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Aspects of Fault-Tolerant Quantum Computation
Get PDFThis thesis is concerned with fault-tolerant quantum information processing using quantum error-correcting codes. It contains two major pieces of work. The first is a study of coherent noise in the context of stabilizer error-correcting codes. The second is a proposed scheme for a universal set of fault-tolerant logical gates in a particular code family built out of the 3D toric code.
Chapter 1 provides an introduction to quantum computation and fault tolerance. Many basic concepts in error-correcting codes are defined. Special attention is paid to the set of code properties that are most likely to determine how easily a given fault-tolerant scheme might be implemented on a physical device. These include the fault-tolerant noise threshold and the overhead.
In Chapters 2 and 3 we study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In Chapter 2 we introduce coherent noise and incoherent noise and a number of methods that are useful for the study of coherent noise. We study the repetition code as a basic example, and we also study a correlated noise model. In Chapter 3 we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.
Chapter 4 is connected to another aspect of fault tolerance, fault-tolerant logical gates. The toric code is a promising candidate for fault-tolerant quantum computation because of its high threshold and low-weight stabilizers. A universal gate set in the toric code generally requires magic state distillation, which can incur a significant qubit overhead. In this work we construct an error-correcting code in three dimensions based on the toric code that features a fault-tolerant T gate with no magic state distillation required. We further describe a subsystem version of our code which supports a universal set of fault-tolerant gates. This code can be converted into the stabilizer version using gauge-fixing. We also argue that our code can be converted to a (2+1)-D protocol, where a 2D lattice undergoes a measurement-based protocol over time. In this way, a fault-tolerant logical T gate can be realized in a 2D toric code structure.</p
Pezo-1 Function in Caenorhabditis elegans
Get PDFThe piezo class of mechanosensative ion channels is a recently discovered class of cation channels with orthologs found in every phylogenetic clade aside from yeast and bacteria. THey are large channels, both in gene length and in overall diameter, with the diameter of the human PIEZO2 protein measuring in at 280 Å in its full homotrimeric form. In addition, like many similar mechanosensitive channels, such as the DEG/ENaC channels, TRP channels, and TREK/TRAAK channels, they have been linked to a number of different functions within drosophila, zebrafish, and mice, including light touch, nociception, blood cell volume regulation, vascular development, and neuropathic pain. Structurally, this channel is intriguing as it possesses no previously categorized structural motifs and is organized into a central pore with a cap, surrounded by three "propeller blade" regions that are theorized to anchor the channel to the membrane and control gating through hydrophobic mismatch based on membrane curvature.
The C. elegans piezo, pezo-1, has not yet been fully characterized, even though a crystal structure of part of this particular piezo was used to assist in the resolution of the first set of cryo-EM images. Here, I generated a number of GFP transcriptional fusions of non-coding potential promoter regions to track the expression of the pezo-1 gene in C. elegans, using these expression patterns to design further experiments. From these expression patterns, I identified expression in a number of neurons of the C. elegans male tail, the primary mating apparatus of the male, and identified these neurons as key neurons as relating to mating. In particular, I identified neurons HOB, PCB, PCC and various ray neurons as potential candidates, which are ciliated neurons theorized to have mechanosensitive properties. In addition, I also identified expression in the vulva muscle and spermatheca of the hermaphrodite, both theorized to be involved in ovulation and egg-laying processes.
From there, I designed CRISPR/Cas9 mutants with defects in pezo-1 in order to investigate the potential link between the pezo-1 expression in those neurons and mating behavior via a mating assay. Similarly, I devised a fecundity assay to investigate the link between pezo-1 expression in ovulation organs and progeny survival. I have discovered that pezo-1 has function in both of these areas, with pezo-1 mutant males demonstrating discrete mating defects that correlate with the expression pattern seen from the GFP transcriptional fusion mutants and with pezo-1 mutant hermaphrodites having much smaller brood sizes than wildtype hermaphrodites. However, I have also discovered that the processes this mechanotransducer is involved in are also more complex than I originally believed, as I discovered that pezo-1 appears to interact with another mechanotransducer, trp-4, illuminating some potentially novel pathway considerations for how these channels overlap in function.</p
Applied Safety Critical Control
Get PDFThere is currently a clear gap between control-theoretical results and the reality of robotic implementation, in the sense that it is very difficult to transfer analytical guarantees to practical ones. This is especially problematic when trying to design safety-critical systems where failure is not an option. While there is a vast body of work on safety and reliability in control theory, very little of it is actually used in practice where safety margins are typically empiric and/or heuristic. Nevertheless, it is still widely accepted that a solution to these problems can only emerge from rigorous analysis, mathematics, and methods. In this work, we therefore seek to help bridge this gap by revisiting and expanding existing theoretical results in light of the complexity of hardware implementation.
To that end, we begin by making a clear theoretical distinction between systems and models, and outline how the two need to be related for guarantees to transfer from the latter to the former. We then formalize various imperfections of reality that need to be accounted for at a model level to provide theoretical results with better applicability. We then discuss the reality of digital controller implementation and present the mathematical constraints that theoretical control laws must satisfy for them to be implementable on real hardware. In light of these discussions, we derive new realizable set-invariance conditions that, if properly enforced, can guarantee safety with an arbitrary high levels of confidence. We then discuss how these conditions can be rigorously enforced in a systematic and minimally invasive way through convex optimization-based Safety Filters. Multiple safety filter formulations are proposed with varying levels of complexity and applicability. To enable the use of these safety filters, a new algorithm is presented to compute appropriate control invariant sets and guarantee feasibility of the optimization problem defining these filters. The effectiveness of this approach is demonstrated in simulation on a nonlinear inverted pendulum and experimentally on a simple vehicle. The aptitude of the framework to handle a system's dynamics uncertainty is illustrated by varying the mass of the vehicle and showcasing when safety is conserved. Then, the aptitude of this approach to provide guarantees that account for controller implementation's constraints is illustrated by varying the frequency of the control loop and again showcasing when safety is conserved.
In the second part of this work, we revisit the safety filtering approach in a way that addresses the scalability issues of the first part of this work. There are two main approaches to safety-critical control. The first one relies on computation of control invariant sets and was presented in the first part of this work. The second approach draws from the topic of optimal control and relies on the ability to realize Model-Predictive-Controllers online to guarantee the safety of a system. In that online approach, safety is ensured at a planning stage by solving the control problem subject for some explicitly defined constraints on the state and control input. Both approaches have distinct advantages but also major drawbacks that hinder their practical effectiveness, namely scalability for the first one and computational complexity for the second one. We therefore present an approach that draws from the advantages of both approaches to deliver efficient and scalable methods of ensuring safety for nonlinear dynamical systems. In particular, we show that identifying a backup control law that stabilizes the system is in fact sufficient to exploit some of the set-invariance conditions presented in the first part of this work. Indeed, one only needs to be able to numerically integrate the closed-loop dynamics of the system over a finite horizon under this backup law to compute all the information necessary for evaluating the regulation map and enforcing safety.
The effect of relaxing the stabilization requirements of the backup law is also studied, and weaker but more practical safety guarantees are brought forward. We then explore the relationship between the optimality of the backup law and how conservative the resulting safety filter is. Finally, methods of selecting a safe input with varying levels of trade-off between conservativeness and computational complexity are proposed and illustrated on multiple robotic systems, namely: a two-wheeled inverted pendulum (Segway), an industrial manipulator, a quadrotor, and a lower body exoskeleton.</p
Additive Manufacturing of 3D Functional Materials: From Surface Chemistry to Combustion-Derived Materials
Get PDFOver the past decade, additive manufacturing has emerged as one of the most powerful manufacturing tools available today. Vat photopolymerization techniques, in particular, are especially promising as they are capable of achieving high resolutions and throughputs. However, the vast majority of materials that are compatible with them only have structural functionality. The fabrication of functional materials still remains a challenge in the field: functional polymers often require a complex multi-step synthesis. Ceramics-based photoresins are limited in composition and are challenging to use or synthesize. Metals have also been hardly explored with vat photopolymerization techniques.
This thesis explores methods of fabricating functional materials with vat photopolymerization. We develop accessible techniques for the fabrication of functional polymers, ceramics, metals, and multimaterials at a variety of length scales, from sub-micron to centimeter scales. On the polymer front, we first explore how surface coatings can be an accessible method of introducing chemical functionality to a material. In particular, we demonstrate the surface coating of genomic DNA on an architected polymeric structure and show how it can be used as a drug capture device to reduce off-target toxicity in chemotherapy. We also explore the use of click chemistry, the thiol-Michael reaction in particular, in the facile synthesis of acrylate monomers with a variety of functional groups. We demonstrate the compatibility of these functionalized monomers with two-photon lithography and highlight some potential applications of these functional polymers structures.
In the fabrication of ceramics and metals, we present a novel technique called photopolymer complex synthesis that combines solution combustion synthesis with vat photopolymerization to enable their fabrication. We illustrate the use of this technique by first fabricating piezoelectric zinc oxide architected structures with sub-micron features using two-photon lithography. Following that, we fabricate lithium cobalt oxide structures using digital light processing printing and highlight their use as architected lithium-ion battery cathodes. Lastly, we show how photopolymer complex synthesis can be expanded to fabricate metal and multimaterial architected structures. Our work highlights the use of polymer chemistry and materials science in expanding the range of materials that are compatible with vat photopolymerization, with the vision of democratizing the fabrication of advanced functional materials and enabling the production of previously impossible 3D devices.</p
The Integral Coefficient Geometric Satake Equivalence in Mixed Characteristic and its Arithmetic Applications
Get PDFThe first main result of this thesis is the proof of the integral coefficient geometric Satake equivalence in mixed characteristic setting. Our proof can be divided into three parts: the construction of the monoidal structure of the hypercohomology functor on the category of integral coefficient equivariant perverse sheaves on the mixed characteristic affine Grassmannian; a generalized Tannakian formalism; and, the identification of group schemes. In particular, our proof does not employ Scholze’s theory of diamonds.
We derive a geometric construction of the Jacquet-Langlands transfer for weighted automorphic forms as an application of the geometric Satake equivalence in the above setting. Our strategy follows the recent work of Xiao-Zhu [XZ17]. We relate the geometry and (ℓ-adic) cohomology of the mod fibers of the canonical smooth integral models of different Hodge type Shimura varieties, and obtain a Jacquet-Langlands transfer for weighted automorphic forms.</p
Towards High Fidelity Quantum Computation and Simulation with Rydberg Atoms
Get PDFIndividually trapped neutral atoms are a promising candidate for use in quantum computing and simulation applications. They are highly scalable, have long coherence times and can be entangled via strong dipole-dipole interactions by driving to highly excited Rydberg states. However, the fidelity of single atom operations as well as two-atom entangling operations is limited by intrinsic sources of decoherence such as atomic motion, as well as technical sources of noise such as laser intensity fluctuations and phase/frequency fluctuations. We study the effect of these factors on single atom Rabi oscillations and two-atom Rydberg blockaded Rabi oscillations, using perturbation theory and numerical simulation. We develop a window function approach which helps us qualitatively understand the significance of the different spectral components of the noise as well as quantitatively understand the dependence of the Rabi oscillation fidelity on Rabi frequency. This allows us to predict the maximum experimentally achievable fidelities using independent measurements of experimental parameters such as noise spectra and atomic temperature. Turning to the question of near-term scalability of the experimental system, we prototype and test a method of generating a ’ladder’ configuration of optical tweezers utilizing two independent lasers. Our setup allows us to fully tune the geometry of the ladder, namely the separation between the two rows, the angle between them, and their relative position along the axis of the ladder. This pseudo-2D configuration enables us to reach larger system sizes in the near future and allows us to access beyond 1D physics
Energy Non-Conservation in Quantum Mechanics
Get PDFConservation of energy is an integral component of modern physics, but questions remain in quantum mechanics. In traditional quantum mechanics, superpositions of energy eigenstates collapse into a single energy eigenstate upon measurement, and any change in energy after measurement is thought to be lost/gained in the measurement process. However, we argue that energy non-conservation in quantum mechanics cannot be entirely accounted for by leakage to the apparatus/environment. We first present a comprehensive Hamiltonian where energy is not conserved in any considered interpretation according to an observer. Next, we present a protocol for developing experiments to observe this non-conservation, then we use our protocol to develop an example thought experiment. In both the comprehensive Hamiltonian and thought experiment examples, energy is not conserved to an observer in all considered interpretations, but it is conserved when considering the universe's global wave function in Everettian quantum mechanics. Finally, we discuss implications for the status of conservation of energy and other conservation laws. The work presented in this thesis will be adapted into an upcoming paper, Carroll and Lodman.</p
Vision for Social Robots: Human Perception and Pose Estimation
Get PDFIn order to extract the underlying meaning from a scene captured from the surrounding world in a single still image, social robots will need to learn the human ability to detect different objects, understand their arrangement and relationships relative both to their own parts and to each other, and infer the dynamics under which they are evolving. Furthermore, they will need to develop and hold a notion of context to allow assigning different meanings (semantics) to the same visual configuration (syntax) of a scene.
The underlying thread of this Thesis is the investigation of new ways for enabling interactions between social robots and humans, by advancing the visual perception capabilities of robots when they process images and videos in which humans are the main focus of attention.
First, we analyze the general problem of scene understanding, as social robots moving through the world need to be able to interpret scenes without having been assigned a specific preset goal. Throughout this line of research, i) we observe that human actions and interactions which can be visually discriminated from an image follow a very heavy-tailed distribution; ii) we develop an algorithm that can obtain a spatial understanding of a scene by only using cues arising from the effect of perspective on a picture of a person’s face; and iii) we define a novel taxonomy of errors for the task of estimating the 2D body pose of people in images to better explain the behavior of algorithms and highlight their underlying causes of error.
Second, we focus on the specific task of 3D human pose and motion estimation from monocular 2D images using weakly supervised training data, as accurately predicting human pose will open up the possibility of richer interactions between humans and social robots. We show that when 3D ground-truth data is only available in small quantities, or not at all, it is possible to leverage knowledge about the physical properties of the human body, along with additional constraints related to alternative types of supervisory signals, to learn models that can regress the full 3D pose of the human body and predict its motions from monocular 2D images.
Taken in its entirety, the intent of this Thesis is to highlight the importance of, and provide novel methodologies for, social robots' ability to interpret their surrounding environment, learn in a way that is robust to low data availability, and generalize previously observed behaviors to unknown situations in a similar way to humans.</p
Exact Bosonization in All Dimensions: the Duality Between Fermionic and Bosonic Phases of Matter
Get PDFWe describe an n-dimensional (n≥2) analog of the Jordan-Wigner transformation, which maps an arbitrary fermionic system to Pauli matrices while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization gives a duality between any fermionic system in arbitrary n spatial dimensions and a new class of (n-1)-form Z₂ gauge theories in n dimensions with a modified Gauss’s law. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model, and 3d bosonization, including a solvable Z₂ lattice gauge theory with Dirac cones in the spectrum. This bosonization formalism has an explicit dependence on the second Stiefel-Whitney class and a choice of spin structure on the manifold, a key feature for defining fermions. A new formula for Stiefel-Whitney homology classes on lattices is derived. We also derive the Euclidean actions for the corresponding lattice gauge theories from the bosonization. The topological actions contain Chern-Simons terms for (2+1)D or Steenrod Square terms for general dimensions. Finally, we apply the bosonization to construct various bosonic or fermionic symmetry-protectedtopological (SPT) phases. It has been shown that supercohomology fermionic SPT phases are dual to bosonic higher-group SPT phases