The Quantum Adiabatic Algorithm applied to random optimization problems: the quantum spin glass perspective

Among various algorithms designed to exploit the specific properties of quantum computers with respect to classical ones, the quantum adiabatic algorithm is a versatile proposition to find the minimal value of an arbitrary cost function (ground state energy). Random optimization problems provide a natural testbed to compare its efficiency with that of classical algorithms. These problems correspond to mean field spin glasses that have been extensively studied in the classical case. This paper reviews recent analytical works that extended these studies to incorporate the effect of quantum fluctuations, and presents also some original results in this direction.Comment: 151 pages, 21 figure

Quantum annealing of a hard combinatorial problem

Projecte Final de Màster Oficial fet en col.laboració amb el Departament de Física Fonamental, Facultat de Física,Universitat de BarcelonaWe present the numerical results obtained using quantum annealing (QA) in a hard combinatorial problem: the coloring problem (q-COL) of an Erd˝os-R´enyi graph. We first propose a quantum coloring Hamiltonian, natural extension of q-COL, based on the quantum Ising model in a transverse field. We then test several QA schemes and find the one that solves the highest number of graphs in the smallest number of iterations. Our results suggest that the computation time of QA scales exponentially in the size and it does not improve the results obtained by thermal annealing (TA) for q-COL

Quantum annealing of a hard combinatorial problem

Projecte Final de Màster Oficial fet en col.laboració amb el Departament de Física Fonamental, Facultat de Física,Universitat de BarcelonaWe present the numerical results obtained using quantum annealing (QA) in a hard combinatorial problem: the coloring problem (q-COL) of an Erd˝os-R´enyi graph. We first propose a quantum coloring Hamiltonian, natural extension of q-COL, based on the quantum Ising model in a transverse field. We then test several QA schemes and find the one that solves the highest number of graphs in the smallest number of iterations. Our results suggest that the computation time of QA scales exponentially in the size and it does not improve the results obtained by thermal annealing (TA) for q-COL

Approximate Approximation on a Quantum Annealer

Many problems of industrial interest are NP-complete, and quickly exhaust resources of computational devices with increasing input sizes. Quantum annealers (QA) are physical devices that aim at this class of problems by exploiting quantum mechanical properties of nature. However, they compete with efficient heuristics and probabilistic or randomised algorithms on classical machines that allow for finding approximate solutions to large NP-complete problems. While first implementations of QA have become commercially available, their practical benefits are far from fully explored. To the best of our knowledge, approximation techniques have not yet received substantial attention. In this paper, we explore how problems' approximate versions of varying degree can be systematically constructed for quantum annealer programs, and how this influences result quality or the handling of larger problem instances on given set of qubits. We illustrate various approximation techniques on both, simulations and real QA hardware, on different seminal problems, and interpret the results to contribute towards a better understanding of the real-world power and limitations of current-state and future quantum computing.Comment: Proceedings of the 17th ACM International Conference on Computing Frontiers (CF 2020

Coupling of hard dimers to dynamical lattices via random tensors

We study hard dimers on dynamical lattices in arbitrary dimensions using a random tensor model. The set of lattices corresponds to triangulations of the d-sphere and is selected by the large N limit. For small enough dimer activities, the critical behavior of the continuum limit is the one of pure random lattices. We find a negative critical activity where the universality class is changed as dimers become critical, in a very similar way hard dimers exhibit a Yang-Lee singularity on planar dynamical graphs. Critical exponents are calculated exactly. An alternative description as a system of `color-sensitive hard-core dimers' on random branched polymers is provided.Comment: 12 page

Critical phenomena in complex networks

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview a wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks. We also discuss strong finite size effects in these systems and highlight open problems and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references, extende

Liouville Field Theory of Fluctuating Loops

Effective field theories of two-dimensional lattice models of fluctuating loops are constructed by mapping them onto random surfaces whose large scale fluctuations are described by a Liouville field theory. This provides a geometrical view of conformal invariance in two-dimensional critical phenomena and a method for calculating critical properties of loop models exactly. As an application of the method, the conformal charge and critical exponents for two mutually excluding Hamiltonian walks on the square lattice are calculated.Comment: 4 RevTex pages, 1 eps figur