Physics
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Item Harnessing Quantum Systems for Sensing, Simulation, and Optimization(2024) Bringewatt, Jacob Allen; Gorshkov, Alexey V; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Quantum information science offers a remarkable promise: by thinking practically about how quantum systems can be put to work to solve computational and information processing tasks, we gain novel insights into the foundations of quantum theory and computer science. Or, conversely, by (re)considering the fundamental physical building blocks of computers and sensors, we enable new technologies, with major impacts for computational and experimental physics. In this dissertation, we explore these ideas through the lens of three different types of quantum hardware, each with a particular application primarily in mind: (1) networks of quantum sensors for measuring global properties of local field(s); (2) analog quantum computers for solving combinatorial optimization problems; and (3) digital quantum computers for simulating lattice (gauge) theories. For the setting of quantum sensor networks, we derive the fundamental performance limits for the sensing task of measuring global properties of local field(s) in a variety of physical settings (qubit sensors, Mach-Zehnder interferometers, quadrature displacements) and present explicit protocols that achieve these limits. In the process, we reveal the geometric structure of the fundamental bounds and the associated algebraic structure of the corresponding protocols. We also find limits on the resources (e.g. entanglement or number of control operations) required by such protocols. For analog quantum computers, we focus on the possible origins of quantum advantage for solving combinatorial optimization problems with an emphasis on investigating the power of adiabatic quantum computation with so-called stoquastic Hamiltonians. Such Hamiltonians do not exhibit a sign problem when classically simulated via quantum Monte Carlo algorithms, suggesting deep connections between the sign problem, the locality of interactions, and the origins of quantum advantage. We explore these connections in detail. Finally, for digital quantum computers, we consider the optimization of two tasks relevant for simulating lattice (gauge) theories. First, we investigate how to map fermionic systems to qubit systems in a hardware-aware manner that consequently enables an improved parallelization of Trotter-based time evolution algorithms on the qubitized Hamiltonian. Second, we investigate how to take advantage of known symmetries in lattice gauge theories to construct more efficient randomized measurement protocols for extracting purities and entanglement entropies from simulated states. We demonstrate how these protocols can be used to detect a phase transition between a trivial and a topologically ordered phase in $Z_2$ lattice gauge theory. Detecting this transition via these randomized methods would not otherwise be possible without relearning all symmetries.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.