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arXiv:quant-ph/0405120 · Physical Review A

Spatial search and the Dirac equation

Andrew M. Childs, Jeffrey Goldstone

Published Oct 19, 2004• 145 citations
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Abstract

We consider the problem of searching a $d$-dimensional lattice of $N$ sites for a single marked location. We present a Hamiltonian that solves this problem in time of order $\sqrt{N}$ for $d>2$ and of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\mathrm{log}\phantom{\rule{0.3em}{0ex}}N$ in the critical dimension $d=2$. This improves upon the performance of our previous quantum walk search algorithm (which has a critical dimension of $d=4$), and matches the performance of a corresponding discrete-time quantum walk algorithm. The improvement uses a lattice version of the Dirac Hamiltonian, and thus requires the introduction of spin (or coin) degrees of freedom.

DOI
10.1103/physreva.70.042312

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We consider the problem of searching a ddd-dimensional lattice of NNN sites for a single marked location. We present a Hamiltonian that solves this problem in time of order N\sqrt{N}N​ for d>2d>2d>2 and of order NlogN\sqrt{N}\phantom{\rule{0.3em}{0ex}}\mathrm{log}\phantom{\rule{0.3em}{0ex}}NN​logN in the critical dimension d=2d=2d=2. This improves upon the performance of our previous quantum walk search algorithm (which has a critical dimension of d=4d=4d=4), and matches the performance of a corresponding discrete-time quantum walk algorithm. The improvement uses a lattice version of the Dirac Hamiltonian, and thus requires the introduction of spin (or coin) degrees of freedom.
We consider the problem of searching a general ddd-dimensional lattice of NNN vertices for a single marked item using a continuous-time quantum walk. We demand locality, but allow the walk to vary periodically on a small scale. By constructing lattice Hamiltonians exhibiting Dirac points in their dispersion relations and exploiting the linear behavior near a Dirac point, we develop algorithms that solve the problem in a time of O(N)O(\sqrt{N})O(N​) for d>2d>2d>2 and O(NlogN)O(\sqrt{N}\phantom{\rule{0.16em}{0ex}}logN)O(N​logN) in d=2d=2d=2. In particular, we show that such algorithms exist even for hypercubic lattices in any dimension. Unlike previous continuous-time quantum walk algorithms on hypercubic lattices in low dimensions, our approach does not use external memory.
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=2d=2d=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)O(\sqrt{N}\mathrm{ln}N)O(N​lnN) algorithm, which is not optimal. The scaling behavior can be improved to O(NlnN)O(\sqrt{N\mathrm{ln}N})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.
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.
We consider the problem of finding a desired item out of NNN items arranged on the sites of a two-dimensional lattice of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$. The previous quantum-walk based algorithms take O(N ln N)O(\sqrt{N}\text{ }\text{ln}\text{ }N)O(N​ ln N) steps to solve this problem, and it is an open question whether the performance can be improved. We present an algorithm which solves the problem in O(N ln N)O(\sqrt{N\text{ }\text{ln}\text{ }N})O(N ln N​) steps, thus giving an O(ln N)O(\sqrt{\text{ln}\text{ }N})O(ln N​) improvement over the known algorithms. The improvement is achieved by controlling the quantum walk on the lattice using an ancilla qubit.
Random walks describe diffusion processes, where movement at every time step is restricted to only the neighboring locations. We construct a quantum random walk algorithm, based on discretization of the Dirac evolution operator inspired by staggered lattice fermions. We use it to investigate the spatial search problem, that is, to find a marked vertex on a ddd-dimensional hypercubic lattice. The restriction on movement hardly matters for d>2d>2d>2, and scaling behavior close to Grover's optimal algorithm (which has no restriction on movement) can be achieved. Using numerical simulations, we optimize the proportionality constants of the scaling behavior, and demonstrate the approach to that for Grover's algorithm (equivalent to the mean-field theory or the $d\ensuremath{\rightarrow}\ensuremath{\infty}$ limit). In particular, the scaling behavior for d=3d=3d=3 is only about 2525%25 higher than the optimal $d\ensuremath{\rightarrow}\ensuremath{\infty}$ value.
Quantum walks have been very useful in developing search algorithms in quantum information, in particular for devising of spatial search algorithms. However, the construction of continuous-time quantum search algorithms in two-dimensional lattices has proved difficult, requiring additional degrees of freedom. Here, we demonstrate that a continuous-time quantum walk search is possible in two dimensions by changing the search topology to a graphene lattice, utilizing the Dirac point in the energy spectrum. This is made possible by making a change to standard methods of marking a particular site in the lattice. Various ways of marking a site are shown to result in successful search protocols. We further establish that the search can be adapted to transfer probability amplitude across the lattice between specific lattice sites thus establishing a line of communication between these sites.