Performance and Robustness Analysis of Co-Prime and Nested Sampling

Coprime and nested sampling are well known deterministic sampling techniques that operate at rates significantly lower than the Nyquist rate, and yet allow perfect reconstruction of the spectra of wide sense stationary signals. However, theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling, and ability to perfectly compute statistical expectations. This thesis studies the performance of coprime and nested samplers in spatial and temporal domains, when these assumptions are violated. In spatial domain, the robustness of these samplers is studied by considering arrays with perturbed sensor locations (with unknown perturbations). Simplified expressions for the Fisher Information matrix for perturbed coprime and nested arrays are derived, which explicitly highlight the role of co-array. It is shown that even in presence of perturbations, it is possible to resolve $O(M^2)$ under appropriate conditions on the size of the grid. The assumption of small perturbations leads to a novel ``bi-affine" model in terms of source powers and perturbations. The redundancies in the co-array are then exploited to eliminate the nuisance perturbation variable, and reduce the bi-affine problem to a linear underdetermined (sparse) problem in source powers. This thesis also studies the robustness of coprime sampling to finite number of samples and sampling jitter, by analyzing their effects on the quality of the estimated autocorrelation sequence. A variety of bounds on the error introduced by such non ideal sampling schemes are computed by considering a statistical model for the perturbation. They indicate that coprime sampling leads to stable estimation of the autocorrelation sequence, in presence of small perturbations. Under appropriate assumptions on the distribution of WSS signals, sharp bounds on the estimation error are established which indicate that the error decays exponentially with the number of samples. The theoretical claims are supported by extensive numerical experiments.

Quantum Gate and Quantum State Preparation through Neighboring Optimal Control

Successful implementation of fault-tolerant quantum computation on a system of qubits places severe demands on the hardware used to control the many-qubit state. It is known that an accuracy threshold Pa exists for any quantum gate that is to be used for such a computation to be able to continue for an unlimited number of steps. Specifically, the error probability Pe for such a gate must fall below the accuracy threshold: Pe < Pa. Estimates of Pa vary widely, though Pa ∼ 10−4 has emerged as a challenging target for hardware designers. I present a theoretical framework based on neighboring optimal control that takes as input a good quantum gate and returns a new gate with better performance. I illustrate this approach by applying it to a universal set of quantum gates produced using non-adiabatic rapid passage. Performance improvements are substantial comparing to the original (unimproved) gates, both for ideal and non-ideal controls. Under suitable conditions detailed below, all gate error probabilities fall by 1 to 4 orders of magnitude below the target threshold of 10−4. After applying the neighboring optimal control theory to improve the performance of quantum gates in a universal set, I further apply the general control theory in a two-step procedure for fault-tolerant logical state preparation, and I illustrate this procedure by preparing a logical Bell state fault-tolerantly. The two-step preparation procedure is as follow: Step 1 provides a one-shot procedure using neighboring optimal control theory to prepare a physical qubit state which is a high-fidelity approximation to the Bell state |β01⟩ = 1/√2(|01⟩ + |10⟩). I show that for ideal (non-ideal) control, an approximate |β01⟩ state could be prepared with error probability ϵ ∼ 10−6 (10−5) with one-shot local operations. Step 2 then takes a block of p pairs of physical qubits, each prepared in |β01⟩ state using Step 1, and fault-tolerantly prepares the logical Bell state for the C4 quantum error detection code.

THE PRICE SENSITIVITY OF DEMAND FOR HIGHER EDUCATION AMONG NON-TRADITIONAL STUDENTS

This dissertation examines the price sensitivity of demand for higher education among non-traditional students in the United States. Chapter 1 discusses the issues related to the demand for higher education. It presents the recent trends and reviews the literature addressing these issues. A major conclusion that emerges from this chapter is that the price sensitivity of demand for higher education appears to depend on the source of the variation in price and the characteristics of the students who face the price change. The baseline estimate for the price sensitivity of demand is that a $1,000 (in year 2000 dollars) decrease in tuition costs should result in a 4 percentage-point increase in enrollment for the traditional 18- to 24-year-old student. Chapter 2 examines the price sensitivity of demand for higher education for military spouses resulting from variation in tuition due to military-mandated moves across states. The data suggest that a $1,000 (in year 2000 dollars) decrease in the cost of 2-year schools is associated with a 1--1.5 percentage-point increase in the probability of attending college. This estimate is less than half the previous estimates due to in-state tuition price differences faced by the civilian 18- to 24-year-old population on a percentage-point basis. However, this represents a 7--10 percent increase for this population, and the magnitude of this metric is in line with previous estimates. This suggests tuition assistance can be an effective means of increasing enrollment for military spouses, but other barriers to education for this population may also need to be addressed. Chapter 3 examines the impact of a change in the tax treatment of savings set aside for higher education by those who decide to suspend their education and enter the workforce. The taxation of these funds appears to have increased the rate at which these funds are included in an employee's initial contract and the quantity of funds allocated. These results are counterintuitive if the tax preference was the primary reason for the savings plan. However, these results suggest the rationale for the savings plan was to offer targeted additional compensation to recruits with greater negotiating power. Taxation of funds previously set aside did not appear to have a statistically significant impact on their utilization. Point estimates of the price sensitivity of demand from changes in the out-of-pocket costs for higher education induced by the taxation of these funds were small and often not statistically significant. The results from this dissertation show responses to changes in the net cost of college that differ by the source of price variation and the population experiencing them. This is consistent with the previous literature. This dissertation contributes to the literature by providing estimates for the price sensitivity of demand for higher education to previously understudied non-traditional students.