arXiv:1312.3852 · Physical Review Letters

Quantum Search on Graphene Lattices

Iain Foulger, Sven Gnutzmann∥, Gregor Tanner

Published Feb 21, 2014• 33 citations

Abstract

We present a continuous-time quantum search algorithm on a graphene lattice. This provides the sought-after implementation of an efficient continuous-time quantum search on a two-dimensional lattice. The search uses the linearity of the dispersion relation near the Dirac point and can find a marked site on a graphene lattice faster than the corresponding classical search. The algorithm can also be used for state transfer and communication.

DOI: 10.1103/physrevlett.112.070504

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arXiv:1312.3852 · Physical Review Letters

Quantum Search on Graphene Lattices

Iain Foulger, Sven Gnutzmann∥, Gregor Tanner

Published Feb 21, 2014• 33 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.112.070504

Related papers

Quantum walks and quantum search on graphene lattices

Iain Foulger, Sven Gnutzmann∥, Gregor Tanner

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.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.

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Simon Apers, Shantanav Chakraborty, Leonardo Novo +1 more

Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.

arXiv:2112.12746Physical Review LettersOct 14, 202265 citations

We consider the problem of searching a general

d

-dimensional lattice of

N

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(\sqrt{N})

for

d>2

and

O(\sqrt{N}\phantom{\rule{0.16em}{0ex}}logN)

d=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.

Quantum-walk-based algorithms that search a marked location among

N

locations on a

d

-dimensional lattice succeeds in time

O(\sqrt{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(\sqrt{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(\sqrt{N})

is achieved on a graph with average degree scaling as a constant.

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.