Quantum walks on networks – A paradigm for quantum simulation and computation [HBNI Th231]

Abstract

Quantum walks are the quantum generalization of classical random walks, and form a powerful yet versatile toolkit for the development of quantum algorithms for quantum simulation and quantum computing applications. One of the most significant differences between quantum walks and classical random walks is the spreading rate of the resulting probability distribution in the position space. A quantum walker spreads quadratically faster than a classical random walker due to quantum phenomena like superposition and interference. The probability distribution of a quantum walk can be controlled and mod- ified by careful choice of evolution operators. This highlights the feasibility of using quantum walk-based approaches for development of quantum algorithms as well as mod- eling dynamics in various quantum systems. <br><br> This thesis reports the utilization of quantum walks on networks as models for the de- sign of quantum algorithms, and for applications in quantum simulation and computation. Quantum walks allow an encoding of a network structure in their position Hilbert space, and this provides one with an additional degree of freedom to tune the dynamics of a quantum walker. We use the continuous-time quantum walk to model the percolation of a quantum particle on a lattice, and report a comparison between percolation on regular and quasicrystalline lattices in two dimensions. Discrete-time quantum walks were used to develop an extension of the classical PageRank algorithm for quantum networks. The results and techniques from both the previous studies were collated and used to study the properties of aromatic molecules. We also discuss a paradigm where single-particle quan- tum walks on networks may be used to design a protocol to achieve multi-qubit universal quantum computation. The tuning of parameters and choice of quantum walk operators is also discussed in the context of designing quantum random number generators capable of generating multi-bit random numbers. <br><br>