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

The corner poset with an application to an n-dimensional hypercube stacking puzzle

For any dimension n ≥ 3, we establish the corner poset, a natural triangular poset structure on the corners of 2-color hypercubes. We use this poset to study a problem motivated by a classical cube stacking puzzle posed by Percy MacMahon as well as Eric Cross’s more recent “Eight Blocks to Madness.” We say that a hypercube is 2-color when each of its facets has one of two colors. Given an arbitrary multiset of 2-color unit n-dimensional hypercubes, we investigate when it is possible to find a submultiset of 2n hypercubes that can be arranged into a larger hypercube of side length 2 with monochrome facets. Through a careful analysis of the poset and its properties, we construct interesting puzzles, find and enumerate solutions, and study the maximum size, S(n), for a puzzle that does not contain a solution. Further, we find bounds on S(n), showing that it grows as Θ(n2n)

Structure in Machine Learning: Graphical Models and Monte Carlo Methods

This thesis is concerned with two main areas: approximate inference in discrete graphical models, and random embeddings for dimensionality reduction and approximate inference in kernel methods. Approximate inference is a fundamental problem in machine learning and statistics, with strong connections to other domains such as theoretical computer science. At the same time, there has often been a gap between the success of many algorithms in this area in practice, and what can be explained by theory; thus, an important research effort is to bridge this gap. Random embeddings for dimensionality reduction and approximate inference have led to great improvements in scalability of a wide variety of methods in machine learning. In recent years, there has been much work on how the stochasticity introduced by these approaches can be better controlled, and what further computational improvements can be made. In the first part of this thesis, we study approximate inference algorithms for discrete graphical models. Firstly, we consider linear programming methods for approximate MAP inference, and develop our understanding of conditions for exactness of these approximations. Such guarantees of exactness are typically based on either structural restrictions on the underlying graph corresponding to the model (such as low treewidth), or restrictions on the types of potential functions that may be present in the model (such as log-supermodularity). We contribute two new classes of exactness guarantees: the first of these takes the form of particular hybrid restrictions on a combination of graph structure and potential types, whilst the second is given by excluding particular substructures from the underlying graph, via graph minor theory. We also study a particular family of transformation methods of graphical models, uprooting and rerooting, and their effect on approximate MAP and marginal inference methods. We prove new theoretical results on the behaviour of particular approximate inference methods under these transformations, in particular showing that the triplet relaxation of the marginal polytope is unique in being universally rooted. We also introduce a heuristic which quickly picks a rerooting, and demonstrate benefits empirically on models over several graph topologies. In the second part of this thesis, we study Monte Carlo methods for both linear dimensionality reduction and approximate inference in kernel machines. We prove the statistical benefit of coupling Monte Carlo samples to be almost-surely orthogonal in a variety of contexts, and study fast approximate methods of inducing this coupling. A surprising result is that these approximate methods can simultaneously offer improved statistical benefits, time complexity, and space complexity over i.i.d. Monte Carlo samples. We evaluate our methods on a variety of datasets, directly studying their effects on approximate kernel evaluation, as well as on downstream tasks such as Gaussian process regression.EPSR

Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration

The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics and the natural sciences. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions represented by diagonals of multivariate rational functions. We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page

Distributed learning and inference in deep models

In recent years, the size of deep learning problems has been increased significantly, both in terms of the number of available training samples as well as the number of parameters and complexity of the model. In this thesis, we considered the challenges encountered in training and inference of large deep models, especially on nodes with limited computational power and capacity. We studied two classes of related problems; 1) distributed training of deep models, and 2) compression and restructuring of deep models for efficient distributed and parallel execution to reduce inference times. Especially, we considered the communication bottleneck in distributed training and inference of deep models. Data compression is a viable tool to mitigate the communication bottleneck in distributed deep learning. However, the existing methods suffer from a few drawbacks, such as the increased variance of stochastic gradients (SG), slower convergence rate, or added bias to SG. In my Ph.D. research, we have addressed these challenges from three different perspectives: 1) Information Theory and the CEO Problem, 2) Indirect SG compression via Matrix Factorization, and 3) Quantized Compressive Sampling. We showed, both theoretically and via simulations, that our proposed methods can achieve smaller MSE than other unbiased compression methods with fewer communication bit-rates, resulting in superior convergence rates. Next, we considered federated learning over wireless multiple access channels (MAC). Efficient communication requires the communication algorithm to satisfy the constraints imposed by the nodes in the network and the communication medium. To satisfy these constraints and take advantage of the over-the-air computation inherent in MAC, we proposed a framework based on random linear coding and developed efficient power management and channel usage techniques to manage the trade-offs between power consumption and communication bit-rate. In the second part of this thesis, we considered the distributed parallel implementation of an already-trained deep model on multiple workers. Since latency due to the synchronization and data transfer among workers adversely affects the performance of the parallel implementation, it is desirable to have minimum interdependency among parallel sub-models on the workers. To achieve this goal, we developed and analyzed RePurpose, an efficient algorithm to rearrange the neurons in the neural network and partition them (without changing the general topology of the neural network) such that the interdependency among sub-models is minimized under the computations and communications constraints of the workers.Ph.D

Computer Aided Verification

The open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Computer Aided Verification

The open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency