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 acquisition of adjunct control: grammar and processing

This dissertation uses children’s acquisition of adjunct control as a case study to investigate grammatical and performance accounts of language acquisition. In previous research, children have consistently exhibited non-adultlike behavior for sentences with adjunct control. To explain children’s behavior, several different grammatical accounts have been proposed, but evidence for these accounts has been inconclusive. In this dissertation, I take two approaches to account for children’s errors. First, I spell out the predictions of previous grammatical accounts, and test these predictions after accounting for some methodological concerns that might have influenced children’s behavior in previous studies. While I reproduce the non-adultlike behavior observed in previous studies, the predictions of previous grammatical accounts are not borne out, suggesting that extragrammatical factors are needed to explain children’s behavior. Next, I consider the role of two different types of extragrammatical factors in predicting children’s non-adultlike behavior. With a new task designed to address the task demands in previous studies, children exhibit significantly higher accuracy than with previous tasks. This suggests that children’s behavior has been influenced by task- specific processing factors. In addition to the task, I also test the predictions of a similarity-based interference account, which links children’s errors to the same memory mechanisms involved in sentence processing difficulties observed in adults. These predictions are borne out, supporting a more continuous developmental trajectory as children’s processing mechanisms become more resistant to interference. Finally, I consider how children’s errors might influence their acquisition of adjunct control, given the distribution in the linguistic input. I discuss the results of a corpus analysis, including the possibility that adjunct control could be learned from the input. The kinds of information that could be useful to a learner become much more limited, however, after considering the processing limitations that would interfere with the representations available to the learner.