A quantum-walk-inspired adiabatic algorithm for solving graph isomorphism problems

Abstract

We present a 2-local quantum algorithm for graph isomorphism GI based on an adiabatic protocol. By exploiting continuous-time quantum-walks, we are able to avoid a mere diffusion over all possible configurations and to significantly reduce the dimensionality of the visited space. Within this restricted space, the graph isomorphism problem can be translated into the search of a satisfying assignment to a 2-SAT formula without resorting to perturbation gadgets or projective techniques. We present an analysis of the execution time of the algorithm on small instances of the graph isomorphism problem and discuss the issue of an implementation of the proposed adiabatic scheme on current quantum computing hardware.

Quantum Hash function and its application to privacy amplification in quantum key distribution, pseudo-random number generation and image encryption

“…In this paper, we construct a quantum Hash function (QHF) by subtly modifying the quantum walks (QW) model 5 6 7 8 9 10 11 12 13 and it can be used for the privacy amplification process of QKD systems with higher security by means of the physical principles of quantum mechanics. As a byproduct, QHF can also be used for pseudo-random number generation due to its inherent chaotic dynamics and further we propose a novel QHF-based image encryption algorithm.…”

An Enhanced Photonic Quantum Finite Automaton

“…In our implementation, a given input word is accepted by the automaton, with a given error probability, whenever a single photon arrives at the output of the device with a specific polarization. In particular, the experimental realization, based on the manipulation of single-photon polarization and photodetection, has demonstrated the possibility of building small quantum computational component to be embedded in more sophisticated and precise quantum finite automata or also in other computational systems and approaches [13][14][15]. Albeit the photonic automaton realized in [6] is fed with single photons, it works in a regime where polarized laser pulses (coherent states) are enough, up to detecting the intensity of the output signals instead of counting the number of photons successfully passing through the device with a given polarization (see in [6] for details).…”

Quantum-classical dynamical distance and quantumness of quantum walks

“…In these contexts, in order to understand the very nature of the underlying dynamics, the question often arises how to compare and assess the different behaviors of classical and quantum walks on a given structure. Quantum walks are also very useful to build quantum algorithms [5][6][7][8], and a comparison with the corresponding classical random walks is crucial to assess the possible quantum enhancement due to the faster spreading of probability distributions. As a consequence, the differences between a classical and a quantum walk have been analyzed quite extensively, with short-and long-time behavior studied in both scenarios [9][10][11][12][13][14].…”