Balanced Judicious Bipartition is Fixed-Parameter Tractable

The family of judicious partitioning problems, introduced by Bollob\u27as and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollob\u27as and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT)

Brief Announcement: Bounded-Degree Cut is Fixed-Parameter Tractable

In the bounded-degree cut problem, we are given a multigraph G=(V,E), two disjoint vertex subsets A,B subseteq V, two functions u_A, u_B:V -> {0,1,...,|E|} on V, and an integer k >= 0. The task is to determine whether there is a minimal (A,B)-cut (V_A,V_B) of size at most k such that the degree of each vertex v in V_A in the induced subgraph G[V_A] is at most u_A(v) and the degree of each vertex v in V_B in the induced subgraph G[V_B] is at most u_B(v). In this paper, we show that the bounded-degree cut problem is fixed-parameter tractable by giving a 2^{18k}|G|^{O(1)}-time algorithm. This is the first single exponential FPT algorithm for this problem. The core of the algorithm lies two new lemmas based on important cuts, which give some upper bounds on the number of candidates for vertex subsets in one part of a minimal cut satisfying some properties. These lemmas can be used to design fixed-parameter tractable algorithms for more related problems

Parameterized Complexity of Multi-Node Hubs

Hubs are high-degree nodes within a network. The examination of the emergence and centrality of hubs lies at the heart of many studies of complex networks such as telecommunication networks, biological networks, social networks and semantic networks. Furthermore, identifying and allocating hubs are routine tasks in applications. In this paper, we do not seek a hub that is a single node, but a hub that consists of k nodes. Formally, given a graph G=(V,E), we a seek a set A subseteq V of size k that induces a connected subgraph from which at least p edges emanate. Thus, we identify k nodes which can act as a unit (due to the connectivity constraint) that is a hub (due to the cut constraint). This problem, which we call Multi-Node Hub (MNH), can also be viewed as a variant of the classic Max Cut problem. While it is easy to see that MNH is W[1]-hard with respect to the parameter k, our main contribution is the first parameterized algorithm that shows that MNH is FPT with respect to the parameter p. Despite recent breakthrough advances for cut-problems like Multicut and Minimum Bisection, MNH is still very challenging. Not only does a connectivity constraint has to be handled on top of the involved machinery developed for these problems, but also the fact that MNH is a maximization problem seems to prevent the applicability of this machinery in the first place. To deal with the latter issue, we give non-trivial reduction rules that show how MNH can be preprocessed into a problem where it is necessary to delete a bounded-in-parameter number of vertices. Then, to handle the connectivity constraint, we use a novel application of the form of tree decomposition introduced by Cygan et al. [STOC 2014] to solve Minimum Bisection, where we demonstrate how connectivity constraints can be replaced by simpler size constraints. Our approach may be relevant to the design of algorithms for other cut-problems of this nature

Probing universality with entanglement entropy via quantum Monte Carlo

Our understanding of physical phenomena hinges on finding universal core mechanisms that unite them. The concept of universality is deeply ingrained in the study of quantum many-body systems. At zero temperature, microscopically different systems with long-range order collapse into their universal state described by a handful of universal parameters. Establishing those parameters implies identification of the universal theory that is effectively describing the system and ultimately providing the desired understanding. We take up this task with the help of Renyi entanglement entropy. Defined with respect to a system bipartition, this measure quantifies information shared between the subsystems. Understanding what insights into universality are encoded within this information-theoretic quantity as well as developing numerical tools to efficiently estimate the Renyi entanglement entropy are the subjects of this thesis. On the computational end of this far-reaching goal, we develop a novel theoretical framework for constructing improved Renyi entanglement entropy estimators in the context of d+1 quantum Monte Carlo methods. The discovery of a connection of this methodology to the well-established Kandel-Domany formalism provides a clear path towards generalization. Additionally, we embrace a data-driven approach towards learning the ground state wavefunction. We demonstrate how a restricted Boltzmann machine can be used to reconstruct the Renyi entanglement entropy from projective measurements of a quantum ground state. Furthermore, we extend this classical generative architecture to a quantum analogue that we call the quantum Boltzmann machine. On the theoretical side of this endeavour, we study the Renyi entanglement entropy scaling terms for two quantum lattice models embedded in two dimensional space via extensive quantum Monte Carlo simulations. For the ground state of the XY model, we provide conclusive numerical evidence for a logarithmic contribution that uniquely characterizes the continuous symmetry of the emerging order parameter. Moreover, we confirm the form of the subleading universal geometric contribution arising due to the bosonic nature of low-energy degrees of freedom in this model. For the critical ground state of the transverse field Ising model, we develop a novel scaling procedure to extract a universal number κ revealed via a cylindrical entangling bipartition in the thin-slice limit. The combined product of our work sheds new light on the entanglement-based classification of universality and brings a suite of new powerful numerical tools to continue illuminating this theoretical program in the future

Entanglement, topology, & renormalization

In this thesis we will explore entanglement of various subsystems in quantum field theory, and its uses for describing underlying structure of states. We begin this journey by focusing on topological quantum field theory in chapters 1 and 2. These chapters are based heavily upon the papers [1] and [2, 3], respectively. In the introduction to chapter 1 we take the opportunity to recall the definition of bipartite entanglement entropy in quantum field theory and describe, briefly, how the entropy associated to connected spatial subregions provides a signature of topological order at the level of the wave-function. In particular, for a gapped phase with topological order, we recapitulate how the sub-leading correction to the “area-law” of the entanglement entropy contains coarse information about the topological phase while also being insensitive to the short-distance details of the underlying theory. We then present results that show that this sub-leading correction can also detect the class of interactions that glue a subregion to its complement. In principle, the subregion could be a different topological phase than its complement, and so we say that the entanglement entropy provides a signature of the interface between topological phases. In showing this, we discuss the classes of interactions between gapped topological phases and how they are represented as boundary conditions in the low-energy effective topological field theory, which in this case is an Abelian Chern-Simons theory. Using these boundary conditions, we perform path-integral calculations of the entanglement entropy using the replica trick and show that the universal contribution does indeed depend on the class of boundary conditions, and the gapped topological phases of the subsystem and its complement. To finish this chapter we reformulate this problem at the level of the Hilbert space and show how these universal contributions are related to resolving a generic ambiguity in defining subsystems of field theories with gauge invariance. We continue in the theme of topological field theories for chapter 2. Here we will shift focus from subregions of a connected spatial slice to disconnected spatial regions in an effort to explore, firstly, how long-range entanglement is encoded in topological field theory, and secondly, how that entanglement is encoded amongst multiple parties. We begin with a brief introduction of multi-party entanglement and recall the classification of three-party entanglement. Moving on to our specific setup, we describe the construction of states on multiple non-intersecting torus boundaries. Generically, these are states on link complements where the torus boundaries can be thought of as the edges of thickened closed strings entwined and knotted inside a compact manifold, which we will take to be the three-sphere. The wave-functions of these states are represented by various link invariants (such as the Gauss linking number for Abelian links, and Jones polynomial for SU(2)). We will exhibit several examples of multi-component links in both the Abelian and the SU(2) theories, and afterwards discuss generic features of the entanglement of link states in Chern-Simons theories with a compact group. In this discussion, we will conjecture a classification (albeit broad) of links via multi-partite entanglement of their corresponding states. We will end the chapter moving away from compact groups and discuss hyperbolic links in SL(2,C) Chern-Simons theory in the semi-classical limit. In particular, utilizing the generalized volume conjecture, we will show that the entanglement structure of hyperbolic links is greatly constrained in this semi-classical limit. Finally, in chapter 3 we switch gears and discuss renormalization of free bosonic field theory. The content of this chapter is drawn from the work of [4]. Motivated by work establishing duality between the singlet sector of O(N) vector models at large N and higher spin gauge theory on Anti de Sitter (AdS) space, we develop an exact renormalization group (ERG) framework for discussing the renormalization of a large class of O(N) singlet states. We will show that for the ground state and the excited states in this class the ERG is implemented by the action of unitary operators (that can be taken to be local). Consequently, the ERG framework can be regarded as a continuum tensor network. We contrast this tensor network with some well known tensor network renormalization schemes such as the muti-scale entanglement renormalization ansatz (MERA) and its proposed continuum version (cMERA). We also comment on the nature of this network with ordinary Wilsonian renormalization. One of the central features of the ERG network is how it acts on the momentum space entanglement of the field theory. In particular, we argue that for excited states, the ERG disentangles modes that lie just above and below the UV cutoff

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