In recent decades enormous progress has been made in studying the geometrical structure of the quantum state space. Far from an abstraction, this geometric struc- ture is defined operationally in terms of the distinguishability of states connected by parameterizations that can be controlled in a laboratory. This geometry is manifest in the kinds of response functions that are measured by well established experimen- tal techniques, such as inelastic neutron scattering. In this thesis we explore the properties of the state space geometry in the vicinity of the ground state of two paradigmatic models of low dimensional magnetism. The first model is the spin-1 anti-ferromagnetic Heisenberg chain, which is a central example of symmetry pro- tected topological physics in one dimension, exhibiting a non-local string order, and symmetry protected short range entanglement. The second is the Kitaev honeycomb model, a rare example of an analytically solvable quantum spin liquid, characterized by long range topological order. In Chapter 2 we employ the single mode approximation to estimate the genuine multipartite entanglement in the spin-1 chain as a function of the unaxial anisotropy up to finite temperature. We find that the genuine multipartite entanglement ex- hibits a finite temperature plateau, and recove the universality class of the phase transition induced by negative anisotropy be examining the finite size scaling of the quantum Fisher information. In Chapter 4 we map out the zero temperature phase diagram in terms of the QFI for a patch of the phase space parameterized by the anisotropy and applied magnetic field, establishing that any non-zero anisotropy en- hances that entanglement of the SPT phase, and the robustness of the phase to finite temperatures. We also establish a connection between genuine multipartite entanglement and state space curvature. In Chapter 3 we turn to the Kitaev honeycomb model and demonstrate that, while the QFI associated to local operators remains trivial, the second derivative of such quantities with respect to the driving parameter exhibit divergences. We characterize the critical exponents associated with these divergences. / Thesis / Doctor of Philosophy (PhD) / Systems composed of many bodies tend to order as their energy is reduced. Steam, a state characterized by the complete disorder of the constituent water molecules, condenses to liquid water as the temperature (energy) decreases, wherein the water molecules are organized enough for insects to walk atop them. Water freezes to ice, which is so ordered that it can hold sleds and skaters. Quantum mechanics allows for patterns of organization that go beyond the solid-liquid-gas states. These patterns are manifest in the smallest degrees of freedom in a solid, the electrons, and are responsible for fridge magnets and transistors. While quantum systems still tend to order at lower energies, they are characterized by omni-present fluctuations that can conceal hidden forms of organization. One can imagine that the states of matter live in a vast space, where each point represents a different pattern. In this thesis we show that by probing the geometry of this space, we can detect hidden kinds of order that would be otherwise invisible to us.

<p>The blooming of quantum computer hardware provokes enormous enthusiasm seeking for applications in various fields.</p> <p>Particularly, it is always of great interest to study the chemical or physical systems with quantum enhanced learning process or quantum simulation in the NISQ era.</p> <p>Here we will present our recent research on chemical or physical systems based on quantum computing. </p> <p><br></p> <p>One main focus of this dissertation is the quantum classification algorithms development, especially for the entanglement classification.</p> <p>As a quantum mechanical property describing the correlation between quantum mechanical systems, entanglement has no classical analog.</p> <p>In the past 100 years, entanglement has been attracting enormous attentions in both the theoretical and experimental research.</p> <p>We investigate the entanglement classification in chemical reactions, generalizing the typical CHSH inequality from discrete measurement results into the continuous measurement results.</p> <p>Furthermore, we develop a quantum classification algorithm based on the typical instance-based learning algorithms, which in turn is applied into the entanglement classification problems.</p> <p>Additionally, the proposed quantum algorithm has a variety of applications, such as the prediction of phase transition. </p> <p><br></p> <p>Quantum-enhanced classification algorithm is never the only practicable application of quantum computer.</p> <p>Moreover, we propose a universal quantum circuit implementation to estimate a given one-dimensional functions with a finite Fourier expansion.</p> <p>We demonstrate the circuit implementation with the application on square wave function.</p> <p>Additionally, we present a quantum circuit for the typical time-independent perturbation theory.</p> <p>Perturbation theory is always one of the most powerful tools for physicists and chemists dealing with the eigenenergy problems in quantum mechanics.</p> <p>Though PT is quite popular today, it seems that the techniques for PT does not take a ride in the era of quantum computing.</p> <p>In this dissertation, we present a a universal quantum circuit implementation  for the time-independent PT method, which is often termed as Rayleigh–Schr\"odinger PT.</p> <p>In order to demonstrate the implementation of the proposed quantum circuit, the extended Fermi Hubbard Model is introduced as an example.</p> <p>In particular, the proposed quantum circuit shows considerable speedup comparing with the typical PT methods.</p>

Quantum Computation and Information requires high accuracy in gate control despite noises and imperfections from the environment and physical implementation. Here we introduce an SCQC Formalism based on dynamical decoupling and reverse engineering. Space Curve Quantum Control Formalism discovers the tight connections between quantum, geometric, and classical systems. We are able to use such connections to build noise-canceling, precise control, and time-optimal arbitrary gates. / Doctor of Philosophy / Quantum Computation and Information is a fast-developing technology and its application is within reach. But errors due to noises in the environment and imperfections from physical implementation are roadblocks to the practical application. In this thesis, we will introduce the Space Curve Quantum Control Formalism, which builds connections between Geometric, Quantum, and Classical pictures. We utilize these connections to build noise-robust quantum gates and time-optimal gates.

Quantum computers have seen rapid development over the last two decades. Despite this, they are not yet scalable or fault-tolerant (i.e. we cannot address arbitrarily many error-corrected qubits). Therefore, improvements that include consideration of the underlying physics are paramount. To do this, we must reduce existing errors and understand how algorithms perform without error correction. In this dissertation, we provide contributions toward these goals. We organize these efforts into three groups. Firstly, we focus on quantum control. We introduce a novel scheme for performing entangling gates on superconducting qubits. We create fast, high-fidelity entangling operations and single-qubit gates to implement arbitrary quantum operations. Then, we implement entangling gates on real transmon qubits. Finally, we develop new techniques for entangling gates on spin qubits. In total, we improve low-level device performance with high-fidelity entangling operations. Secondly, we focus on quantum simulation algorithms. First, we apply error mitigation techniques to a quantum simulation algorithm while simultaneously performing device characterization. Then we take advantage of known symmetries of the input Hamiltonian to improve the same algorithm. Then, we demonstrate that this reduces resources compared to other approaches in the presence of noise. Then we compare this technique with state-of-the-art approaches. Then, we improve this algorithm with approaches from quantum control. Finally, we develop a novel algorithm to simulate spin chains on a quantum processor with improved resources compared to other techniques. In total, we improve quantum simulation algorithms, with the aim of better utilizing current devices. Thirdly, we consider the ADAPT-VQE algorithm, which is used to construct quantum circuits for preparing trial states in quantum simulation. In total, we improve gate counts for the algorithm, improve a separate algorithm by utilizing the gradient criterion, and leverage the repeating structure of an input Hamiltonian to improve performance. Finally, we provide a deeper understanding of ADAPT-VQE and demonstrate its robustness to scaling issues of competing algorithms. In total, we improve the algorithm and its applicability. Thus, we improve quantum simulation algorithms that can be run in the near term. / Doctor of Philosophy / The computers that we interact with every day rely on the processing of bits, represented as 1's or 0's. The rules that govern how they operate mostly rely on classical physics (i.e. discovered before quantum physics), which does not include any quantum effects. If we instead allow for quantum rules and quantum bits ("qubits"'), new types of algorithms are possible. That is to say, quantum computers can do some things more efficiently than classical computers. As such, there is a massive effort to build these devices. Because these devices are so delicate and in their early stages, this requires an understanding of the algorithm and the physical device performing it. Therefore, improving the overall performance requires taking high and low-level aspects of this design into consideration. In this dissertation, we provide three groups of contributions to achieving this goal. In the first group, we improve the device performance by considering how operations are performed on qubits, primarily in terms of producing quantum operations that have no classical analog. In the second group, we improve the simulation of quantum systems on quantum devices with a focus on how existing imperfections in the device impact the results. In the third group, we make improvements to an algorithm used to simulate quantum systems like molecules, while also developing a deeper understanding of how the algorithm functions. In each of these parts, we develop novel techniques to improve device and algorithm performance, contributing to the applicability and utility of current and future quantum devices.

The highly parallel nature of the fundamental principles of quantum mechanics means that certain key resource-intensive tasks --- including searching, code decryption and medical, chemical and material simulations --- can be computed polynomially or even exponentially faster with a quantum computer. In spite of its remarkably fast development, the field of quantum computing is still young, and a large-scale prototype using any one of the candidate quantum bits (or 'qubits') under investigation has yet to be developed. Spin-based qubits in condensed matter systems are excellent candidates. Spins controlled using magnetic resonance have provided the first, most advanced, and highest fidelity experimental demonstrations of quantum algorithms to date. Despite having highly promising control characteristics, most physical ensembles investigated using magnetic resonance are unable to produce entanglement, a critical missing ingredient for a pure-state quantum computer. Quantum objects are said to be entangled if they cannot be described individually: they remain fundamentally linked regardless of their physical separation. Such highly non-classical states can be exploited for a host of quantum technologies including teleportation, metrology, and quantum computation. Here I describe how to experimentally create, control and characterise entangled quantum ensembles using magnetic resonance. I first explore the relationship between entanglement and quantum metrology and demonstrate a sensitivity enhancement over classical resources using molecular sensors controlled with liquid-state nuclear magnetic resonance. I then examine the computational potential of irreversible relaxation processes in combination with traditional reversible magnetic resonance control techniques. I show how irreversible processes can polarise both nuclear and electronic spins, which improves the quality of qubit initialisation. I discuss the process of quantum state tomography, where an arbitrary quantum state can be accurately measured and characterised, including components which go undetected using traditional magnetic resonance techniques. Lastly, I combine the above findings to initialise, create and characterise entanglement between an ensemble of electron and nuclear spin defects in silicon. I further this by generating pseudo-entanglement between an ensemble of nuclear spins mediated by a transient electron spin in a molecular system. These findings help pave the way towards a particular architecture for a scalable, spin-based quantum computer.

Spin properties of donor impurities in silicon have been investigated by electron spin resonance (ESR) techniques for more than sixty years. These studies gave us a contribution towards understanding some of the physics of doped semiconductor materials in general, which is the platform for much of our current technology. Despite the fact that donor electron and nuclear spins have been researched for so long, ESR studies of their properties are still giving us interesting insights. With the introduction of the concept of quantum information in the 1980s, some properties of donor spins in silicon, that were known from the fifties (such as long relaxations), have been reinterpreted for their potential application in this field. Since then, incredible experimental results have been achieved with magnetic resonance control, including manipulation and read-out of individual spins. However, some open questions are still to be answered before the realisation of a spin-based silicon quantum architecture will be achieved. Currently, ESR studies still contribute to help answering some of those questions. In this thesis, we demonstrate electrical and optical methods for donor charge state manipulation measured by ESR. Recent experiments have demonstrated that coherence time of nuclear spins may be enhanced by manipulating the state of donors from neutral to singly charged. We investigate electric field ionisation/neutralisation of arsenic donors in a silicon SOI device measured by ESR. Below ionisation threshold, we also measure the hyperfine Stark shift of arsenic donors spins in silicon. These results have, for instance, implications on how fast individual addressability of donor spins may be achieved in certain quantum computer architectures. Here, we also study optical-driven charge state manipulation of selenium impurities in silicon. Selenium has two additional electrons when it replaces an atom in the silicon crystal (i.e. double donor). The electronic properties of singly-ionised selenium make it potentially advantageous as spin qubit, compared to the more commonly studied group-V donors. For instance, we find here that the electron spin relaxation and coherence times of selenium are up to two orders of magnitude longer than phosphorus at the same temperature. Finally, we demonstrate that it is possible to bring selenium impurity in singly-charged state and subsequently re-neutralise them leaving a potential long-lived ⁷⁷Se nuclear spin.

The present work, briefly described below, consists of implementation of several quantum algorithms in an NMR Quantum Information Processor. Game theory gives us mathematical tools to analyze situations of conflict between two or more players who take decisions that influence their welfare. Classical game theory has been applied to various fields such as market strategy, communication theory, biological processes, foreign policies. It is interesting to study the behaviour of the games when the players share certain quantum correlations such as entanglement. Various games have been studied under the quantum regime with the hope of obtaining some insight into designing new quantum algorithms. Chapter 2 presents the NMR implementation of three such algorithms. Experimental NMR implementation given in this chapter are: (i) Three qubit ‘Dilemma’ game with corrupt sources’. The Dilemma game deals with the situation where three players have to choose between going/not going to a bar with a seating capacity of two. It is seen that in the players have a higher payoff if they share quantum correlations. However, the pay-off falls rapidly with increasing corruption in the source qubits. Here we report the experimental NMR implementation of the quantum version of the Dilemma game with and without corruption in the source qubits. (ii) Two qubit ‘Ulam’s game’. This is a two player game where one player has to find out the binary number thought by the other player. This problem can be solved with one query if quantum resources are used. This game has been implemented in a two qubit system in an NMR quantum information processor. (iii) Two qubit ‘Battle of Sexes’ game. This game deal with a situation where two players have conflicting choices but a deep desire to be together. This leads to a dilemma in the classical case. Quantum mechanically this dilemma is resolved and a unique solution emerges. The NMR implementation of the quantum version of this game is also given in this chapter. Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover’s search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, if one uses local adiabatic evolutions, their complexity is of the order √N (where N=2n) [37, 38]. In Chapter 3, the NMR implementation of (i) the Deutsch-Jozsa and the (ii) Grover’s search algorithm using local adiabatic evolution has been presented. In adiabatic algorithm, the system is first prepared in the equal superposition of all the possible states which is the ground state of the beginning Hamiltonian. The solution is encoded in the ground state of the final Hamiltonian. The system is evolved under a linear combination of the beginning and the final Hamiltonian. During each step of the evolution the interpolating Hamiltonian slowly changes from the beginning to the final Hamiltonian, thus evolving the ground state of the beginning Hamiltonian towards the ground state of the final Hamiltonian. At the end of the evolution the system is in the ground state of the final Hamiltonian which is the solution. The final Hamiltonian, for each of the two cases of adiabatic algorithm described in this chapter, are constructed depending on the problem definition. Adiabatic algorithms have been proved to be equivalent to standard quantum algorithms with respect to complexity [39]. NMR implementation of adiabatic algorithms in homonuclear spin systems face problems due to decoherence and complicated pulse sequences. The decoherence destroys the answer as it causes the final state to evolve to a mixed state and in homonuclear systems there is a substantial evolution under the internal Hamiltonian during the application of the soft pulses which prevents the initial state to converge to the solution state. The resolution of these issues are necessary before one can proceed for the implementation of an adiabatic algorithm in a large system. Chapter 4 demonstrates that by using ‘strongly modulated pulses’ for creation of interpolating Hamiltonian, one can circumvent both the problems and thus successfully implement the adiabatic SAT algorithm in a homonuclear three qubit system. The ‘strongly modulated pulses’ (SMP) are computer optimized pulses in which the evolution under the internal Hamiltonian of the system and RF inhomogeneities associated with the probe is incorporated while generating the SMPs. This results in precise implementation of unitary operators by these pulses. This work also demonstrates that the strongly modulated pulses tremendously reduce the time taken for the implementation of the algorithm, can overcome problems associated with decoherence and will be the modality in future implementation of quantum information processing by NMR. Quantum search algorithm, involving a large number of qubits, is highly sensitive to errors in the physical implementation of the unitary operators. This can put an upper limit to the size of the data base that can be practically searched. The lack of robustness of the quantum search algorithm for a large number of qubits, arises from the fact that stringent ‘phase-matching’ conditions are imposed on the algorithm. To overcome this problem, a modified operator for the search algorithm has been suggested by Tulsi [40]. He has theoretically shown that even when there are errors in implementation of the unitary operators, the search algorithm with his modified operator converges to the target state while the original Grover’s algorithm fails. Chapter 5, presents the experimental NMR implementation of the modified search algorithm with errors and its comparison with the original Grover’s search algorithm. We experimentally validate the theoretical predictions made by Tulsi that the introduction of compensatory Walsh-Hadamard and phase-flip operations refocuses the errors. Experimental Quantum Information Processing is in a nascent stage and it would be too early to predict its future. The excitement on this topic is still very prevalent and many options are being explored to enhance the hardware and software know-how. This thesis endeavors in this direction and probes the experimental feasibility of the quantum algorithms in an NMR quantum information processor.

Electron spin resonance (ESR) experiments are used to assess the possibilities for processing quantum information in the electronic and nuclear spins of endohedral fullerenes. It is shown that ¹⁵N@C₆₀ can be used for universal two-qubit quantum computing. The first step in this scheme is to initialize the nuclear and electron spins that each store one qubit. This was achieved with a magnetic field of 8.6 T at 3 K, by applying resonant RF and microwave radiation. This dynamic nuclear polarization technique made it possible to show that the nuclear T₁ time of ¹⁵N@C₆₀ is on the order of twelve hours at 4.2 K. The electronic T₂ is the limiting decoherence time for the system. At 3.7 K, this can be extended to 215 μs by using amorphous sulphur as the solvent. Pulse sequences are described that could perform all single-qubit gates to the two qubits independently, as well as CNOT gates. After these manipulations, the value of the qubits should be measured. Two techniques are demonstrated for this, by measuring the nuclear spin. Sc@C₈₂ could also be useful for quantum computation. By comparing ESR measurements with density functional theory calculations, it is shown how the orientation of a Sc@C₈₂ molecule in an applied magnetic field affects the molecule's Zeeman and hyperfine coupling. Hence the g- and A-tensors are written in the coordinate frame of the molecule. Pulsed ESR measurements show that the decoherence time at 20 K is 13 μs, which is 20 times longer than had been previously reported. Carbon nanotubes have been filled with endohedral fullerenes, forming 1D arrays that could lead to a scalable quantum computer. N@C₀₆ and Sc@C₈₂ have been used for this filling in various concentrations. ESR measurements of these samples are consistent with simulations of the dipolar coupling.

This thesis describes experimental demonstrations of high-fidelity readout of trapped ion quantum bits ("qubits") for quantum information processing. We present direct single-shot measurement of an "optical" qubit stored in a single calcium-40 ion by the process of resonance fluorescence with a fidelity of 99.991(1)% (surpassing the level necessary for fault-tolerant quantum computation). A time-resolved maximum likelihood method is used to discriminate efficiently between the two qubit states based on photon-counting information, even in the presence of qubit decay from one state to the other. It also screens out errors due to cosmic ray events in the detector, a phenomenon investigated in this work. An adaptive method allows the 99.99% level to be reached in 145us average detection time. The readout fidelity is asymmetric: 99.9998% is possible for the "bright" qubit state, while retaining 99.98% for the "dark" state. This asymmetry could be exploited in quantum error correction (by encoding the "no-error" syndrome of the ancilla qubits in the "bright" state), as could the likelihood values computed (which quantify confidence in the measurement outcome). We then extend the work to parallel readout of a four-ion string using a CCD camera and achieve the same 99.99% net fidelity, limited by qubit decay in the 400us exposure time. The behaviour of the camera is characterised by fitting experimental data with a model. The additional readout error due to cross-talk between ion images on the CCD is measured in an experiment designed to remove the effect of qubit decay; a spatial maximum likelihood technique is used to reduce this error to only 0.2(1)x10^{-4} per qubit, despite the presence of ~4% optical cross-talk between neighbouring qubits. Studies of the cross-talk indicate that the readout method would scale with negligible loss of fidelity to parallel readout of ~10,000 qubits with a readout time of ~3us per qubit. Monte-Carlo simulations of the readout process are presented for comparison with experimental data; these are also used to explore the parameter space associated with fluorescence detection and to optimise experimental and analysis parameters. Applications of the analysis methods to readout of other atomic and solid-state qubits are discussed.

Quantum information processing (QIP), which inherently utilizes quantum mechanical phenomena to perform information processing, may outperform its classical counterpart at certain tasks. As one of the physical implementations of QIP, the electron-spin based architecture has recently attracted great interests. Endohedral fullerenes with unpaired electrons, such as N@C₆₀, are promising candidates to embody the qubits because of their long spin decoherence time. This thesis addresses several fundamental aspects of the strategy of engineering the N@C₆₀ molecules for applications in QIP. Chemical functionalization of N@C₆₀ is investigated and several different derivatives of N@C₆₀ are synthesized. These N@C₆₀ derivatives exhibit different stability when they are exposed to ambient light in a degassed solution. The cyclopropane derivative of N@C60 shows comparable stability to pristine N@C₆₀, whereas the pyrrolidine derivatives demonstrate much lower stability. To elucidate the effect of the functional groups on the stability, an escape mechanism of the encapsulated nitrogen atom is proposed based on DFT calculations. The escape of nitrogen is facilitated by a 6-membered ring formed in the decomposition of the pyrrolidine derivatives of N@C₆₀. In contrast, the 4-membered ring formed in the cyclopropane derivative of N@C₆₀ prohibits such an escape through the addends. Two N@C₆₀-porphyrin dyads are synthesized. The dyad with free base porphyrin exhibits typical zero-field splitting (ZFS) features due to functionalization in the solid-state electron spin resonance (ESR) spectrum. However, the nitrogen ESR signal in the second dyad of N@C₆₀ and copper porphyrin is completely suppressed at a wide range of sample concentrations. The dipolar coupling between the copper spin and the nitrogen spins is calculated to be 27.0 MHz. To prove the presence of the encapsulated nitrogen atom in the second dyad, demetallation of the copper porphyrin moiety is carried out. The recovery of approximately 82% of the signal intensity confirms that the dipolar coupling suppresses the ESR signal of N@C₆₀. To prepare ordered structure of N@C₆₀, the nematic matrix MBBA is employed to align the pyrrolidine derivatives of N@C₆₀. Orientations of these derivatives are investigated through simulation of their ESR spectra. The derivatives with a –CH3 or phenyl group derived straightforward from the N-substituent of the pyrrolidine ring are preferentially oriented based on their powder-like ESR spectra in the MBBA matrix. An angle of about is also found between the directors of fullerene derivatives and MBBA. In contrast, the derivatives with a –CH₂ group inserted between the phenyl group and the pyrrolidine ring are nearly randomly distributed in MBBA. These results illustrate the applicability of liquid crystal as a matrix to align N@C₆₀ derivatives for QIP applications.