Unconventional computing platforms and nature-inspired methods for solving hard optimisation problems

The search for novel hardware beyond the traditional von Neumann architecture has given rise to a modern area of unconventional computing requiring the efforts of mathematicians, physicists and engineers. Many analogue physical systems, including networks of nonlinear oscillators, lasers, condensates, and superconducting qubits, are proposed and realised to address challenging computational problems from various areas of social and physical sciences and technology. Understanding the underlying physical process by which the system finds the solutions to such problems often leads to new optimisation algorithms. This thesis focuses on studying gain-dissipative systems and nature-inspired algorithms that form a hybrid architecture that may soon rival classical hardware. Chapter 1 lays the necessary foundation and explains various interdisciplinary terms that are used throughout the dissertation. In particular, connections between the optimisation problems and spin Hamiltonians are established, their computational complexity classes are explained, and the most prominent physical platforms for spin Hamiltonian implementation are reviewed. Chapter 2 demonstrates a large variety of behaviours encapsulated in networks of polariton condensates, which are a vivid example of a gain-dissipative system we use throughout the thesis. We explain how the variations of experimentally tunable parameters allow the networks of polariton condensates to represent different oscillator models. We derive analytic expressions for the interactions between two spatially separated polariton condensates and show various synchronisation regimes for periodic chains of condensates. An odd number of condensates at the vertices of a regular polygon leads to a spontaneous formation of a giant multiply-quantised vortex at the centre of a polygon. Numerical simulations of all studied configurations of polariton condensates are performed with a mean-field approach with some theoretically proposed physical phenomena supported by the relevant experiments. Chapter 3 examines the potential of polariton graphs to find the low-energy minima of the spin Hamiltonians. By associating a spin with a condensate phase, the minima of the XY model are achieved for simple configurations of spatially-interacting polariton condensates. We argue that such implementation of gain-dissipative simulators limits their applicability to the classes of easily solvable problems since the parameters of a particular Hamiltonian depend on the node occupancies that are not known a priori. To overcome this difficulty, we propose to adjust pumping intensities and coupling strengths dynamically. We further theoretically suggest how the discrete Ising and $n$-state planar Potts models with or without external fields can be simulated using gain-dissipative platforms. The underlying operational principle originates from a combination of resonant and non-resonant pumping. Spatial anisotropy of pump and dissipation profiles enables an effective control of the sign and intensity of the coupling strength between any two neighbouring sites, which we demonstrate with a two dimensional square lattice of polariton condensates. For an accurate minimisation of discrete and continuous spin Hamiltonians, we propose a fully controllable polaritonic XY-Ising machine based on a network of geometrically isolated polariton condensates. In Chapter 4, we look at classical computing rivals and study nature-inspired methods for optimising spin Hamiltonians. Based on the operational principles of gain-dissipative machines, we develop a novel class of gain-dissipative algorithms for the optimisation of discrete and continuous problems and show its performance in comparison with traditional optimisation techniques. Besides looking at traditional heuristic methods for Ising minimisation, such as the Hopfield-Tank neural networks and parallel tempering, we consider a recent physics-inspired algorithm, namely chaotic amplitude control, and exact commercial solver, Gurobi. For a proper evaluation of physical simulators, we further discuss the importance of detecting easy instances of hard combinatorial optimisation problems. The Ising model for certain interaction matrices, that are commonly used for evaluating the performance of unconventional computing machines and assumed to be exponentially hard, is shown to be solvable in polynomial time including the Mobius ladder graphs and Mattis spin glasses. In Chapter 5 we discuss possible future applications of unconventional computing platforms including emulation of search algorithms such as PageRank, realisation of a proof-of-work protocol for blockchain technology, and reservoir computing

Codes, arrangements, matroids, and their polynomial links

Codes, arrangements, matroids, and their polynomial links Many mathematical objects are closely related to each other. While studying certain aspects of a mathematical object, one tries to find a way to "view" the object in a way that is most suitable for a specific problem. Or, in other words, one tries to find the best way to model the problem. Many related fields of mathematics have evolved from one another this way. In practice, it is very useful to be able to transform a problem into other terminology: it gives a lot more available knowledge and that can be helpful to solve a problem. This thesis deals with various closely related fields in discrete mathematics, starting from linear error-correcting codes and their weight enumerator. We can generalize the weight enumerator in two ways, to the extended and generalized weight enumerators. The set of generalized weight enumerators is equivalent to the extended weight enumerator. Summarizing and extending known theory, we define the two-variable zeta polynomial of a code and its generalized zeta polynomial. These polynomials are equivalent to the extended and generalized weight enumerator of a code. We can determine the extended and generalized weight enumerator using projective systems. This calculation is explicitly done for codes coming from finite projective and affine spaces: these are the simplex code and the first order Reed-Muller code. As a result we do not only get the weight enumerator of these codes, but it also gives us information on their geometric structure. This is useful information in determining the dimension of geometric designs. To every linear code we can associate a matroid that is representable over a finite field. A famous and well-studied polynomial associated to matroids is the Tutte polynomial, or rank generating function. It is equivalent to the extended weight enumerator. This leads to a short proof of the MacWilliams relations for the extended weight enumerator. For every matroid, its flats form a geometric lattice. On the other hand, every geometric lattice induces a simple matroid. The Tutte polynomial of a matroid determines the coboundary polynomial of the associated geometric lattice. In the case of simple matroids, this becomes a two-way equivalence. Another polynomial associated to a geometric lattice (or, more general, to a poset) is the Möbius polynomial. It is not determined by the coboundary polynomial, neither the other way around. However, we can give conditions under which the Möbius polynomial of a simple matroid together with the Möbius polynomial of its dual matroid defines the coboundary polynomial. The proof of these relations involves the two-variable zeta polynomial, that can be generalized from codes to matroids. Both matroids and geometric lattices can be truncated to get an object of lower rank. The truncated matroid of a representable matroid is again representable. Truncation formulas exist for the coboundary and Möbius polynomial of a geometric lattice and the spectrum polynomial of a matroid, generalizing the known truncation formula of the Tutte polynomial of a matroid. Several examples and counterexamples are given for all the theory. To conclude, we give an overview of all polynomial relations

An operational information decomposition via synergistic disclosure

Abstract: Multivariate information decompositions hold promise to yield insight into complex systems, and stand out for their ability to identify synergistic phenomena. However, the adoption of these approaches has been hindered by there being multiple possible decompositions, and no precise guidance for preferring one over the others. At the heart of this disagreement lies the absence of a clear operational interpretation of what synergistic information is. Here we fill this gap by proposing a new information decomposition based on a novel operationalisation of informational synergy, which leverages recent developments in the literature of data privacy. Our decomposition is defined for any number of information sources, and its atoms can be calculated using elementary optimisation techniques. The decomposition provides a natural coarse-graining that scales gracefully with the system’s size, and is applicable in a wide range of scenarios of practical interest

M\"{o}bius transforms, cycles and q-triplets in statistical mechanics

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analyse observed sequences of q-triplets, or q-doublets if one of them is the unity, in terms of cycles of successive M\"obius transforms of the line preserving unity ( q=1 corresponds to the BG theory). Such transforms have the form q --> (aq + 1-a)/[(1+a)q -a], where a is a real number; the particular cases a=-1 and a=0 yield respectively q --> (2-q) and q --> 1/q, currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties.Comment: Keyword: non-additive entropy; q-statistics; M\"obius transform; complex system

Lattice QCD Simulations towards Strong and Weak Coupling Limits

Lattice gauge theory is a special regularization of continuum gauge theories and the numerical simulation of lattice quantum chromodynamics (QCD) remains as the only first principle method to study non-perturbative QCD at low energy. The lattice spacing a, which serves as the ultraviolet cut off, plays a significant role in determining error on any lattice simulation results. Physical results come from extrapolating a series of simulations with different values for a to a=0. Reducing the size of these errors for non-zero a improves the extrapolation and minimizes the error. In the strong coupling limit the coarse lattice spacing pushes the analysis of the finite lattice spacing error to its limit. Section 4 measures two renormalized physical observables, the neutral kaon mixing parameter BK and the Delta I=3/2 K pi pi decay amplitude A2 on a lattice with coarse lattice spacing of a ~ 1GeV and explores the a^2 scaling properties at this scale. In the weak coupling limit the lattice simulations suffer from critical slowing down where for the Monte Carlo Markov evolution the cost of generating decorrelated samples increases significantly as the lattice spacing decreases, which makes reliable error analysis on the results expensive. Among the observables the topological charge of the configurations appears to have the longest integrated autocorrelation time. Based on a previous work where a diffusion model is proposed to describe the evolution of the topological charge, section 2 extends this model to lattices with dynamical fermions using a new numerical method that captures the behavior for different Fourier modes. Section 3 describes our effort to find a practical renormalization group transformation to transform lattice QCD between two different scales, whose knowledge could ultimately leads to a multi-scale evolution algorithm that solves the problem of critical slowing down. For a particular choice of action, we have found that doubling the lattice spacing of a fine lattice yields observables that agree at the few precent level with direct simulations on the coarser lattice. Section 5 aims at speeding up the lattice simulations in the weak coupling limit from the numerical method and hardware perspective. It proposes a preconditioner for solving the Dirac equation targeting the ensemble generation phase and details its implementation on currently the fastest supercomputer in the world

One-way communication complexity and non-adaptive decision trees

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. More generally, we show the following when the gadget is Inner Product on 2b input bits for all b ≥ 2, denoted IP. If f is a total Boolean function that depends on all of its n input bits, then the bounded-error one-way quantum communication complexity of f ◦ IP equals Ω(n(b - 1)). If f is a partial Boolean function, then the deterministic one-way communication complexity of f ◦ IP is at least Ω(b · D→dt (f)), where D→dt (f) denotes non-adaptive decision tree complexity of f. To prove our quantum lower bound, we first show a lower bound on the VC-dimension of f ◦ IP. We then appeal to a result of Klauck [STOC’00], which immediately yields our quantum lower bound. Our deterministic lower bound relies on a combinatorial result independently proven by Ahlswede and Khachatrian [Adv. Appl. Math.’98], and Frankl and Tokushige [Comb.’99]. It is known due to a result of Montanaro and Osborne [arXiv’09] that the deterministic one-way communication complexity of f ◦ XOR equals the non-adaptive parity decision tree complexity of f. In contrast, we show the following when the inner gadget is the AND function on 2 input bits. There exists a function for which even the quantum non-adaptive AND decision tree complexity of f is exponentially large in the deterministic one-way communication complexity of f ◦ AND. However, for symmetric functions f, the non-adaptive AND decision tree complexity of f is at most quadratic in the (even two-way) communication complexity of f ◦ AND. In view of the first bullet, a lower bound on non-adaptive AND decision tree complexity of f does not lift to a lower bound on one-way communication complexity of f ◦ AND. The proof of the first bullet above uses the well-studied Odd-Max-Bit function. For the second bullet, we first observe a connection between the one-way communication complexity of f and the Möbius sparsity of f, and then give a lower bound on the Möbius sparsity of symmetric functions. An upper bound on the non-adaptive AND decision tree complexity of symmetric functions follows implicitly from prior work on combinatorial group testing; for the sake of completeness, we include a proof of this result. It is well known that the rank of the communication matrix of a function F is an upper bound on its deterministic one-way communication complexity. This bound is known to be tight for some F. However, in our final result we show that this is not the case when F = f ◦ AND. More precisely we show that for all f, the deterministic one-way communication complexity of F = f ◦ AND is at most (rank(MF))(1 - Ω(1)), where MF denotes the communication matrix of F

An operational information decomposition via synergistic disclosure

Abstract Multivariate information decompositions hold promise to yield insight into complex systems, and stand out for their ability to identify synergistic phenomena. However, the adoption of these approaches has been hindered by there being multiple possible decompositions, and no precise guidance for preferring one over the others. At the heart of this disagreement lies the absence of a clear operational interpretation of what synergistic information is. Here we fill this gap by proposing a new information decomposition based on a novel operationalisation of informational synergy, which leverages recent developments in the literature of data privacy. Our decomposition is defined for any number of information sources, and its atoms can be calculated using elementary optimisation techniques. The decomposition provides a natural coarse-graining that scales gracefully with the system’s size, and is applicable in a wide range of scenarios of practical interest.</jats:p