron, correctly providing interleaved schedules for both. Applying the algorithm to the small stellated dodecahedron reduces the number of time steps required from 10 to 6, significantly boosting error correction performance. Simulations under a depolarizing noise model confirm the fault tolerance of these new schedules and demonstrate a 2.6x improvement in the code’s pseudo-threshold compared to the original sequential schedule. While the surface code remains more effective at protecting against errors, performing 1.26 times better, the stellated dodecahedron code offers a compelling alternative for near-term research, requiring fewer qubits while maintaining strong performance.

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes.

First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes.

Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode.

Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which conditions these two code constructions and the lifted product construction coincide.

Lastly, we consider a specific example of fibre bundle product codes: the twisted toric code. We determine an analytic expression for the distance of such a code and verify this expression using numerical simulations. Furthermore, we perform an extensive numerical study on these codes to determine how performing a twist alters the scaling of the logical error rate of such codes. We present a new analytic method to explain the scaling of these codes on a small domain, and verify the validity of these calculations by comparing them with the results of our numerical simulations.

In this thesis, the repetition code for bit flip errors is examined. Based the stabilizer measurements outcome of a run of the repetition code, one does not know exactly which errors have occurred. Statistics can be used to estimate the probability of all possible error events. This probability estimation is investigated for simulated data assuming phenomenological and circuit level noise. Experiments on the repetition code are performed by the DiCarlo group at the Delft University of Technology on a superconducting quantum computer. For this data, some error probabilities are estimated to be nonphysical. The goal of this thesis is to investigate these nonphysical probabilities and to determine whether they result from sampling noise or a problem with the assumed error model that determines the estimation of the probabilities. By estimating the standard deviation using the bootstrap method it is shown that the nonphysical probabilities are not due to sampling noise. Therefore, it is possible that non-conventional errors, such as leakage or crosstalk, are affecting these estimates.

In a further analysis on the error probabilities and their standard deviation, it is shown that the standard error for space and time edges in the experiment is consistent with the standard error from phenomenological noise initialised with the average error data and ancilla error probabilities, even when
these may be affected by non-conventional errors.

In this work, we first analyse a theoretical technique based on entropic inequalities, which in principle can be used to compare classical and quantum algorithms running on noisy quantum devices. The technique has been used in the past for depolarizing noise, and has shown that for NISQ variational quantum algorithms (VQAs) to give an advantage over purely classical methods, it is essential that the structure of the problem is close to the quantum hardware the VQA is run on. Here, we use this technique for relaxation and pure dephasing noise and examine whether we can deduce the same results for the Quantum Approximate Optimization Algorithm (QAOA) running on a noisy quantum device. In the later part of the thesis, we study the properties of the steady state of the Lindblad Master Equation for a 1D array of transmon qubits coupled by cross-resonance type interactions for relaxation and pure dephasing noise. For n=2,3 number of qubits we solve exactly the master equation, while for a general number of n qubits we approximate the solution by using two different approaches, namely perturbation theory and mean-field theory. Finally, we assess how well those approximation methods compare to the exact solution for n=2,3 qubits.

In this research, the implementations of quantum random walks in superconducting circuit-QED are studied. In particular, a walk that moves across the Fock states of a quantum harmonic oscillator by a Jaynes-Cummings model is investigated, which is difficult to implement because of different Rabi frequencies for different Fock states. Theoretically, the lower boundary vacuum state of the harmonic oscillator causes a reflection of the probability amplitudes in the distribution. A walk that moves across a grid of coherent states |nα+imα〉 in phase space is then investigated. A setup for a 1D and 2D quantum random walk is suggested, using controlled displacements of a resonator dispersively coupled to one or two superconducting transmon qubits in circuit-QED, followed by Hadamard gates. From numerical simulations it was observed that the 2D walk commutes for α·β = 0 mod π/2 for which the variance is proportional to the number of steps t squared. For other values of α·β the horizontal and vertical displacements do not commute, resulting in extra phase factors. The numerical simulations showed that for most values of α· β with a larger distance to 0 mod π/2 than 0.01, the probability distribution of the
walk collapses to a distribution centered around origin within t = 100 steps, similar to a classical random walk. Exceptions are the 2D walks for α·β = ±π/6 mod π/2 or ±
π/4 mod π/2, for which the variance is still proportional to the number of steps squared.

Quantum systems are in general not e_ciently simulatable by classical means. If one wishes to determine (some of) the eigenvalues of a Hamiltonian H that is associated with a quantum system, there are two favoured strategies: Quantum simulation and quantum Monte Carlo schemes. The former strategy uses an experimentally well-controllable quantum system that emulates the system of interest, in a digital (i.e. universal) or analog manner. The latter, albeit with a limited range of applicability, uses classical stochastic processes to e_ciently obtain (often low-lying) eigenvalues of H. Quantum Monte Carlo methods may su_er from a sign problem when simulating fermionic or frustrated bosonic systems. This yields, for a given accuracy, a simulation time that scales exponentially in the system size and the inverse temperature.

The concept of entanglement is one of the distinguishing features in quantum mechanics. Information about one particle can determine the state of another particle. This information and entanglement in subsystems is quantified by the entanglement entropy. The entanglement entropy has become an important research topic in recent years. Entanglement entropy is expected to grow with the boundary size for systems described by Hamiltonians with local interactions, described by the area law. For one dimension there exists a bound quantifying this area law S = O (2^1/ΔE) in terms of the energy gap ΔE between the ground-state energy and the first excited-state energy. It is an open problem whether the area law holds in higher spatial dimensions. Therefore it is of great interest to study the entanglement entropy in systems that violate the area law. In this thesis spin chain models with local interactions are studied for arbitrary spin. Hamiltonians with a unique ground state are constructed by mapping spin chains onto (coloured) Dyck and (coloured) Motzkin paths. We show that these models express logarithmic or power law violations of the area law for the bipartite entanglement entropy. These results are presented in the table below. The known bound of the area law in one dimensional systems is dependent on the energy gap of these models. The energy gap of the Motzkin-path model is investigated by constructing the an orthogonal excited state with small energy. In doing so we associate the Hamiltonian with the Laplacian of a graph. We present an alternative proof for the bound on the energy gap of the colourless Motzkin-path model than S. Bravyi et al and R. Movassagh et al. We show that the energy gap ΔE scales in terms of the chain length n as ΔE = O(n^-2) in the limit n →∞. Therefore ΔE → 0 in this limit, resulting in area-law violations of the entanglement entropy while maintaining the known bound on the entanglement entropy.

Bell inequalities are certain probabilistic inequalities that should hold in the context of quantum measurement under assumption of a local hidden variable model. These inequalities can be violated according to the theory of quantum mechanics and have also been violated experimentally. Bell inequalities were therefore historically used to disprove local hidden variable models as an interpretation of quantum mechanics. An interesting question is to what degree Bell inequalities can be violated according to the theory of quantum mechanics. The degree to which a particular Bell inequality can be violated is quantified by the largest violation of the Bell inequality. A central question we address in this thesis is how large the largest violation can become when considering all possible Bell inequalities. In particular the CHSH-inequality, a well-known Bell inequality, has a largest violation equal to the square root of 2 and we want to find a Bell inequality with largest violation exceeding the square root of 2. This thesis consists of two main parts. In the first part we formally define Bell inequalities and the largest violation and prove several theorems about Bell inequalities. In the second part we search for Bell inequalities with largest violation exceeding the square root of 2. We do this by first approximating the largest violation of a given Bell inequality using numerical optimization. Next, using this approximation, we use numerical optimization to maximize the largest violation. Due to run-time restrictions we were only able to consider Bell inequalities with a relatively small number of terms and were unable to find any among them with largest violation exceeding the square root of 2.

Electron spins trapped in quantum dots have recently proven to be a promising technology for the implementation of qubits, already demonstrating high fidelity single- and two-qubits gates. The next step towards fault-tolerant quantum computing is to increase the number of so-called spin qubits on the processor. However, this poses several challenges, one of which being how to implement single-qubit gates in multi-qubit systems, with high fidelity. This work focused on two aspects. The first one was to identify the physical phenomena and limitations that hinder the realisation of high fidelity single-qubit gates in multi-qubit environments. The second objective was to investigate, implement and optimise methods in order to minimise the impact of these perturbing phenomena. The identified perturbing effects include unwanted driving between a pulse and the neighbouring qubits, as well as frequency shifts induced by drive-spin coupling and non-ideal pulse in experimental setups. Crosstalk was addressed using pulse shaping (i.e. amplitude or phase modulation of the driving pulse to tailor its spectral characteristics and reduce the energy transmitted at unwanted frequencies). The performance of shapes taken from both NMR and signal processing literature was investigated, for various pulse parameters, through an extensive grid search. In addition, a frequency correction algorithm was devised: off-resonant drive frequencies are selected so that the shifted qubits are resonantly driven. It currently accounts for two phenomena: the AC Stark and Bloch-Siegert shifts. The algorithm furthermore tracks the changes caused by the time-dependent characteristics of the shaped pulses. The proposed frequency correction algorithm was shown to almost entirely negate the effects of the considered shifts, leading to unitary fidelity improvements by up to 55%. In addition, pulse shaping was demonstrated to noticeably improve the fidelity of simultaneous single-qubit rotations compared to unshaped driving. Rotations with fidelities as high as 99.98% were obtained for pi/2 rotations on two-qubits systems. Moreover, shapes whose Fourier transform is narrow and sharp, associated with low Rabi frequencies, were demonstrated to generally provide the highest fidelities of the tested configurations. Lastly, the trends and guidelines highlighted by these results were shown to scale to systems with larger numbers of qubits. The correction techniques investigated in this work have proven promising for the implementation of high fidelity single-qubit gates in multi-qubit systems. In particular, the guidelines for selecting a well-performing pulse shape should also be useful for the design of optimised driving schemes, regardless of the number of qubits involved. Additionally, the proposed frequency shift correction algorithm is expected to be able to handle arbitrary shifts, and so to be easily adaptable to use in experiments.

In this report the goals is to theoretically construct a shift resistant encoding in a rotor system. The report starts with an overview of the rotor space. In the context of which the difference between PVM and POVM is highlighted. After this overview, the shift resistant encodings are discussed. These two concepts are then combined and a shift resistant encoding in the rotor space is presented. Lastly a system which hints at an implementation is presented.. Of this system the Hamiltonian is discussed.