Technical Report: Using Loop Scopes with for-Loops

Loop scopes have been shown to be a helpful tool in creating sound loop invariant rules which do not require program transformation of the loop body. Here we extend this idea from while-loops to for-loops and also present sound loop unrolling rules for while- and for-loops, which require neither program transformation of the loop body, nor the use of nested modalities. This approach allows for-loops to be treated as first-class citizens -- rather than the usual approach of transforming for-loops into while-loops -- which makes semi-automated proofs easier to follow for the user, who may need to provide help in order to finish the proof

Development of a Unique Whole-Brain Model for Upper Extremity Neuroprosthetic Control

Neuroprostheses are at the forefront of upper extremity function restoration. However, contemporary controllers of these neuroprostheses do not adequately address the natural brain strategies related to planning, execution and mediation of upper extremity movements. These lead to restrictions in providing complete and lasting restoration of function. This dissertation develops a novel whole-brain model of neuronal activation with the goal of providing a robust platform for an improved upper extremity neuroprosthetic controller. Experiments (N=36 total) used goal-oriented upper extremity movements with real-world objects in an MRI scanner while measuring brain activation during functional magnetic resonance imaging (fMRI). The resulting data was used to understand neuromotor strategies using brain anatomical and temporal activation patterns. The study\u27s fMRI paradigm is unique and the use of goal-oriented movements and real-world objects are crucial to providing accurate information about motor task strategy and cortical representation of reaching and grasping. Results are used to develop a novel whole-brain model using a machine learning algorithm. When tested on human subject data, it was determined that the model was able to accurately distinguish functional motor tasks with no prior knowledge. The proof of concept model created in this work should lead to improved prostheses for the treatment of chronic upper extremity physical dysfunction

Simulating Quantum Dynamics On A Quantum Computer

We present efficient quantum algorithms for simulating time-dependent Hamiltonian evolution of general input states using an oracular model of a quantum computer. Our algorithms use either constant or adaptively chosen time steps and are significant because they are the first to have time-complexities that are comparable to the best known methods for simulating time-independent Hamiltonian evolution, given appropriate smoothness criteria on the Hamiltonian are satisfied. We provide a thorough cost analysis of these algorithms that considers discretizion errors in both the time and the representation of the Hamiltonian. In addition, we provide the first upper bounds for the error in Lie-Trotter-Suzuki approximations to unitary evolution operators, that use adaptively chosen time steps.Comment: Paper modified from previous version to enhance clarity. Comments are welcom

Higher Order Decompositions of Ordered Operator Exponentials

We present a decomposition scheme based on Lie-Trotter-Suzuki product formulae to represent an ordered operator exponential as a product of ordinary operator exponentials. We provide a rigorous proof that does not use a time-displacement superoperator, and can be applied to non-analytic functions. Our proof provides explicit bounds on the error and includes cases where the functions are not infinitely differentiable. We show that Lie-Trotter-Suzuki product formulae can still be used for functions that are not infinitely differentiable, but that arbitrary order scaling may not be achieved.Comment: 16 pages, 1 figur

Exponential quantum speedup in simulating coupled classical oscillators

We present a quantum algorithm for simulating the classical dynamics of $2^n$ coupled oscillators (e.g., $2^n$ masses coupled by springs). Our approach leverages a mapping between the Schr\"odinger equation and Newton's equation for harmonic potentials such that the amplitudes of the evolved quantum state encode the momenta and displacements of the classical oscillators. When individual masses and spring constants can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in $n$, almost linear in the evolution time, and sublinear in the sparsity. As an example application, we apply our quantum algorithm to efficiently estimate the kinetic energy of an oscillator at any time. We show that any classical algorithm solving this same problem is inefficient and must make $2^{\Omega(n)}$ queries to the oracle and, when the oracles are instantiated by efficient quantum circuits, the problem is BQP-complete. Thus, our approach solves a potentially practical application with an exponential speedup over classical computers. Finally, we show that under similar conditions our approach can efficiently simulate more general classical harmonic systems with $2^n$ modes.Comment: 43 pages, 4 figures. v3 changes include improved presentation, discussion of applications related to potential energies, and new appendix discussing relation to prior wor