One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a lattice or graph containing a marked site. In most conceivable physical applications, however, one would rather expect to have multiple interacting particles. In bosonic systems at zero temperature, the dynamics would be well-described by a discrete non-linear Schrodinger equation. In this thesis we investigate the role of non-linearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. Our analytical results indicate that the search time for this non-linear quantum search scales with size of the database N like square root of N, equivalent to linear spatial search time. The analytical results will be compared with numerical calculations of multiple interacting quantum walkers.
One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a lattice or graph containing a marked site. In most conceivable physical applications, however, one would rather expect to have multiple interacting particles. In bosonic systems at zero temperature, the dynamics would be well-described by a discrete non-linear Schrodinger equation. In this thesis we investigate the role of non-linearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. Our analytical results indicate that the search time for this non-linear quantum search scales with size of the database N like square root of N, equivalent to linear spatial search time. The analytical results will be compared with numerical calculations of multiple interacting quantum walkers.
One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a graph containing a marked site. In many physical implementations, however, one might expect to have multiple particles. In interacting bosonic systems at zero temperature, the dynamics is well described by a discrete nonlinear Schr\"odinger equation. We investigate the role of nonlinearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. The analytical calculations reveal that the nonlinear search time scales with the size of the search space N like N, equivalent to the linear case, though with a different overall constant. The results indicate that interacting Bose-Einstein condensates at zero temperature could be natural systems for implementation of the quantum search algorithm.
Many-body quantum systems with effective nonlinearities have been shown to speed up quantum search on the complete graph, \textit{i.e.}, the combinatorial version of Grover's algorithm, at the expense of the number of particles needed for the effective nonlinearity to hold. Physically, however, data may not be arranged in an all-to-all network, and the task of searching incomplete graphs is the spatial search problem. We explore spatial search using a continuous-time nonlinear quantum walk on a variety of graphs. First, we consider incomplete graphs that are ``sufficiently complete'' so as to asymptotically search like the complete graph under a continuous-time (linear) quantum walk, which includes strongly regular graphs such as Paley graphs, regular graphs such as hypercubes, and irregular graphs such as complete bipartite graphs. For these sufficiently complete graphs, we analytically prove nonlinear speedups for Paley graphs and for complete bipartite graphs whose two partite sets both have size Θ(N), for suitable cubic and cubic-quintic nonlinearities, and we give numerical evidence for stronger nonlinearities and for hypercubes. Second, we explore arbitrary-dimensional cubic lattices, and we numerically show that certain nonlinearities speed up search on sufficiently high dimensional lattices. Thus, nonlinear quantum search can remain viable even when the underlying graph is incomplete.
Spatial search by continuous-time quantum walks on complex networks is focused on using a quantum walk in continuous time in order to find a single or multiple marked vertices within the complex network. The specific formalism used here is to consider a coupling constant that shifts the state of the quantum walker from the initial state to the target state, which is the marked vertex. The thesis begins with establishing the mathematical framework of network theory, quantum walks and numerical methods that will be used in the remainder of the thesis. Then spatial search by continuous-time quantum walk is studied on regular and semi-regular graphs, where most analytical results can be found. This will get us acquainted with spatial search by quantum walk. The complex networks studied are Barabasi-Albert graphs and the Internet network on the level of autonomous systems. Different renormalized and pruned versions of the Internet network are studied. The parameters of the quantum walk that are focused on are the optimal values for the coupling constant, success probability, time and search time.
A quantum walk on a lattice is a paradigm of a quantum search in a database. The database qubit strings are the lattice sites, qubit rotations are tunneling events, and the target site is tagged by an energy shift. For quantum walks on a continuous time, the walker diffuses across the lattice and the search ends when it localizes at the target site. The search time T can exhibit Grover's optimal scaling with the lattice size N , namely, T ∼ N , on an all-connected, complete lattice. For finite-range tunneling between sites, instead, Grover's optimal scaling is warranted when the lattice is a hypercube of d > 4 dimensions. Here, we show that Grover's optimum can be reached in lower dimensions on lattices of long-range interacting particles, when the interaction strength scales algebraically with the distance r as 1 / r α and 0 < α < 3 d / 2 . For α < d the dynamics mimics the one of a globally connected graph. For d < α < d + 2 , the quantum search on the graph can be mapped to a short-range model on a hypercube with spatial dimension d s = 2 d / ( α − d ) , indicating that the search is optimal for d s > 4 . Our work identifies an exact relation between criticality of long-range and short-range systems, it provides a quantitative demonstration of the resources that long-range interactions provide for quantum technologies, and indicates when existing experimental platforms can implement efficient analog quantum search algorithms.
Abstract The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π N / ( 2 2 ) steps, where N is the number of vertices of the Johnson graph.
Quantum-walk-based algorithms that search a marked location among N locations on a d-dimensional lattice succeeds in time O(N) for d>2, while this is not found to be possible when d=2. In this paper, we consider a spatial search algorithm using continuous-time quantum walk on a two-dimensional square lattice with the existence of additional long-range edges. We examined such a search on a probabilistic graph model where an edge connecting non-nearest-neighbor lattice points i and j apart by a distance $|i\ensuremath{-}j|$ is added by probability ${p}_{ij}={|i\ensuremath{-}j|}^{\ensuremath{-}\ensuremath{\alpha}}\phantom{\rule{1em}{0ex}}(\ensuremath{\alpha}\ensuremath{\ge}0)$. Through numerical analysis, we found that the search succeeds in time O(N) when $\ensuremath{\alpha}\ensuremath{\le}{\ensuremath{\alpha}}_{c}=2.4\ifmmode\pm\else\textpm\fi{}0.1$. For $\ensuremath{\alpha}>2$, the expectation value of the additional long-range edges on each node scales as a constant when $N\ensuremath{\rightarrow}\ensuremath{\infty}$, which means that search time of O(N) is achieved on a graph with average degree scaling as a constant.