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Item Quantum Compiling Methods for Fault-Tolerant Gate Sets of Dimension Greater than Two(2019) Glaudell, Andrew Noble; Taylor, Jacob M; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Fault-tolerant gate sets whose generators belong to the Clifford hierarchy form the basis of many protocols for scalable quantum computing architectures. At the beginning of the decade, number-theoretic techniques were employed to analyze circuits over these gate sets on single qubits, providing the basis for a number of state-of-the-art quantum compiling algorithms. In this dissertation, I further this program by employing number-theoretic techniques for higher-dimensional gate sets on both qudit and multi-qubit circuits. First, I introduce canonical forms for single qutrit Clifford+T circuits and prove that every single-qutrit Clifford+T operator admits a unique such canonical form. I show that these canonical forms are T-optimal and describe an algorithm which takes as input a Clifford+T circuit and outputs the canonical form for that operator. The algorithm runs in time linear in the number of gates of the circuit. Our results provide a higher-dimensional generalization of prior work by Matsumoto and Amano who introduced similar canonical forms for single-qubit Clifford+T circuits. Finally, we show that a similar extension of these normal forms to higher dimensions exists, but do not establish uniqueness. Moving to multi-qubit circuits, I provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/√2, i]. We focus on the subrings Z[1/2], Z[1/√2], Z[1/√−2], and Z[1/2, i], and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X, CX, CCX} with an analogue of the Hadamard gate and an optional phase gate. I then establish the existence and uniqueness of a normal form for one of these gate sets, the two-qubit gate set of Clifford+Controlled Phase gate CS. This normal form is optimal in the number of CS gates, making it the first normal form that is non-Clifford optimal for a fault tolerant universal multi-qubit gate set. We provide a synthesis algorithm that runs in a time linear in the gate count and outputs the equivalent normal form. In proving the existence and uniqueness of the normal form, we likewise establish the generators and relations for the two-qubit Clifford+CS group. Finally, we demonstrate that a lower bound of 5 log2 (1/ε) + O(1) CS gates are required to ε-approximate any 4 × 4 unitary matrix. Lastly, using the characterization of circuits over the Clifford+CS gate set and the existence of an optimal normal form, I provide an ancilla-free inexact synthesis algorithm for two-qubit unitaries using the Clifford+SC gate set for Pauli-rotations. These operators require 6 log2 (1/ε) + O(1) CS gates to synthesize in the typical case and 8 log2 (1/ε) + O(1) in the worst case.Item STUDIES IN NONEQUILIBRIUM QUANTUM THERMODYNAMICS(2019) Smith, Andrew Maven; Jarzynski, Christopher; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The first part of this thesis focuses on verifying the quantum nonequilibrium work relation in the presence of decoherence. The nonequilibrium work relation is a generalization of the second law of thermodynamics that links nonequilibrium work measurements to equilibrium free energies via an equality. Despite being well established for classical systems, a quantum work relation is conceptually difficult to construct for systems that interact with their environment. We argue that for a quantum system which undergoes decoherence but not dissipation, these conceptual difficulties do not arise and the work relation can be proven similarly to the case of an isolated system. This result is accompanied by an experimental demonstration using trapped ions. The second part of this thesis examines the relationship between quantum work and coherence by constructing analogous quantities in classical physics. It has recently been shown that quantum coherence can function as a resource for work extraction. Furthermore, it has been suggested that this property could be a truly quantum aspect of thermodynamics with no classical analog. We examine this assertion within the framework of classical Hamiltonian mechanics and canonical quantization. For classical states we define a so called non-uniformity measure and show that it is a resource for work extraction similar to quantum coherence. Additionally, we show that work extracted from non-uniformity and coherence agree in the classical limit. This calls into question the idea that coherence qualitatively separates classical and quantum thermodynamics. The final part of this thesis explores the connection between decoherence and adiabatic (quasistatic) driving. This topic is inspired by an experiment where it was seen that strong dephasing suppressed energy level transitions. Using a perturbative method we investigate this mechanism in the regime of small to moderate decoherence rate and ask if decoherence can help suppress energy transitions when compared with an adiabatic process without decoherence. We find that strategies that include decoherence are inferior to those where decoherence is absent.Item Topological dispersion relations in spin-orbit coupled Bose gases(2019) Valdes Curiel, Ana; Spielman, Ian; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Quantum degenerate gases have proven to be an ideal platform for the simulation of complex quantum systems. Due to their high level of control it is possible to readily design and implement systems with effective Hamiltonians in the laboratory. This thesis presents new tools for the characterization and control of engineered quantum systems and describes their application in the realization of a topological system with Rashba-type spin-orbit coupling. The underlying properties of these engineered systems depend on their underlying energies. I describe a Fourier transform spectroscopy technique for characterizing the single particle spectrum of a quantum system. We tested Fourier spectroscopy by measuring the dispersion relation of a spin-1 spin-orbit coupled Bose-Einstein condensate (BEC) and found good agreement with our predictions. Decoherence due to uncontrolled fluctuations of the environment presents fundamental obstacles in quantum science. I describe an implementation of continuous dynamical decoupling (CDD) in a spin-1 BEC. We applied a strong radio-frequency (RF) magnetic field to the ground state hyperfine manifold of Rubidium-87 atoms, generating a dynamically protected dressed system that was first-order insensitive to changes in magnetic field. The CDD states constitute effective clock states and we observed a reduction in sensitivity to magnetic field of up to four orders of magnitude. We additionally show that the CDD states can be coupled in a fully connected geometry and thus enable the implementation of new models not possible using the bare atomic states. Finally, I describe a new realization of Rashba-type SOC using Raman coupled CDD states. Our system had non-trivial topology but no underlying crystalline structure that yields integer valued Chern numbers in conventional materials. We validated our procedure using Fourier transform spectroscopy to measure the full dispersion relation containing only a single Dirac point. We measured the quantum geometry underlying the dispersion relation and obtained the topological index using matter-wave interferometry. In contrast to crystalline materials, where topological indices take on integer values, our continuum system reveals an unconventional half-integer Chern number.Item QUANTUM ALGORITHMS FOR DIFFERENTIAL EQUATIONS(2019) Ostrander, Aaron Jacob; Childs, Andrew; Monroe, Chris; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs). Product formulas are used to simulate Hamiltonians which can be expressed as a sum of terms which can each be simulated individually. By simulating each of these terms in sequence, the net effect approximately simulates the total Hamiltonian. We find that the error of product formulas can be improved by randomizing over the order in which the Hamiltonian terms are simulated. We prove that this approach is asymptotically better than ordinary product formulas and present numerical comparisons for small numbers of qubits. The ODE algorithm applies to the initial value problem for time-independent first order linear ODEs. We approximate the propagator of the ODE by a truncated Taylor series, and we encode the initial value problem in a large linear system. We solve this linear system with a quantum linear system algorithm (QLSA) whose output we perform a post-selective measurement on. The resulting state encodes the solution to the initial value problem. We prove that our algorithm is asymptotically optimal with respect to several system parameters. The PDE algorithms apply the finite difference method (FDM) to Poisson's equation, the wave equation, and the Klein-Gordon equation. We use high order FDM approximations of the Laplacian operator to develop linear systems for Poisson's equation in cubic volumes under periodic, Neumann, and Dirichlet boundary conditions. Using QLSAs, we output states encoding solutions to Poisson's equation. We prove that our algorithm is exponentially faster with respect to the spatial dimension than analogous classical algorithms. We also consider how high order Laplacian approximations can be used for simulating the wave and Klein-Gordon equations. We consider under what conditions it suffices to use Hamiltonian simulation for time evolution, and we propose an algorithm for these cases that uses QLSAs for state preparation and post-processing.Item Nonequilibrium Dynamics in Open Quantum Systems(2019) Young, Jeremy; Rolston, Steven L; Gorshkov, Alexey V; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Due to the variety of tools available to control atomic, molecular, and optical (AMO) systems, they provide a versatile platform for studying many-body physics, quantum simulation, and quantum computation. Although extensive efforts are employed to reduce coupling between the system and the environment, the effects of the environment can never fully be avoided, so it is important to develop a comprehensive understanding of open quantum systems. The system-environment coupling often leads to loss via dissipation, which can be countered by a coherent drive. Open quantum systems subject to dissipation and drive are known as driven-dissipative systems, and they provide an excellent platform for studying many-body nonequilibrium physics. The first part of this dissertation will focus on driven-dissipative phase transitions. Despite the nonequilibrium nature of these systems, the corresponding phase transitions tend to exhibit emergent equilibrium behavior. However, we will show that in the vicinity of a multicritical point where multiple phase transitions intersect, genuinely nonequilibrium criticality can emerge, even though the individual phase transitions on their own exhibit equilibrium criticality. These nonequilibrium multicritical points can exhibit a variety of exotic phenomena not possible for their equilibrium counterparts, including the emergence of complex critical exponents, which lead to discrete scale invariance and spiraling phase boundaries. Furthermore, the Liouvillian gap can take on complex values, and the fluctuation-dissipation theorem is violated, corresponding to an effective temperature which gets "hotter" and "hotter" at longer and longer wavelengths. The second part of this dissertation will focus on Rydberg atoms. In particular, we study how the spontaneous generation of contaminant Rydberg states drastically modifies the behavior of a driven-dissipative Rydberg system due to the resultant dipole-dipole interactions. These interactions lead to a complicated competition of both blockade and anti-blockade effects, leading to strongly enhanced Rydberg populations for far-detuned drive and reduced Rydberg populations for resonant drive.Item Many-Body Dephasing in a Cryogenic Trapped Ion Quantum Simulator(2019) Kaplan, Harvey B.; Monroe, Christopher R; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)While realizing a fully functional quantum computer presents a long term technical goal, in the present, there are small to mid-sized quantum simulators (up to $\sim 100$ qubits), that are capable of approaching specialized problems. The quantum simulator discussed here uses trapped ions to act as qubits and is housed in a cryogenically cooled vacuum chamber in order to reduce the background pressure, thereby increasing ion chain length and life-time. The details of performance and characterization of this cryogenic apparatus are discussed, and this system is used to study many-body dephasing in a finite-sized quantum spin system. How a closed quantum many-body system relaxes and dephases as a function of time is important to understand when dealing with many-body spin systems. In this work, the first experimental observation of persistent temporal fluctuations after a quantum quench is presented with a tunable long-range interacting transverse-field Ising Hamiltonian. The fluctuations in the average magnetization of a finite-size system of spin-$1/2$ particles are measured presenting a direct measurement of relaxation dynamics in a non-integrable system. This experiment is in the regime where the properties of the system are closely related to the integrable Hamiltonian with global coupling. The system size is varied in order to investigate the dependence on finite-size scaling, and the system size scaling exponent extracted from the measured fluctuations is consistent with theoretical prediction.Item BOSE EINSTEIN CONDENSATES IN DYNAMICALLY CONTROLLED OPTICAL LATTICES(2019) Smith, Zachary Schutz; Rolston, Steven L; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Ultracold atomic gases are often used as quantum simulators, where the ability to precisely control and modulate the potential landscape allows many model Hamiltonians to be realized. Recently, dynamic control over these potentials has been leveraged to extend the kinds of systems that can be explored. Our lab has constructed an optical lattice generator capable of dynamically altering the spacing, phase, and amplitude of an optical lattice at RF timescales. The rapid timescales allows the construction of a time-averaged disordered potential from individual flashes of optical lattice, producing layered system exhibiting a Griffiths phase. A future experiment where the optical lattice is periodically expanded and contracted is proposed, and a numerical treatment is presented suggesting interesting structure in the tunneling dynamics.Item ATOMICALLY PRECISE FABRICATION AND CHARACTERIZATION OF DONOR-BASED QUANTUM DEVICES IN SILICON(2019) Wang, Xiqiao; Silver, Richard M; Appelbaum, Ian; Chemical Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Atomically precise donor-based quantum devices in silicon are a promising candidate for scalable solid-state quantum computing and analog quantum simulation. This thesis demonstrates success in fabricating state-of-the-art silicon-phosphorus (Si:P) quantum devices with atomic precision. We present critical advances towards fabricating high-fidelity qubit circuitry for scalable quantum information processing that demands unprecedented precision and reproducibility to control and characterize precisely placed donors, electrodes, and the quantum interactions between them. We present an optimized atomically precise fabrication scheme with improved process control strategies to encapsulate scanning tunneling microscope (STM)-patterned devices and technological advancements in device registration and electrical contact formation that drastically increase the yield of atomic-precision fabrication. We present an atomic-scale characterization of monolayer step edges on Si (100) surfaces using spatially resolved scanning tunneling spectroscopy and quantitatively determine the impact of step edge density of states on the local electrostatic environment. Utilizing local band bending corrections, we report a significant band gap narrowing behavior along rebonded SB step edges on a degenerately boron-doped Si substrate. We quantify and control atomic-scale dopant movement and electrical activation in silicon phosphorus (Si:P) monolayers using room-temperature grown locking layers (LL), sputter profiling simulation, and magnetotransport measurements. We explore the impact of LL growth conditions on dopant confinement and show that the dopant segregation length can be suppressed below one Si lattice constant while maintaining good epitaxy. We demonstrate weak-localization measurement as a high-resolution, high-throughput, and non-destructive method in determining the conducting layer thickness in the sub-nanometer thickness regime. Finally, we present atomic-scale control of tunnel coupling using STM-patterned Si:P single electron transistors (SET). We demonstrate the exponential scaling of tunnel coupling down to the atomic limit by utilizing the Si (100) 2×1 surface reconstruction lattice as a natural ruler with atomic-accuracy and varying the number of lattices counts in the tunnel gaps. We analyze resonant tunneling spectroscopy through atomically precise tunnel gaps as we scale the SET islands down to the few-donor quantum dot regime. Finally, by combining single/few-donor quantum dots with atomically defined single electron transistors as charge sensors, we demonstrate single electron charge sensing in few-donor quantum dots and characterize the tunnel coupling between few-donor quantum dots and precision-aligned single electron charge sensors.Item Measuring topology of BECs in a synthetic dimensions lattice(2019) Genkina, Dina; Spielman, Ian B.; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We describe several experiments performed on a two species apparatus capable of producing Bose-Einstein condensates (BECs) of rubidium 87 and degenerate Fermi gases (DFGs) of potassium 40. We first describe computational results for observed optical depths with absorption imaging, in a regime where imaging times are long enough that recoil-induced detuning introduces significant corrections. We report that the obseved optical depth depends negligibly on the cloud shape. We also find that the signal-to-noise(SNR) ratio for low atom numbers can be significantly improved by entering this regime and applying the appropriate corrections. We take advantage of this SNR improvement in our subsequent experiment colliding two clouds of potassium 40 for different values of background magnetic field in the vicinity of a Feshbach resonance. We directly imaged the fraction of scattered atoms, which was low and difficult to detect. We used this method to measure the resonance location to be B_0 = 202.06(15) Gauss with a width of 10.(5) Gauss, in good agreement with accepted values. Next, we describe experiments creating an elongated effectively 2D lattice for a BEC of rubidium 87 with non-trivial topological structure using the technique of synthetic dimensions. We set up the lattice by applying a 1D optical lattice to the atoms along one direction, and treating the internal spin states of the atoms as lattice sites in the other direction. This synthetic direction is therefore very short, creating a strip geometry. We then induce tunneling along the synthetic direction via Raman coupling, adding a phase term to the tunneling coefficient. This creates an effective magnetic flux through each lattice plaquette, in the Hofstadter regime, where the flux is of order the flux quantum $h/e$. We detect the resulting eigenstate structure, and observe chiral currents when atom are loaded into the central synthetic site. We further launch analogues of edge magnetoplasmons and image the resulting skipping orbits along each edge of the strip. We then applied a force along the real dimension of the 2D lattice and directly imaged the resulting motion in the transverse, synthetic, direction. We performed these measurements with 3 and 5-site width lattices along the synthetic direction. We used these measurements to identify the value of the Chern number, the topological invariant in 2D, by leveraging the Diophantine equation derived by Thouless, Kohomoto, Nightingale, and den Nijs. We measure Chern numbers with typical uncertainty of 5%, and show that although band topology is only properly defined in infinite systems, its signatures are striking even in extremely narrow systems.Item PHASE MEASUREMENTS WITH A TWO-MODE SQUEEZED STATE OF LIGHT.(2019) Gupta, Prasoon; Lett, Paul D.; Rolston, Steven; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Introducing squeezed states of light into interferometers can increase the phase sensitivity of the device beyond the standard quantum limit (SQL). We discuss an SU(1,1) interferometer, where nonlinear optical elements replace the beam splitters in a Mach-Zehnder interferometer. A two-mode squeezed state of light is generated inside of such an interferometer. We talk about the phase sensitivities of an SU(1,1) interferometer with different detection schemes and their improvement over the SQL. We also discuss the concept of an optimal detection scheme for phase measurement in an interferometer. We describe a modification of the SU(1,1) interferometer which reduces the experimental complexities while giving the same phase sensitivity. We call the design a truncated SU(1,1) interferometer. We show our experimental results of 4 dB improvement in phase sensitivity over the SQL using a truncated SU(1,1) interferometer. We also compare the theoretical sensitivities of vacuum-seeded configurations of a conventional and a truncated SU(1,1) interferometer, and show our experimental results for the phase sensitivity of the truncated version. We explain the dependence of phase sensitivity on the measurement of squeezing. We talk about the methods to improve the measurement of squeezing in a 4-wave mixing experiment, and our efforts in implementing them. Finally, we mention our progress in measuring a big phase shift using an adaptive algorithm in the truncated SU(1,1) interferometer, and discuss the technicalities involved in big phase measurements.