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We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential…
In this article, we give a first example of a pair of quantum isomorphic, non-isomorphic strongly regular graphs, that is, non-isomorphic strongly regular graphs having the same homomorphism counts from all planar graphs. The pair consists…
Quantum annealing has emerged as a powerful platform for simulating and optimizing classical and quantum Ising models. Quantum annealers, like other quantum and/or analog computing devices, are susceptible to nonidealities including…
In contrast to the usual quantum systems which have at most a finite number of open spectral gaps if they are periodic in more than one direction, periodic quantum graphs may have gaps arbitrarily high in the spectrum. This property of…
Classical and quantum annealing is discussed for a kinetically constrained chain of $N$ non-interacting asymmetric double wells, represented by Ising spins in a longitudinal field $h$. It is shown that in certain cases, where the kinetic…
Computational advantages gained by quantum algorithms rely largely on the coherence of quantum devices and are generally compromised by decoherence. As an exception, we present a quantum algorithm for graph isomorphism testing whose…
We present an efficient quantum algorithm for some independent set problems in graph theory, based on non-abelian adiabatic mixing. We illustrate the performance of our algorithm with analysis and numerical calculations for two different…
Quantum computation offers exciting new possibilities for statistics. This paper explores the use of the D-Wave machine, a specialized type of quantum computer, which performs quantum annealing. A general description of quantum annealing…
We summarize different approaches to the theory of quantum graphs and provide several ways to construct concrete examples. First, we classify all undirected quantum graphs on the quantum space $M_2$. Secondly, we apply the theory of…
Determining whether two graphs are structurally identical is a fundamental problem with applications spanning mathematics, computer science, chemistry, and network science. Despite decades of study, graph isomorphism remains a challenging…
We present a general error-correcting scheme for quantum annealing that allows for the encoding of a logical qubit into an arbitrarily large number of physical qubits. Given any Ising model optimization problem, the encoding replaces each…
Significant efforts are being directed towards developing a quantum annealer capable of solving combinatorial optimization problems. The challenges are Hamiltonian programming and large-scale implementations. Here we report quantum…
We perform an in-depth comparison of quantum annealing with several classical optimisation techniques, namely thermal annealing, Nelder-Mead, and gradient descent. We begin with a direct study of the 2D Ising model on a quantum annealer,…
Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that play an important role in networking, computational biology, and biochemistry. We show an explicit construction…
We address quantum systems isospectral to the harmonic oscillator, as those found within the framework of supersymmetric quantum mechanics, as potential resources for continuous variable quantum information. These deformed oscillator…
Graph structure of quantum spin networks plays an essential role in applying quantum annealing (QA). The Ising model is the typical choice to describe the interactions between the spins in the networks. Here, we explore the interplay of…
We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built…
We present an algorithm for learning a latent variable generative model via generative adversarial learning where the canonical uniform noise input is replaced by samples from a graphical model. This graphical model is learned by a…
Given the extensive application of classical random walks to classical algorithms in a variety of fields, their quantum analogue in quantum walks is expected to provide a fruitful source of quantum algorithms. So far, however, such…
Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their…