EFFICIENT COMPUTER SEARCH FOR MULTIPLE RECURSIVE GENERATORS

Pseudo-random numbers (PRNs) are the basis for almost any statistical simulation and thisdepends largely on the quality of the pseudo-random number generator(PRNG) used. In this study, we used some results from number theory to propose an efficient method to accelerate the computer search of super-order maximum period multiple recursive generators (MRGs). We conduct efficient computer searches and successfully found prime modulus p, and the associated order k; (k = 40751; k = 50551; k = 50873) such that R(k; p) is a prime. Using these values of ks, together with the generalized Mersenne prime algorithm, we found and listed many efficient, portable, and super-order MRGs with period lengths of approximately 10e 380278.1;10e 471730.6; and 10e 474729.3. In other words, using the generalized Mersenne prime algorithm, we extended some known results of some efficient, portable, and maximum period MRGs. In particular, the DX/DL/DS/DT large order generators are extended to super-order generators.For r k, super-order generators in MRG(k,p) are quite close to an ideal generator. Forr \u3e k; the r-dimensional points lie on a relatively small family of equidistant parallel hyperplanesin a high dimensional space. The goodness of these generators depend largely on the distance between these hyperplanes. For LCGs, MRGs, and other generators with lattice structures, the spectral test, which is a theoretical test that gives some measure of uniformity greater than the order k of the MRG, is the most perfect figure of merit. A drawback of the spectral test is its computational complexity. We used a simple and intuitive method that employs the LLL algorithm, to calculate the spectral test. Using this method, we extended the search for better DX-k-s-t farther than the known value of k = 25013: In particular, we searched and listed better super-order DX-k-s-t generators for k = 40751; k = 50551, and k = 50873.Finally, we examined, another special class of MRGs with many nonzero terms known as the DW-k generator. The DW-k generators iteration can be implemented efficiently and in parallel, using a k-th order matrix congruential generator (MCG) sharing the same characteristic polynomial. We extended some known results, by searching for super-order DW-k generators, using our super large k values that we obtained in this study. Using extensive computer searches, we found and listed some super-order, maximum period DW(k; A, B, C, p = 2e 31 - v) generators

PAN: Pulse Ansatz on NISQ Machines

Variational quantum algorithms (VQAs) have demonstrated great potentials in the NISQ era. In the workflow of VQA, the parameters of ansatz are iteratively updated to approximate the desired quantum states. We have seen various efforts to draft better ansatz with less gates. In quantum computers, the gate ansatz will eventually be transformed into control signals such as microwave pulses on transmons. And the control pulses need elaborate calibration to minimize the errors such as over-rotation and under-rotation. In the case of VQAs, this procedure will introduce redundancy, but the variational properties of VQAs can naturally handle problems of over-rotation and under-rotation by updating the amplitude and frequency parameters. Therefore, we propose PAN, a native-pulse ansatz generator framework for VQAs. We generate native-pulse ansatz with trainable parameters for amplitudes and frequencies. In our proposed PAN, we are tuning parametric pulses, which are natively supported on NISQ computers. Considering that parameter-shift rules do not hold for native-pulse ansatz, we need to deploy non-gradient optimizers. To constrain the number of parameters sent to the optimizer, we adopt a progressive way to generate our native-pulse ansatz. Experiments are conducted on both simulators and quantum devices to validate our methods. When adopted on NISQ machines, PAN obtained improved the performance with decreased latency by an average of 86%. PAN is able to achieve 99.336% and 96.482% accuracy for VQE tasks on H2 and HeH+ respectively, even with considerable noises in NISQ machines.Comment: 13 pages, 13 figure

UTILIZING DESIGN STRUCTURE FOR IMPROVING DESIGN SELECTION AND ANALYSIS

Recent work has shown that the structure for design plays a role in the simplicity or complexity of data analysis. To increase the knowledge of research in these areas, this dissertation aims to utilize design structure for improving design selection and analysis. In this regard, minimal dependent sets and block diagonal structure are both important concepts that are relevant to the orthogonality of the columns of a design. We are interested in finding ways to improve the data analysis especially for active effect detection by utilizing minimal dependent sets and block diagonal structure for design. We introduce a new classification criterion for minimal dependent sets to enhance existing criteria for design selection. The block diagonal structure of certain nonregular designs will also be discussed as a means of improving model selection. In addition, the block diagonal structure and the concept of parallel flats will be utilized to construct three-quarter nonregular designs. Based on the literature review on the effectiveness of the simulation study for slight the light on the success or failure of the proposed statistical method, in this dissertation, simulation studies were used to evaluate the efficacy of our proposed methods. The simulation results show that the minimal dependent sets can be used as a design selection criterion, and block-diagonal structure can also help to produce an effective model selection procedure. In addition, we found a strategy for constructing three-quarters of nonregular designs which depend on the orthogonality of the design columns. The results indicate that the structure of the design has an impact on developing data analysis and design selections. On this basis, it is recommended that analysts consider the structure of the design as a key factor in order to improve the analysis. Further research is needed to determine more concepts related to the structure of the design, which could help to improve data analysis

Robust and Efficient Hamiltonian Learning

With the fast development of quantum technology, the sizes of both digital and analog quantum systems increase drastically. In order to have better control and understanding of the quantum hardware, an important task is to characterize the interaction, i.e., to learn the Hamiltonian, which determines both static and dynamic properties of the system. Conventional Hamiltonian learning methods either require costly process tomography or adopt impractical assumptions, such as prior information on the Hamiltonian structure and the ground or thermal states of the system. In this work, we present a robust and efficient Hamiltonian learning method that circumvents these limitations based only on mild assumptions. The proposed method can efficiently learn any Hamiltonian that is sparse on the Pauli basis using only short-time dynamics and local operations without any information on the Hamiltonian or preparing any eigenstates or thermal states. The method has a scalable complexity and a vanishing failure probability regarding the qubit number. Meanwhile, it performs robustly given the presence of state preparation and measurement errors and resiliently against a certain amount of circuit and shot noise. We numerically test the scaling and the estimation accuracy of the method for transverse field Ising Hamiltonian with random interaction strengths and molecular Hamiltonians, both with varying sizes and manually added noise. All these results verify the robustness and efficacy of the method, paving the way for a systematic understanding of the dynamics of large quantum systems.Comment: 41 pages, 6 figures, Open source implementation available at https://github.com/zyHan2077/HamiltonianLearnin

Study of Λc+ → Λμ+νμ and test of lepton flavor universality with Λc+ → Λℓ+νℓ decays

The study of the Cabibbo-favored semileptonic decay Λc+→Λμ+νμ is reported using 4.5 fb-1 of e+e- annihilation data collected at center-of-mass energies ranging from 4.600 to 4.699 GeV. The branching fraction of the decay is measured to be B(Λc+→Λμ+νμ)=(3.48±0.14stat±0.10syst)%, 3 times more precise than the prior world average result. Tests of lepton flavor universality using Λc+→Λℓ+νℓ (ℓ=e, μ) decays are reported for the first time, based on measurements of the differential decay rates and the forward-backward asymmetries in separate four-momentum transfer regions. The results are compatible with Standard Model predictions. Furthermore, we improve the determination of the form-factor parameters in Λc+→Λℓ+νℓ decays, which provide stringent tests and calibration for lattice quantum chromodynamics calculations

New Materials and Methods towards High-Energy Lithium Metal Batteries

The sluggish progress of battery technologies has drastically hindered the rapid development of electric vehicles and next-generation portable electronics. Improving the energy density requires breakthroughs in materials for both cathode and anode, and new characterization methods to accurately correlate the materials with their performances.For cathodes, lithium (Li) rich layered oxides exhibit high reversible specific capacities over 300 mAh g-1, attributing to the oxygen redox reaction. However, oxygen activity comes with instability in the form of oxygen loss, which is associated with irreversible voltage decay and capacity fading. Calculations suggest that incorporating 4d elements, such as Mo, enhances the structural stability by altering the local band structure and impeding oxygen vacancy formation. Driven by these findings, Mo is co-doped with Co into Li[Li0.2Ni0.2Mn0.6]O2, showing notably reduced voltage decay and capacity fading without sacrificing energy density and cycle life.The Li metal anode is critical to break the energy-density bottleneck of current Li-ion chemistry. Inactive Li formation is the immediate cause of capacity loss and catastrophic failure of Li metal batteries. However, its composition has not yet been quantitatively studied due to the lack of effective diagnosis tools that can accurately differentiate Li+ in solid electrolyte interphase (SEI) components and the electrically isolated unreacted metallic Li0, which together comprise the inactive Li. By establishing a new analytical method, Titration Gas Chromatography (TGC), we accurately quantify the contribution from unreacted metallic Li0 to the total amount of inactive Li. We identify the Li0, rather than the (electro)chemically formed Li+ in SEI, as the dominating cause for the inactive Li and capacity loss. Coupling the measurements of the unreacted metallic Li0 global content to the observations of its local micro- and nano-structure by cryogenic electron microscopies, we also reveal the formation mechanism of inactive Li in different types of electrolytes, and determine the true underlying cause of low CE in Li metal deposition and stripping. We ultimately propose strategies for highly efficient Li deposition and stripping to enable Li metal anode for next generation high-energy batteries

Metal Hydride Based Materials for Advanced Lithium Storage Applications

The state of the art, commercial used graphite anode materials are far from meeting the increasing demand for high-energy density devices. It is necessary to develop anode materials with high energy density, low-cost and superior safety. Generally, compared with classical intercalation anodes, conversion type anodes display higher theoretical capacity have been paid widely attention. Some of the emerging metal hydrides demonstrate high specific capacity, small polarization and suitable working potential. This thesis focusing on two metal hydrides both with relatively high specific capacity, sodium alanate (NaAlH4) and mageneisum hydride (MgH2) as anodes materials in LIBs. In order to enhance the electronic conductivity of the material and relieve the volume variation, four methods of carbon doping, nano crystallization, surface modification and process modification were used to raise the lithium storage performance of metal hydride-based anode materials (NaAlH4 and MgH2). By way of self-assembly, gas-solid reaction and other synthesis methods, the multi nanostructure of metal hydride-based anode materials (NaAlH4 and MgH2) were designed, and a variety of composite materials of metal hydride and graphene with different structures were successfully prepared. Through a facile solvent evaporation induced deposition method, NaAlH4 nanoparticles with an average size of ~ 12 nm encapsulated in graphene nanosheets has been developed. The SAH@G-50 electrode exhibits an discharge capacity about 1995 mAh g-1 at 100 mAh g−1 at first cycle, with a coulombic efficiency (CE) of 85.7%. The specific discharge capacity slowly decayed and then was stabilized at ~698 mAh g-1 after 200 cycles. It has been founded in this thesis, graphene could act as an effective platform to tailor the metal-hydrogen bonds of NaAlH4 through their favorable molecular interaction. Theoretical and experimental results confirm that graphene is capable of weakening the Al-H bonds of NaAlH4, thus facilitating the breaking and recombination of Al-H bonds towards advanced lithium storage performance. In addition, The synergistic effects of the favorable molecular interaction between graphene and NaAlH4, and the noticeable decrease in particle size significantly boost the lithium storage performances of NaAlH4

Quantum computational chemistry methods for early-stage quantum computers

One of the first practical applications of quantum computers is expected to be molecular modelling. Performing this task would profoundly affect areas such as chemistry, materials science and drug synthesis. Modelling of molecules, which are classically intractable, can be achieved with just over $30$ qubits, whereas state of the art quantum computers already have more than $50$ qubits. The Variational Quantum Eigensolver (VQE) algorithm and VQE based protocols, are promising candidates to enable this task on emerging Noisy Intermediate-Scale Quantum (NISQ) computers. These protocols require short quantum circuits and short coherence times, and are particularly resilient to quantum errors. Nevertheless, there is still a significant gap between the accuracy and the coherence times of current NISQ computers, and the hardware requirements of VQE protocols to simulate practically interesting molecules. In this thesis, I present my contribution to narrowing this gap by developing VQE protocols for molecular modelling that are less demanding on quantum hardware. The VQE relies on the Rayleigh-Ritz variational principle to estimate the eigenvalues of a Hamiltonian operator, by minimizing its expectation value with respect to a trial quantum state, prepared by an ansatz. A major challenge for the practical realisation of VQE protocols on NISQ computers is to construct an ansatz that: (1) can accurately approximate the eigenstates of the Hamiltonian; (2) is easy to optimize; and (3) can be implemented by a shallow circuit, within the capabilities of a NISQ computer. The most widely used, unitary coupled cluster (UCC), type of ans\"atze mathematically correspond to a product of unitary evolutions of fermionic excitation operators. Owing to their fermionic structure, UCC ans\"atze preserve the symmetries of electronic wavefunctions, and thus are accurate and easy to optimize. Nevertheless, UCC ans\"atze are implemented by high depth circuits, which severely limit the size of the molecules that can be reliably simulated on NISQ computers. In this thesis, I begin by constructing efficient quantum circuits to perform evolutions of fermionic excitation operators. The circuits are optimized in the number of two-qubit entangling gates, which are the current bottleneck of NISQ computers. Compared to the standard circuits used to implement evolutions of fermionic excitation operators, the circuits derived in this thesis reduce the number of two-qubit entangling gates by more than $70\%$ on average. As an intermediate result, I also derive efficient circuits to perform evolutions of qubit excitation operators (excitation operators that account for qubit, rather than fermionic commutation relations). Even with the fermionic-excitation-evolution circuits derived here, UCC ans\"atze still require very long circuits, with a particularly large number of two-qubit entangling gates. In this thesis, I consider the use of alternative VQE ans\"atze, based on evolutions of qubit excitation operators. Due to not accounting for fermionic anticommutation, evolutions of qubit excitation operators can be performed by circuits that require asymptotically fewer two-qubit entangling gates. Furthermore, qubit excitation operators preserve many of the physical properties of fermionic excitation operators. Performing a number of classical numerical VQE simulations for small molecules, I show that qubit-excitation-based ans\"atze can approximate molecular electronic wavefunctions almost as accurately as fermionic-excitation-based ans\"atze. Hence, I argue that evolutions of qubit excitation operators are more suitable to construct molecular ans\"atze than evolutions of fermionic excitation operators, especially in the era of NISQ computers. Motivated by the advantage of qubit-excitation-based ans\"atze, I introduce the qubit-excitation-based adaptive variational quantum eigensolver (QEB-ADAPT-VQE). The QEB-ADAPT-VQE belongs to a family of ADAPT-VQE protocols for molecular modelling that grow a problem-tailored ansatz by iteratively appending unitary operators sampled from a predefined finite-size pool of operators. The operator at each iteration is sampled based on an ansatz-growing strategy, which aims to achieve the lowest estimate for the Hamiltonian expectation value at each iteration. In this way, ADAPT-VQE protocols construct shallow-circuit, few-parameter ans\"atze tailored specifically to the molecular systems of interest. In the case of the QEB-ADAPT-VQE, the operator pool is defined by a set of evolutions of single and double qubit excitation operators. I benchmark the performance of the QEB-ADAPT-VQE, by performing classical numerical simulations. I demonstrate that it can construct ans\"atze that are several orders of magnitude more accurate, and require significantly shallower circuits, than standard UCC ans\"atze. I also compare the QEB-ADAPT-VQE against the original fermionic-ADAPT-VQE, which utilizes a pool of fermionic excitation evolutions, and the qubit-ADAPT-VQE, which utilizes a pool of Pauli-string evolutions. I demonstrate that, in terms of circuit efficiency and convergence speed, the QEB-ADAPT-VQE systematically outperforms the qubit-ADAPT-VQE, which to my knowledge was the previous most circuit-efficient, scalable VQE protocol for molecular modeling. The QEB-ADAPT-VQE protocol, therefore represents a significant improvement in the field of VQE protocols for molecular modelling and brings us closer to achieving practical quantum advantage. Lastly, I outline a modified version of the QEB-ADAPT-VQE, the excited-QEB-ADAPT-VQE, designed to estimate energies of excited molecular states. The excited-QEB-ADAPT-VQE is more robust to initial simulation conditions, at the expense of increased computational complexity.I acknowledge the funding I received from the Engineering and Physical Sciences Research Council, and Hitachi Cambridge Laborator