Related papers: Approximate locality for quantum systems on graphs
Quantum walk algorithms can speed up search of physical regions of space in both the discrete-time [arXiv:quant-ph/0402107] and continuous-time setting [arXiv:quant-ph/0306054], where the physical region of space being searched is modeled…
Quantum walks provide a natural framework to approach graph problems with quantum computers, exhibiting speedups over their classical counterparts for tasks such as the search for marked nodes or the prediction of missing links.…
Fault-tolerant quantum computing requires classical hardware to perform the decoding necessary for error correction. The Union-Find decoder is one of the best candidates for this. It has remarkably organic characteristics, involving the…
A new approach to quantum walks is presented. Considering a quantum system undergoing some unitary discrete-time evolution in a directed graph G, we think of the vertices of G as sites that are occupied by the quantum system, whose internal…
We derive an intuitive and novel method to represent nodes in a graph with special unitary operators, or quantum operators, which does not require parameter training and is competitive with classical methods on scoring similarity between…
We consider a graph with a single quantum system at each node. The entire compound system evolves in discrete time steps by iterating a global evolution $U$. We require that this global evolution $U$ be unitary, in accordance with quantum…
Continuous-time quantum walks (CTQWs) play a crucial role in quantum computing, especially for designing quantum algorithms. However, how to efficiently implement CTQWs is a challenging issue. In this paper, we study implementation of CTQWs…
The Quantum Approximate Optimization Algorithm can naturally be applied to combinatorial search problems on graphs. The quantum circuit has p applications of a unitary operator that respects the locality of the graph. On a graph with…
The interest in quantum walks has been steadily increasing during the last two decades. It is still worth to present new forms of quantum walks that might find practical applications and new physical behaviors. In this work, we define…
For a locally compact group $G$, let $A^n(G)$ denote the multidimensional Fourier algebra given by $ \otimes_{n}^{eh} A(G).$ This work explores the approximation identity and operator amenability of the algebra $A^n(G)$. Further, we study…
We analyze effective approximation of unitary matrices. In our formulation, a unitary matrix is represented as a product of rotations in two-dimensional subspaces, so-called Givens rotations. Instead of the quadratic dimension dependence…
Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity,…
We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an…
In this paper we show that inverses of well-conditioned, finite-time Gramians and impulse response matrices of large-scale interconnected systems described by sparse state-space models, can be approximated by sparse matrices. The…
The quantum approximate optimization algorithm (QAOA) has become a cornerstone of contemporary quantum applications development. Here we show that the \emph{density} of problem constraints versus problem variables acts as a performance…
We initiate the study of approximation algorithms and computational barriers for constructing sparse $\alpha$-navigable graphs [IX23, DGM+24], a core primitive underlying recent advances in graph-based nearest neighbor search. Given an…
The global macroscopic behaviour of a dynamical system is encoded in the eigenfunctions of a certain transfer operator associated to it. For systems with low dimensional long term dynamics, efficient techniques exist for a numerical…
The goal of the paper is to lay the foundation for the qualitative analogue of the classical, quantitative sparse graph limit theory. In the first part of the paper we introduce the qualitative analogues of the Benjamini-Schramm and…
The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the…
We investigate numerically efficient approximations of eigenspaces associated to symmetric and general matrices. The eigenspaces are factored into a fixed number of fundamental components that can be efficiently manipulated (we consider…