Classical randomized algorithms use a coin toss instruction to explore different evolutionary branches of a problem. Quantum algorithms, on the other hand, can explore multiple evolutionary branches by mere superposition of states. Discrete quantum random walks, studied in the literature, have nonetheless used both superposition and a quantum coin toss instruction. This is not necessary, and a discrete quantum random walk without a quantum coin toss instruction is defined and analyzed here. Our construction eliminates quantum entanglement from the algorithm, and the results match those obtained with a quantum coin toss instruction.
The spatial search problem on regular lattice structures in integer number of dimensions d ≥ 2 has been studied extensively, using both coined and coinless quantum walks. The relativistic Dirac operator has been a crucial ingredient in these studies. Here we investigate the spatial search problem on fractals of non-integer dimensions. Although the Dirac operator cannot be defined on a fractal, we construct the quantum walk on a fractal using the flip-flop operator that incorporates a Klein-Gordon mode. We find that the scaling behavior of the spatial search is determined by the spectral (and not the fractal) dimension. Our numerical results have been obtained on the well-known Sierpinski gaskets in two and three dimensions.
We investigate the spatial search problem on the two-dimensional square lattice, using the Dirac evolution operator discretised according to the staggered lattice fermion formalism. d = 2 is the critical dimension for the spatial search problem, where infrared divergence of the evolution operator leads to logarithmic factors in the scaling behaviour. As a result, the construction used in our accompanying article [6] provides an O( √ N ln N ) algorithm, which is not optimal. The scaling behaviour can be improved to O( √ N ln N ) by cleverly controlling the massless Dirac evolution operator by an ancilla qubit, as proposed by Tulsi [4]. We reinterpret the ancilla control as introduction of an effective mass at the marked vertex, and optimise the proportionality constants of the scaling behaviour of the algorithm by numerically tuning the parameters.
Random walks describe diffusion processes, where movement at every time step is restricted to only the neighbouring locations. We construct a quantum random walk algorithm, based on discretisation of the Dirac evolution operator inspired by staggered lattice fermions. We use it to investigate the spatial search problem, i.e. find a marked vertex on a d-dimensional hypercubic lattice. The restriction on movement hardly matters for d > 2, and scaling behaviour close to Grover's optimal algorithm (which has no restriction on movement) can be achieved. Using numerical simulations, we optimise the proportionality constants of the scaling behaviour, and demonstrate the approach to that for Grover's algorithm (equivalent to the mean field theory or the d → ∞ limit). In particular, the scaling behaviour for d = 3 is only about 25% higher than the optimal d → ∞ value. The spatial search problem is to find a marked object from an unsorted database of size N spread over distinct locations. Its characteristic is that one can proceed from any location to only its neighbours, while inspecting the objects. The problem is conventionally represented by a graph, with the vertices denoting the locations of objects and the edges labeling the connectivity of neighbours. Classical algorithms for this problem are O(N ), since they can do no better than inspect one location after another until reaching the marked object. On the other hand, quantum algorithms can do better in search problems by working with a superposition of states, Grover's algorithm being the prime example [1]. The spatial search problem has been a focus of investigation in recent years using different quantum algorithmic techniques, both analytical and numerical, and in a variety of spatial geometries ranging from a single hypercube to regular lattices [2][3][4][5][6][7]. These studies have mainly looked at the asymptotic scaling forms of the algorithms, and have not varied the database size N and its dimension d independently. In this work, we study the specific case of searching for a marked vertex on a d-dimensional hypercubic lattice with N = L d vertices. We let N and d be independent, as well as determine the scaling prefactors, and thereby develop a broad picture of how the dimension of the database (or the connectivity of the graph) influences the spatial search problem.The quantum algorithmic strategy for spatial search is to construct a Hamiltonian evolution, where the kinetic part of the Hamiltonian diffuses the amplitude distribution all over the lattice and the potential part of the Hamiltonian attracts the amplitude distribution toward * Electronic address: adpatel@cts.iisc.ernet.in † Electronic address: aminoorrahaman@yahoo.com the marked vertex [8]. The optimization criterion for the algorithm is to concentrate the amplitude distribution toward the marked vertex as quickly as possible. Grover constructed the optimal global diffusion operator, but it requires movement from any vertex to any other vertex in just one step. That corresponds to a fully co...
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