Quantum simulation and computation using discrete-time quantum walk [HBNI Th200]

Abstract

Quantum walk is a quantum analogue of classical random walk that has been extensively used for developing quantum algorithms for quantum simulations and quantum computation. The speed-up observed in the spread of probability distribution of quantum walk compared to its classical counterpart can be attributed to quantum phenomenon such as superposition, and coherence in position space of the walker (also referred as particle). In this thesis we have made use of unique features of the variants of discrete-time quantum walk namely, directed quantum walk, standard quantum walk, and split-step quantum walk for developing new protocols and efficient quantum circuits for quantum simulation. We will also present a new way of realizing universal quantum computation on a single particle quantum walk. Discrete-time quantum walk has been used for simulation of many quantum phenomenon. We present a scheme to incorporate acceleration into the dynamics of one- and two- particle discrete-time quantum walk in position space. We show the positive effect of acceleration on the entanglement between the particle and position space in single-particle quantum walk and in generation of entanglement between the two initially separated particle in two-particle quantum walk. By introducing the disorder in the form of phase operator we study the localization of the particle in the position space and then we show the transition from localization to de-localization as a function of acceleration. These inter- winding connection between acceleration, entanglement generation and localization along with well established connection of quantum walks with relativistic quantum mechanics and quantum field theory, one can use it to get a better understanding of system in high energy physics. Expansion of operational tools for quantum simulations and modelling the dynamics of accelerated particle is an other direction where accelerated quantum walk can play an important role. From the protocols based on discrete-time quantum walk and the work on accelerated quantum walk, the importance of evolution parameter is already established. The evolution parameter not only plays a major role in the dynamics of discrete-time quantum walk but in quantum simulation protocols too. It is mapped to the important quantities of the simulated system e.g., in simulation of Dirac cellular automata, it is mapped to the masses of the Dirac particle. Therefore, we have used quantum estimation theory to give an optimal probe for the estimation of evolution parameter of discrete-time quantum walk. This in turn will help in estimating the parameters associated with the dynamics of the system to be simulated on quantum walk. Our approach is based on the fact that in discrete-time quantum walk, the walkers coin space entangles with the position space after the very first step of the evolution. This phenomenon may be exploited to estimate the value of the evolution parameter by performing measurements on the sole position space of the walker. We find that the quantum Fisher information of the walker’s position space increases with time which, in turn, may be seen as a metrological resource. Discrete-time quantum walk also represents an important test case for the application of quantum computers. In this thesis we show the equivalence of the variants of discrete- time quantum walk for physical realizations. Using an appropriate digital mapping of the position state to the multi-qubit state of a quantum processor, we present different con- figurations of quantum circuits for the implementation of discrete-time quantum walks in one-dimensional position space. We have provided quantum circuits on five qubit quantum processor and addressed the scalability to higher dimensions. Quantum circuit for one such configuration has been implemented on an ion-trap quantum processor and quantum algorithm to simulate Dirac cellular automata is also realised for this configuration. Beyond its applications in quantum simulations and quantum algorithm, quantum walks itself has been regarded as a primitive to universal quantum computation. By using the operations required to describe the single particle discrete-time quantum walk on a position space we demonstrate the realization of the universal set of quantum gates on two and three-qubit systems. The idea is to utilize the effective Hilbert space of the single particle and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates are realized by exploiting the evolution operations on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum walk model for physical implementation. We have also discussed the scalability of the model and some propositions for using lesser number of qubits in realizing larger qubit systems