29 research outputs found
Approximate Transversals of Latin Squares
A latin square of order n is an n by n array whose entries are drawn from an n-set of symbols such that each symbol appears precisely once in each row and column. A transversal of a latin square is a subset of cells that meets each row, column, and symbol precisely once.
There are many open and difficult questions about the existence and prevalence of transversals. We undertake a systematic study of collections of cells that exhibit regularity properties similar to those of transversals and prove numerous theorems about their existence and structure. We hope that our results and methods will suggest new strategies for the study of transversals.
The main topics we investigate are partial and weak transversals, weak orthogonal mates, integral weight functions on the cells of a latin square, applications of Alon\u27s Combinatorial Nullstellensatz to latin squares, and complete mappings of finite loops
Topological States of Matter in Frustrated Quantum Magnetism
Frustrated quantum magnets may exhibit fascinating collective phenomena. The
main goal of this dissertation is to provide conclusive evidence for the
emergence of novel phases of matter like quantum spin liquids in local quantum
spin models. We develop novel algorithms for large-scale Exact Diagonalization
computations. Sublattice coding methods for efficient use of lattice symmetries
in the procedure of diagonalizing the Hamiltonian matrix are proposed and
suggest a randomized distributed memory parallelization strategy. Benchmarks of
computations on various supercomputers with system size up to 50 spin-1/2
particles have been performed. Results concerning the emergence of a chiral
spin liquid in a frustrated kagome Heisenberg antiferromagnet are presented.
The stability and extent of this phase are discussed. In an extended Heisenberg
model on the triangular lattice, we establish another chiral spin liquid phase.
We discuss the special case of the Heisenberg - model and present a
scenario where the critical point of phase transition from the 120-degree
N\'eel to a putative spin liquid is described by a Dirac spin
liquid. A generalization of the SU(2) Heisenberg model with SU(N) degrees of
freedom on the triangular lattice with an additional ring-exchange term is
discussed. We present our contribution to the project and the final results
that suggest a series of chiral spin liquid phases in an extended parameter
range. Finally, we present preliminary data from a Quantum Monte Carlo study of
an SU(N) version of the J-Q model on a square lattice for N=2,...,10, and
multi-column representations. We establish the phase boundary between the
N\'eel ordered phase and the disordered phases. The disordered phase in the
four-column representation is expected to be a two-dimensional analog of the
Haldane phase for the spin-1 Heisenberg chain.Comment: Ph.D. thesis, 161 page
Tools and Algorithms for the Construction and Analysis of Systems
This open access two-volume set constitutes the proceedings of the 26th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The total of 60 regular papers presented in these volumes was carefully reviewed and selected from 155 submissions. The papers are organized in topical sections as follows: Part I: Program verification; SAT and SMT; Timed and Dynamical Systems; Verifying Concurrent Systems; Probabilistic Systems; Model Checking and Reachability; and Timed and Probabilistic Systems. Part II: Bisimulation; Verification and Efficiency; Logic and Proof; Tools and Case Studies; Games and Automata; and SV-COMP 2020
On Differentiable Interpreters
Neural networks have transformed the fields of Machine Learning and Artificial Intelligence with the ability to model complex features and behaviours from raw data. They quickly became instrumental models, achieving numerous state-of-the-art performances across many tasks and domains. Yet the successes of these models often rely on large amounts of data. When data is scarce, resourceful ways of using background knowledge often help. However, though different types of background knowledge can be used to bias the model, it is not clear how one can use algorithmic knowledge to that extent. In this thesis, we present differentiable interpreters as an effective framework for utilising algorithmic background knowledge as architectural inductive biases of neural networks. By continuously approximating discrete elements of traditional program interpreters, we create differentiable interpreters that, due to the continuous nature of their execution, are amenable to optimisation with gradient descent methods. This enables us to write code mixed with parametric functions, where the code strongly biases the behaviour of the model while enabling the training of parameters and/or input representations from data. We investigate two such differentiable interpreters and their use cases in this thesis. First, we present a detailed construction of ∂4, a differentiable interpreter for the programming language FORTH. We demonstrate the ability of ∂4 to strongly bias neural models with incomplete programs of variable complexity while learning missing pieces of the program with parametrised neural networks. Such models can learn to solve tasks and strongly generalise to out-of-distribution data from small datasets. Second, we present greedy Neural Theorem Provers (gNTPs), a significant improvement of a differentiable Datalog interpreter NTP. gNTPs ameliorate the large computational cost of recursive differentiable interpretation, achieving drastic time and memory speedups while introducing soft reasoning over logic knowledge and natural language
Multipartite Quantum States and their Marginals
Subsystems of composite quantum systems are described by reduced density
matrices, or quantum marginals. Important physical properties often do not
depend on the whole wave function but rather only on the marginals. Not every
collection of reduced density matrices can arise as the marginals of a quantum
state. Instead, there are profound compatibility conditions -- such as Pauli's
exclusion principle or the monogamy of quantum entanglement -- which
fundamentally influence the physics of many-body quantum systems and the
structure of quantum information. The aim of this thesis is a systematic and
rigorous study of the general relation between multipartite quantum states,
i.e., states of quantum systems that are composed of several subsystems, and
their marginals. In the first part, we focus on the one-body marginals of
multipartite quantum states; in the second part, we study general quantum
marginals from the perspective of entropy.Comment: PhD thesis, ETH Zurich. The first part contains material from
arXiv:1208.0365, arXiv:1204.0741, and arXiv:1204.4379. The second part is
based on arXiv:1302.6990 and arXiv:1210.046