4 research outputs found
Search on a Hypercubic Lattice using a Quantum Random Walk: I. d>2
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. finding a marked vertex on a -dimensional hypercubic
lattice. The restriction on movement hardly matters for , 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 limit). In particular, the scaling behaviour for is only
about 25% higher than the optimal value.Comment: 11 pages, Revtex (v2) Introduction and references expanded. Published
versio
Search on a Hypercubic Lattice through a Quantum Random Walk: II. d=2
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. 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 \cite{dgt2search} provides an
algorithm, which is not optimal. The scaling behaviour can
be improved to by cleverly controlling the massless Dirac
evolution operator by an ancilla qubit, as proposed by Tulsi \cite{tulsi}. 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.Comment: Revtex4, 5 pages (v2) Introduction and references expanded. Published
versio
Search on a Hypercubic Lattice using Quantum Random Walk
We investigate the spatial search problem on the two-dimensional square lattice, using the Dirac evolution operator discretized 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 behavior. As a result, the construction used in our accompanying article [ A. Patel and M. A. Rahaman Phys. Rev. A 82 032330 (2010)] provides an O(√NlnN) algorithm, which is not optimal. The scaling behavior can be improved to O(√NlnN) by cleverly controlling the massless Dirac evolution operator by an ancilla qubit, as proposed by Tulsi Phys. Rev. A 78 012310 (2008). We reinterpret the ancilla control as introduction of an effective mass at the marked vertex, and optimize the proportionality constants of the scaling behavior of the algorithm by numerically tuning the parameters
Search on a hypercubic lattice using a quantum random walk. II. d = 2
We investigate the spatial search problem on the two-dimensional square lattice, using the Dirac evolution operator discretized 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 behavior. As a result, the construction used in our accompanying article A. Patel and M. A. Rahaman, Phys. Rev. A 82, 032330 (2010)] provides an O(root N ln N) algorithm, which is not optimal. The scaling behavior can be improved to O(root N ln N) by cleverly controlling the massless Dirac evolution operator by an ancilla qubit, as proposed by Tulsi Phys. Rev. A 78, 012310 (2008)]. We reinterpret the ancilla control as introduction of an effective mass at the marked vertex, and optimize the proportionality constants of the scaling behavior of the algorithm by numerically tuning the parameters