Formal Methods in Quantum Circuit Design

The design and compilation of correct, efficient quantum circuits is integral to the future operation of quantum computers. This thesis makes contributions to the problems of optimizing and verifying quantum circuits, with an emphasis on the development of formal models for such purposes. We also present software implementations of these methods, which together form a full stack of tools for the design of optimized, formally verified quantum oracles. On the optimization side, we study methods for the optimization of Rz and CNOT gates in Clifford+Rz circuits. We develop a general, efficient optimization algorithm called phase folding, which reduces the number of Rz gates without increasing any metrics by computing its phase polynomial. This algorithm can further be combined with synthesis techniques for CNOT-dihedral operators to optimize circuits with respect to particular costs. We then study the optimal synthesis problem for CNOT-dihedral operators from the perspectives of Rz and CNOT gate optimization. In the case of Rz gate optimization, we show that the optimal synthesis problem is polynomial-time equivalent to minimum-distance decoding in certain Reed-Muller codes. For the CNOT optimization problem, we show that the optimal synthesis problem is at least as hard as a combinatorial problem related to Gray codes. In both cases, we develop heuristics for the optimal synthesis problem, which together with phase folding reduces T counts by 42% and CNOT counts by 22% across a suite of real-world benchmarks. From the perspective of formal verification, we make two contributions. The first is the development of a formal model of quantum circuits with ancillary bits based on the Feynman path integral, along with a concrete verification algorithm. The path integral model, with some syntactic sugar, further doubles as a natural specification language for quantum computations. Our experiments show some practical circuits with up to hundreds of qubits can be efficiently verified. Our second contribution is a formally verified, optimizing compiler for reversible circuits. The compiler compiles a classical, irreversible language to reversible circuits, with a formal, machine-checked proof of correctness written in the proof assistant F*. The compiler is structured as a partial evaluator, allowing verification to be carried out significantly faster than previous results

Quantum circuit analysis using analytic functions

In this thesis, classical computation is first introduced. Finite quantum systems are considered with D-dimensional Hilbert space, and position x and momentum p taking values in Z(D) (the integers modulo D). An analytic rep resentation of finite quantum systems that use Theta function is presented and considered. The first novel part of this thesis is contribution to study reversible classical CNOT gates and their binary inputs and outputs with reversible cir cuits. Furthermore, a reversible classical Toffoli gates are considered, as well as implementation of a Boolean expression with classical CNOT and Toffoli gates. Reversible circuits with classical CNOT and Toffoli gates are also considered. The second novel part of this thesis the study of quantum computation in terms of CNOT and Toffoli gates. Analytic representations and their zeros are considered, while zeros of the inputs and outputs for quantum CNOT and Toffoli gates are studied. Also, approximate computation of their zeros on the output are calculated. Finally, some quantum circuits are discussed.

Mechanical Computing in Microelectromechanical Systems (MEMS)

Mechanical computing devices in polysilicon-based microelectromechanical systems (MEMS) were designed with the goal of developing computing devices for harsh environments, such as those with high dose radiation and high temperatures, as well as devices that may be able to interface with molecular or biological computer systems. The devices that were designed include both analog and digital computing devices. The analog devices include integrators, differentials (summers), multipliers, and those that perform trigonometric functions. The digital devices that were designed are inverters, NAND, NOR, and XOR logic gates. Analog-to-digital (A-to-D) and digital-to-analog (D-to-A) converters were also designed. The designs were submitted to a commercial surface micromachining foundry to be fabricated. The completed MEMS devices were then released and tested to determine proper operation. Of the mechanical devices that have been fabricated and tested, a functioning inverter, sine function device, cosine function device, and digital-to-analog converter have been demonstrated

Development Of Molecular Biosensors And Multifunctional Graphene-Based Nanomaterials

In the first project, a simple, rapid, and reversible fluorescent DNA INHIBIT logic gate has been developed for sensing mercury (Hg2+) and iodide (I-) ions based on a molecular beacon (MB). In this logic gate, a mercury ion was introduced as the first input into the MB logic gate system to assist in the hybridization of the MB with an assistant DNA probe through the thymine–Hg2+–thymine interaction, which eventually restored the fluorescence of MB as the output. With this signal-on process, mercury ions can be detected with a limit of detection as low as 7.9 nM. Furthermore, when iodide ions were added to the Hg2+/MB system as the second input, the fluorescence intensity decreased because Hg2+ in the thymine–Hg2+–thymine complex was grabbed by I- due to a stronger binding force. Iodide ions can be detected with a limit of detection of 42 nM. Meanwhile, we studied the feasibility and basic performance of the DNA INHIBIT logic gate, optimized the logic gate conditions, and investigated its sensitivity and selectivity. The results showed that the MB based logic gate is highly selective and sensitive for the detection of Hg2+ and I- over other interfering cations and anions. In the second project, an ultrasensitive and rapid turn-on fluorescence assay has been developed for the detection of 3’-5’ exonuclease activity of exonuclease III (Exo III) using molecular beacons (MBs). This method has a linear detection range from 0.04 to 8.00 U mL-1 with a limit of detection of 0.01 U mL-1. In order to improve the selectivity of the method, a dual-MB system has been developed to distinguish between different exonucleases. With the introduction of two differently designed MBs which respond to different exonucleases, the T5 exonuclease, Exo III and RecJf exonucleases can be easily distinguished from each other. Furthermore, fetal bovine serum and fresh mouse serum were used as complex samples to investigate the feasibility of the dual-MB system for the detection of the enzymatic activity of Exo III. As a result, the dual-MB system showed a similar calibration curve for the detection of Exo III as in the ideal buffer solution. The designed MB probe could be a potential sensor for the detection of Exo III in biological samples. In the third project, A sensitive label-free fluorescence assay for monitoring T4 polynucleotide kinase (T4 PNK) activity and inhibition was developed based on a coupled λ exonuclease cleavage reaction and SYBR Green I. In this assay, a double-stranded DNA (dsDNA) was stained with SYBR Green I and used as a substrate for T4 PNK. After the 5´-hydroxyl termini of the dsDNA was phosphorylated by the T4 PNK, the coupled λ exonuclease began to digest the dsDNA to form mononucletides and single-stranded DNA (ssDNA). At this moment, the fluorescence intensity of the SYBR Green I decreased because less affinity with ssDNA than dsDNA. The decrease extent was proportional to the concentration of the T4 PNK. After optimization of the detection conditions, including the concentration of ATP, amount of λ exonuclease and reaction time, the activity of T4 PNK was monitored by the fluorescence measurement, with the limit of detection of 0.11 U/mL and good linear correlation between 0.25-1.00 U/mL (R2=0.9896). In this assay, no label was needed for the fluorescence detection. Moreover, the inhibition behaviours of the T4 PNK’s inhibitors were evaluated by this assay. The result indicated a potential of using this assay for monitoring of phosphorylation-related process. In the fourth project, a facile bottom-up method for the synthesis of highly fluorescent graphene quantum dots (GQDs) has been developed using a one-step pyrolysis of a natural amino acid, L-glutamic acid, with the assistance of a simple heating mantle device. The developed GQDs showed strong blue, green and red luminescence under irradiation with ultra-violet, blue and green light, respectively. Moreover, the GQDs emitted near-infrared (NIR) fluorescence in the range 800–850 nm with an excitation-dependent manner. This NIR fluorescence has a large Stokes shift of 455 nm, providing a significant advantage for the sensitive determination and imaging of biological targets. The fluorescence properties of the GQDs, such as the quantum yields, fluorescence life times, and photostability, were measured and the fluorescence quantum yield was as high as 54.5%. The morphology and composites of the GQDs were characterized using TEM, SEM, EDS, and FT-IR. The feasibility of using the GQDs as a fluorescent biomarker was investigated through in vitro and in vivo fluorescence imaging. The results showed that the GQDs could be a promising candidate for bioimaging. Most importantly, compared to the traditional quantum dots (QDs), the GQDs are chemically inert. Thus, the potential toxicity of the intrinsic heavy metal in the traditional QDs would not be a concern for GQDs. In addition, the GQDs possessed an intrinsic peroxidase-like catalytic activity that was similar to graphene sheets and carbon nanotubes. Coupled with 2,20-azino-bis(3-ethylbenzothiazoline-6-sulphonic acid) (ABTS), the GQDs can be used for the sensitive detection of hydrogen peroxide with a limit of detection of 20 mM. In the fifth project, a general, environmental-friendly, one-pot method for the fabrication of reduced graphene oxide (RGO)/metal (oxide) (e.g. RGO/Au, RGO/Cu2O, and RGO/Ag) nanocomposties was developed using glucose as the reducing agent and stabilizer. The RGO/metal (oxide) nanocomposites were characterized using STEM, FE-SEM, EDS, UV-vis absorption spectroscopy, XRD, FT-IR and Raman spectroscopy. The reducing agent, glucose, not only reduced GO effectively to RGO, but it also reduced the metal precursors to form metal (oxide) nanoparticles on the surface of RGO. Moreover, the RGO/metal (oxide) nanocomposites were stabilized by gluconic acid on the surface of RGO. Finally, the developed nanomaterials were successfully applied to simultaneous electrochemical analysis of L-ascorbic acid (L-AA), dopamine (DA) and uric acid sing RGO/Au nanocomposite as an electrode catalyst. In the sixth project, a reduced graphene oxide/silver nanoparticle (RGO/Ag) nanocomposite using glucose as the environmental-friendly reducing agent was developed. The antibacterial activity of RGO/Ag nanocomposite was carefully investigated using Escherichia coli (E. coli) and Klebsiella pneumoniae (Kp) as bacterial models. We found that, compared with AgNPs, graphene oxide (GO) and RGO, RGO/Ag nanocomposite had higher antibacterial efficiency. Furthermore, under the near-infrared (NIR) irradiation, RGO/Ag nanocomposite demonstrated enhanced synergetic antibacterial activity through the photothermal effect. Almost 100 % of E. coli and Kp were killed by the treatment of 15 µg/mL and 20 µg/mL, respectively, with NIR irradiation. Moreover, the membrane integrity assay and ROS species assay demonstrated that RGO/Ag nanocomposite under NIR irradiation caused the cell membranes disruption and generation of ROS species, providing other possible mechanisms for their high antibacterial activity besides photothermal effect. In the seventh project, a rigid distance spacer, silica shell, was used between GO and dyes in this work to elucidate the quenching ability of GO. First, an organic dye was doped in silica nanoparticles, followed by the modification of another layer of silica shell with a different thickness. Due to the electrostatic interaction between GO and positively charged silica nanoparticles, GO wrapped the silica nanoparticles when they were mixed together. Therefore, the distance between GO and organic dyes was adjusted by the thickness of the silica shell. The quenching efficiency of GO to two different organic dyes, including Tetramethylrhodamine (TAMRA) and Tris(bipyridine)ruthenium(II) chloride (Rubpy), was measured at various distances. This quenching ability investigation of GO to dyes with distance-dependent manner would provide a guideline for the design of the fluorescent functional composite using GO in the future. In the eighth project, we characterized the antibacterial activity of GO in both cell culture and animal models. Klebsiella pneumoniae (Kp) is one of the most common multidrug resistant (MDR) pathogens in causing persistent nosocomial infections and is very difficult to eradicate once established in the host. First, we demonstrated that GO exerted direct killing of Kp in agar dishes and afforded the protection of alveolar macrophages (AM) from Kp infection in culture. We then evaluated the mortality, tissue damage, polymorphonuclear neutrophil (PMN) penetration, and bacterial dissemination in Kp-infected mice. Our results revealed that GO can counteract the invasive ability of Kp in vivo, resulting in lessened tissue injury, significant but subdued inflammatory response, and prolonged mouse survival. These findings indicate that GO may be an alternative agent for controlling MDR pathogens in clinics

Quantum Algorithms for Scientific Computing and Approximate Optimization

Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations

Nucleic Acid Architectures for Therapeutics, Diagnostics, Devices and Materials

Nucleic acids (RNA and DNA) and their chemical analogs have been utilized as building materials due to their biocompatibility and programmability. RNA, which naturally possesses a wide range of different functions, is now being widely investigated for its role as a responsive biomaterial which dynamically reacts to changes in the surrounding environment. It is now evident that artificially designed self-assembling RNAs, that can form programmable nanoparticles and supra-assemblies, will play an increasingly important part in a diverse range of applications, such as macromolecular therapies, drug delivery systems, biosensing, tissue engineering, programmable scaffolds for material organization, logic gates, and soft actuators, to name but a few. The current exciting Special Issue comprises research highlights, short communications, research articles, and reviews that all bring together the leading scientists who are exploring a wide range of the fundamental properties of RNA and DNA nanoassemblies suitable for biomedical applications

Resource optimization for fault-tolerant quantum computing

In this thesis we examine a variety of techniques for reducing the resources required for fault-tolerant quantum computation. First, we show how to simplify universal encoded computation by using only transversal gates and standard error correction procedures, circumventing existing no-go theorems. We then show how to simplify ancilla preparation, reducing the cost of error correction by more than a factor of four. Using this optimized ancilla preparation, we develop improved techniques for proving rigorous lower bounds on the noise threshold. Additional overhead can be incurred because quantum algorithms must be translated into sequences of gates that are actually available in the quantum computer. In particular, arbitrary single-qubit rotations must be decomposed into a discrete set of fault-tolerant gates. We find that by using a special class of non-deterministic circuits, the cost of decomposition can be reduced by as much as a factor of four over state-of-the-art techniques, which typically use deterministic circuits. Finally, we examine global optimization of fault-tolerant quantum circuits under physical connectivity constraints. We adapt techniques from VLSI in order to minimize time and space usage for computations in the surface code, and we develop a software prototype to demonstrate the potential savings.Comment: 231 pages, Ph.D. thesis, University of Waterlo

Measurements, Models, Systems and Design

531 s.

Power Management ICs for Internet of Things, Energy Harvesting and Biomedical Devices

This dissertation focuses on the power management unit (PMU) and integrated circuits (ICs) for the internet of things (IoT), energy harvesting and biomedical devices. Three monolithic power harvesting methods are studied for different challenges of smart nodes of IoT networks. Firstly, we propose that an impedance tuning approach is implemented with a capacitor value modulation to eliminate the quiescent power consumption. Secondly, we develop a hill-climbing MPPT mechanism that reuses and processes the information of the hysteresis controller in the time-domain and is free of power hungry analog circuits. Furthermore, the typical power-performance tradeoff of the hysteresis controller is solved by a self-triggered one-shot mechanism. Thus, the output regulation achieves high-performance and yet low-power operations as low as 12 µW. Thirdly, we introduce a reconfigurable charge pump to provide the hybrid conversion ratios (CRs) as 1⅓× up to 8× for minimizing the charge redistribution loss. The reconfigurable feature also dynamically tunes to maximum power point tracking (MPPT) with the frequency modulation, resulting in a two-dimensional MPPT. Therefore, the voltage conversion efficiency (VCE) and the power conversion efficiency (PCE) are enhanced and flattened across a wide harvesting range as 0.45 to 3 V. In a conclusion, we successfully develop an energy harvesting method for the IoT smart nodes with lower cost, smaller size, higher conversion efficiency, and better applicability. For the biomedical devices, this dissertation presents a novel cost-effective automatic resonance tracking method with maximum power transfer (MPT) for piezoelectric transducers (PT). The proposed tracking method is based on a band-pass filter (BPF) oscillator, exploiting the PT’s intrinsic resonance point through a sensing bridge. It guarantees automatic resonance tracking and maximum electrical power converted into mechanical motion regardless of process variations and environmental interferences. Thus, the proposed BPF oscillator-based scheme was designed for an ultrasonic vessel sealing and dissecting (UVSD) system. The sealing and dissecting functions were verified experimentally in chicken tissue and glycerin. Furthermore, a combined sensing scheme circuit allows multiple surgical tissue debulking, vessel sealer and dissector (VSD) technologies to operate from the same sensing scheme board. Its advantage is that a single driver controller could be used for both systems simplifying the complexity and design cost. In a conclusion, we successfully develop an ultrasonic scalpel to replace the other electrosurgical counterparts and the conventional scalpels with lower cost and better functionality

Advances in Solid State Circuit Technologies

This book brings together contributions from experts in the fields to describe the current status of important topics in solid-state circuit technologies. It consists of 20 chapters which are grouped under the following categories: general information, circuits and devices, materials, and characterization techniques. These chapters have been written by renowned experts in the respective fields making this book valuable to the integrated circuits and materials science communities. It is intended for a diverse readership including electrical engineers and material scientists in the industry and academic institutions. Readers will be able to familiarize themselves with the latest technologies in the various fields