Quantum walks and quantum search on graphene lattices

2.1 MB·English

by  Iain Foulger

Quantum walks and quantum search on graphene lattices
Iain Foulger, Sven Gnutzmann, and Gregor Tanner
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
Quantum walks have been very useful in developing search algorithms in quantum information,
inparticularfordevisingofspatialsearchalgorithms. However,theconstructionofcontinuous-time
quantum search algorithms in two-dimensional lattices has proved difficult, requiring additional
degrees of freedom. Here, we demonstrate that continuous-time quantum walk search is possible in
two-dimensions by changing the search topology to a graphene lattice, utilising the Dirac point in
theenergyspectrum. Thisismadepossiblebymakingachangetostandardmethodsofmarkinga
particularsiteinthelattice. Variouswaysofmarkingasiteareshowntoresultinsuccessfulsearch
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.
5
PACSnumbers: 03.67.Hk,72.80.Vp,03.65.Sq,03.67.Lx
1
0
2
I. INTRODUCTION ingtwo-dimensionalcontinuous-timequantumsearching,

l butalsopavesthewayforlookingfornoveleffectsinthe
u
J Quantum walks have been a particularly fruitful field material graphene.
1 of research in quantum information going back to ideas Inthispaper,weofferadetailedaccountofthetheory
from Feynman [1], Meyer [2], and Aharonov [3]. One and numerical results thus expanding on the findings as

] focus of recent work on quantum walks is their applica- presented in [10]. Generalisations of the results in [10]
h
p tion to spatial search algorithms. It is well-known that have been given in [11], where a approach to solving the
- Grover’s algorithm [4, 5] for searching on unstructured problem has been taken through encoding extra degrees
nt databases offers a quadratic improvement in search time of freedom into crystal lattices. Recent experiments on
a over classical models. Grover’salgorithm isnot designed artificial microwave graphene [12] have shown that ad-
u to search physical systems where only local operations ditional site perturbations, as discussed in Sec. VII, can
q arepossibleandquantumwalkalgorithmshavebeenem- be used to create a search protocol. Searching on hon-
[ ployed for this purpose. In developing such algorithms, eycomb lattices in discrete-time setting has been consid-
2 themajorconsiderationisthesearchtimeattemptingto ered in [13]; however, no improvement over discrete time
v match the quadratic improvement over the classical case searches on cubic lattices was found.
3 offeredbyGrover’salgorithm. Thefirstspatialsearchal- The paper is structured as follows: In Section II we
4 gorithmtodothis,usingthestandardmodelofquantum giveanintroductiontotheformalismofcontinuous-time
5
walks, was developed by Shenvi, Kempe and Whaley [6] quantum walks and the construction of quantum walk
7
0 for searches on a hypercube. search algorithms. We also explain why previous search
algorithms struggled on lower-dimensional lattices and
. While there exist discrete-time searches on d-
1
why graphene offers a solution. Section III will detail
dimensional cubic lattices which are faster than classical
0
the relevant properties of graphene and set-up the no-
5 searchesford≥2[7],effectivecontinuous-timequantum
tation we shall use throughout. Sections IV & V con-
1 searches only exist for d ≥ 4 [8] or else they require ad-
: ditionalmemory(intheformofspindegreesoffreedom) tain our main analytical results, where we detail the
v
specifics of our search algorithm and offer an analysis
in order to improve their search time in lower dimen-
i
X sions[9]. In[10],wedemonstratedthateffectivesearches of the search running time and success probability. We
shallthenshowinSectionVIhowthissearchprotocolcan
r over two-dimensional lattices may be achieved in an ar-
a be adapted to demonstrate novel communication setups.
guably simpler way which does not require extra degrees
SectionsVII&VIIIcontainnumericalworkdemonstrat-
of freedom, and could, therefore, be viewed as more effi-
ing the possibility of using alternative methods of mark-
cient. This is achieved through the choice of a different
ing to create search behaviour and other carbon nanos-
lattice, specifically, a honeycomb lattice which is the un-
tructures. We conclude with a review and discussion of
derlyinglatticestructureofcarbonatomsinthematerial
our results in Section IX.
graphene.
The association with graphene is important as, al-
though we first study a purely theoretical problem in
quantum information, the use of a graphene lattice also II. QUANTUM WALKS AND SEARCHING
offers a potential physical realisation. We thus envis-
age using the quantum walk and quantum search algo- Continuous-time quantum walks (CTQW), first de-
rithmframeworktoinvestigatetheeffectofperturbations fined in [14], are the quantum analogue of continuous-
on the dynamics on graphene and other carbon struc- time Markov chains. They are defined purely on the
tures. Thisoffersnotonlythepossibilityfordemonstrat- state space, that is, the Hilbert space H , spanned by
p
2
the states |j(cid:105) which represent the jth site of the lattice. and that the corresponding eigenstates are of the form
√
Thus, the time-evolution of such systems is defined by (|s(cid:105)±|w(cid:105))/ 2. By preparing the system initially in |s(cid:105)
the Sch¨odinger equation and allowing it to evolve for a set period of time T =
√
π/∆E ∝ N ,ameasurementofthesystemwillresultin
N
d (cid:88) thestate|w(cid:105)beingmeasuredwithhighprobability. This
α (t)=−i H α (t) , (1)
dt j jl l explains the speed-up of the quantum walk search. The
l=1
Grover algorithm and its speed-up can be understood in
where αj = (cid:104)j|ψ(t)(cid:105) is the probability amplitude at analogous terms and one can show that all other states
the jth vertex of a system described by the state vector are indeed energetically very well separated.
|ψ(t)(cid:105), and H is the Hamiltonian describing the connec- In the present case this separation only holds above
tivityofthelattice. Notethatweareusingadimension- a critical dimension d = 4. A simple argument for the
less description setting e.g. (cid:126)=1. criticaldimensioncancbeobtainedbycomparingthescal-
The dynamics of a quantum walk over a network is ing behaviour of the energy splitting ∆E ∼ N−1/2 with
defined by the nature of the interaction between con- the energy separation of the first excited state from the
nected sites. Therefore, the Hamiltonian is generally ground state in the unperturbed lattice. For a cubic lat-
constructed from the adjacency matrix of the underly- ticewehaveaquadraticdispersionrelationwhichallows
ing lattice. The adjacency matrix A is defined as us to estimate the energy separation
(cid:26)
1 if j and l are connected
A = (2) E(k)−E ∼|k|2 ∼N−2/d
jl 0 if j and l are not connected. 0
Typically, the Hamiltonian is chosen as H = (cid:15) I+vA. (as |k| ∼ N−1/d for the first excited state in a d-
D
The parameter v determines the coupling strength be- dimensionallattice). Ford>4onethenhas∆E/(E(k)−
tweenconnectedsitesandtheparameter(cid:15)D isanon-site E0)→0asN →∞andadetailedanalysisindeedproves
energythatonlyentersthedynamicsinatrivialwayand thatthequantumwalksearchworkswithoptimalspeed-
thus can be set to a desired value. If v = −1 and (cid:15) is up [8]. At the critical dimension d = 4 the two energy
D
equaltothevalencyofthelatticeHisthediscreteLapla- scalesscaleinthesameway–thedetailedanalysisshows
cian. This form of Hamiltonian is closely related to the thatthesearchstillworksbutthedynamicsismorecom-
tight-binding model for condensed matter systems [15]. plicated due to the interference of excited states. While
Asfirstexplainedin[16,17], aquantumwalkistrans- there is still a speed-up it is only almost optimal; the
formed into a search protocol by introducing a localised optimal search time T ∝ N1/2 gets multiplied with a
perturber state, forming an avoided crossing in the spec- logarithmicallyincreasingfactor. Fortheexperimentally
trumofthesearchHamiltonianbetweenanunperturbed relevantregimesof2-and3-dimensionalcubiclattices,all
eigenstate and the localised perturber. Thus, initialising statesparticipateinthedynamicsasanyavoidedcrossing
thesystem inthe unperturbedeigenstate involved inthe gets dissolved completely as the number of sites grows.
crossing and allowing the system to evolve in time, one As a result no speed-up over classical searches is found.
findsthatthesystemrotatesintothelocalisedperturber The above estimate offers a simple way how to reduce
state. the critical dimension. If one can construct a search
The first CTQW search over d-dimensional cubic lat- around a uniformly distributed state at an energy E
0
tices[8]introducedthelocalisedperturberstateasapro- wherethedispersionrelationisconic(linearin|k|), then
jector onto a single site w, resulting in the search Hamil- in d dimensions
tonian
E−E ∼|k|∼N1/d
0
H =−γA+|w(cid:105)(cid:104)w| . (3)
γ
which results in a critical dimension d = 2. In [9]
Here, γ is a parameter governing the strength of inter- c
this was implemented using a modified Dirac Hamilto-
actions between sites in the lattice. This parameter is
nian. However, this requires the addition of a spin de-
chosen carefully [8] such that the perturber state |w(cid:105) is
gree of freedom, essentially a doubling of memory, with
brought into resonance with the ground state of the un-
the added complication that it is not immediately clear
perturbed Hamiltonian, the uniform superposition |s(cid:105).
how such a system would be physically realised.
Let us assume for the moment that all other unper-
Instead, oursolutionhereistochangethelatticefrom
turbedeigenstatesareenergeticallysufficientlyseparated
a square (cubic) to a graphene lattice. This change of
fromthegroundstate–wewillcomebacktothisassump-
topology automatically implies the first step in our so-
tion later. Then there will be an avoided crossing of two
lution, as one of the important electronic features of
eigenvaluescorrespondingtotheperturberstateandthe
grapheneistheconicdispersionrelationaroundtheDirac
groundstate. Perturbationtheoryestimatesthattheen-
energy. This arises naturally from the tight-binding de-
ergy splitting of these resonant states is proportional to
scriptions of graphene. Note that graphene has the crit-
the overlap of the localised perturber state |w(cid:105) and the
ical dimension d = 2. Indeed our construction as pre-
uniform ground state |s(cid:105), which scales as
sented in [10] has an almost optimal speed-up with log-
∆E ∼|(cid:104)w|s(cid:105)|∼N−1/2, (4) arithmic corrections that need to be evaluated in a de-
3
FIG. 1. Left: Graphene with lattice vectors a , translation
1/2
vectors δ and unit cell (dashed lines). Right: Reciprocal
i
latticewithbasisvectorsb ,symmetrypointsΓ,K,K(cid:48),M
1/2
and first Brillouin zone (hexagon).
tailed analysis that goes beyond the simple perturbative
description given above. FIG. 2. (Color online) Dispersion relation for infinite
graphene sheet ((cid:15) =0 and v=−1).
D
III. RELEVANT PROPERTIES OF GRAPHENE
with a reduced density of states, a necessary feature for
the creation of the search dynamics.
Graphene is a single layer of carbon atoms arranged
As the lattice possesses a translational symmetry the
in a honeycomb lattice. The lattice is bipartite with two
Hamiltonian can be solved using linear superpositions
sublattices, labelled A and B, and a unit cell containing
of Bloch functions over both sublattices. As a basis we
two carbon atoms. The spatial and reciprocal lattices use the orthonormal states {|α,β(cid:105)A,|α,β(cid:105)B} to denote
are shown in Figure 1. The primitive vectors describing
states on either the A or B sublattice in the cell at posi-
the lattice are a , such that the position of a unit cell
1(2) tionR(α,β). Forthemajorityofwhatfollows(exceptin
in the lattice is given by R(α,β) = αa +βa . We use
1 2 Section VIII), we will focus on finite-sized lattices with
dimensionless units in space where the distance between
assumed periodic boundary conditions along the axes of
nearest neighbor sites is a = 1. The reciprocal lattice
bothprimitivevectorssothatthetopologyofourlattice
shows two important points, the Dirac points K and K(cid:48)
isatorus;thatis,ourwavefunctionisofthegeneralform
at the two inequivalent corners of the Brillouin zone. (cid:16) (cid:17)
|ψ(cid:105) = (cid:80)m (cid:80)n ψA |α,β(cid:105)A+ψB |α,β(cid:105)B . Our
Theenergyspectrumofelectronsingraphenewasfirst α=1 β=1 α,β α,β
derivedbyWallace[18]whenconsideringthebandstruc- boundary conditions imply that the state vector must
ture of graphite using a tight-binding Hamiltonian satisfyψA(B) =ψA(B) =ψA(B) wherem,ndenotethe
α,β α+m,β α,β+n
period of the lattice. Thus, the wavefunctions on the
H=(cid:15) 1+vA (5)
D torus take the Bloch function form
(cid:20)
where (cid:15)D is the on-site energy (which we will identify |ψ(cid:105)= (cid:88) √1 eik·R(α,β)|α,β(cid:105)A
as the energy of the Dirac points) and v the hopping N
strength (both dimensionless in our setting). The tight- (α,β)
(cid:21)
binding model for graphene and the derivation of the +C√(k)eik·(R(α,β)+δ1)|α,β(cid:105)B , (8)
solution are well-known [19, 20] and give rise to the dis- N
persion relation
where N is the number of sites in the lattice, k is the
(cid:15)(k)=(cid:15) ± (6) momentum, and C(k) is a relative phase contribution
D
v(cid:118)(cid:117)(cid:117)(cid:116)1+4cos2(cid:18)kx(cid:19)+4cos(cid:18)kx(cid:19)cos(cid:32)√3ky(cid:33), dtieopnenodrevnatleunpcoenbwahnedtsh;eirttchaenstbaetecablecluolnagtsedtobtyheexcpolnicdiutlcy-
2 2 2 working through the tight-binding model.
Applicationoftheperiodicboundaryconditionsresults
in the following quantised momenta
shown in Figure 2 for an infinite graphene lattice.
As there are two atoms per unit cell the spectrum has 2πp 1 (cid:18)4πq (cid:19)
two branches, the upper branch being the conduction kx = m , ky = √3 n −kx , (9)
band and the lower the valence band, which meet at the
corners of the Brillouin zone, the K-points. The energy where p ∈ {0,1,...m−1} and q ∈ {0,1,...n−1}. In
at the K-points is (cid:15)D which we name the Dirac energy. the following and whenever we consider quantum walk
Itisaroundthesepoints, thatthebehaviourofthespec- dynamicsonatorus,thenumberofcellsineachdirection
trum is conical, that is, (cid:113)
isgenerallychosentobethesame,thatis,m=n= N.
√ √ 2
3(cid:113) 3 This choice is purely for simplifying the notation; alter-
(cid:15)(k)≈(cid:15)D±v 2 δkx2+δky2 =(cid:15)D±v 2 |δk| , (7) native torus dimensions are possible as are other choices
4
of boundary conditions corresponding to alternative car-
bon structures (e.g. nanotubes or a graphene sheet) as 3 100
∆
will be demonstrated in Section VIII. For our choice of 10−1
2
torus dimensions, we find there are momenta equal to
the K-points and, consequently, eigenstates with ener- γE/ 1 10−2 102 N 103 104
gies equal to the Dirac energy when both m and n are
some multiple of 3. 0
In fact, using the quantised momenta in Eq. (9) ob-
−1
tained for periodic boundary conditions, we find that
therearefourdegenerateeigenstateswithanenergythat 0.4 0.6 0.8 1 1.2 1.4
γ
coincides exactly with the Dirac energy when m and n
are both multiples of 3. These four states, known as the
Dirac states, can be constructed to live only on one of FIG. 3. (Color online) Spectrum of Hγ in Equation (13) as
a function of γ for a 12 ×12 cell torus (N = 288). The
the sublattices, and are given by
spectrum is symmetric around (cid:15)D = 0. Inse√t: Scaling of the
|K(cid:105)A(B) =(cid:114) 2 (cid:88) ei23π(α+2β+2σ)|α,β(cid:105)A(B) gca/p√∆Nl=ogEN˜+(−daEs˜h−ed(dreodt)s)foarncdomcupravreissocn1./ N (solid blue),
N 2
(α,β)
(cid:114)
(cid:12)(cid:12)K(cid:48)(cid:11)A(B) = N2 (cid:88) ei23π(2α+β)|α,β(cid:105)A(B) , (10) This leaves us with the search Hamiltonian
(α,β)
where σ = 0 for states on the A-sublattice or σ = 1 for H =−γA+W, (13)
γ
states on the B-sublattice.
where γ is a free parameter. In what follows, we always
set the on-site energy (cid:15) = 0. Considering our search
IV. QUANTUM SEARCH ON GRAPHENE - D
REDUCED MODEL Hamiltonian, we can see that setting γ = 1 corresponds
toacouplingstrengthofv =0fromtheperturbedvertex
and its nearest-neighbors; our perturbation essentially
In this section we will describe how a site is marked
removes the marked vertex (α ,β )A from the lattice.
and will derive our optimal search starting state. Our o o
Note that vacancies are a common, naturally occurring,
approach will be to analyse the system’s spectrum and
defects in graphene lattices [22].
its dynamics in a reduced Hamiltonian model involving
onlytherelevantstatesfromtheavoidedcrossing. Inthe In order to establish the critical value of γ, we numer-
next section we will then validate this approach with a ically calculate the spectrum of Hγ as a function of γ,
moredetailedanalysis,usingtheresultsobtainedhereas plottedinFigure3foratorusofdimensionsm=n=12.
an initial guide. As W is a rank-2 perturbation, we see in Figure 3 two
Asalreadyestablished, weintroducethelocalisedper- perturberstatesworkingtheirwaythroughthespectrum
turber state in a region of the spectrum with a conic to an avoided crossing around (cid:15)D = 0 when γ = 1, that
dispersionrelationandthusalowdensityofstates. Sim- is, when the perturbed vertex is removed from the un-
ply altering the on-site energy in Eq. (3) as done in [8] derlying lattice.
does, however, not work here. As the on-site energy (cid:15)D As well as establishing a critical value for γ, we also
and the Dirac energy are equal, one finds that the per- find from this figure the states involved in the avoided
turbationonlyinteractswiththeDiracstatesinthelimit crossing: the four Dirac states at the Dirac energy and
ofzeroperturbationstrength,returningtheunperturbed the two perturber states which form our perturbation
lattice. Therefore, we choose an alternative perturba- matrixW. However, withfurtherconsideration,wemay
tion method: namely, we modify the coupling strength reduce the number of states involved further. As we
between the site we wish to mark and its neighboring haveremovedthesite(α ,β )A fromthelatticeitcanno
0 0
vertices. We focus in this section on changing the cou- longer interact with the rest of the lattice and the cor-
pling to all three neighboring vertices of a particular site responding state drops out. Also, by direct calculation
equally. Our choice of perturbation matrix W to mark one finds that the B-type Dirac states do not interact
the A-type vertex (αo,βo)A is then withtheperturbation, thatis, W|K(cid:105)B =W(cid:12)(cid:12)K(cid:48)(cid:11)B =0.
√ √
W= 3|(cid:96)(cid:105)(cid:104)α ,β |A+ 3|α ,β (cid:105)A(cid:104)(cid:96)| , (11) Thus, they remain an eigenstate of the search Hamilto-
o o o o nian and do not interact with the perturbation. We are
where the state |(cid:96)(cid:105) is the symmetric superposition over leftwiththreestatestakingpartintheavoidedcrossing:
the three neighbors of the perturbed site {|K(cid:105)A,(cid:12)(cid:12)K(cid:48)(cid:11)A,|(cid:96)(cid:105)}.
|(cid:96)(cid:105)= √1 (cid:16)|α −1,β (cid:105)B +|α −1,β +1(cid:105)B +|α ,β (cid:105)B(cid:17) . We reduce our search Hamiltonian in this three state
3 o o o o o o basis at the critical point γ = 1 to obtain the following
(12) reducedHamiltoniandescribingthelocaldynamicsatthe
5
preparedin|s(cid:105)andallowedtoevolveunderthefullsearch
Sum of neighbors
y0.4 |α ,β >B Hamiltonian from Eq. (13) at the critical value γ = 1.
bilit o o One finds that the system localises on the neighbors of
a the marked vertex and, for this particular system size,
ob0.2 theprobabilityforameasurementofthesystemtoreturn
r
P one of the neighbors is around 45% at its peak. This is
orders of magnitude higher than the average probability
0
0 20 40 60 80 100 to measure other vertices in the lattice, 100/N, which
Time
for this system size is around 0.35%. The localised state
interactingwiththespectrumunderthefullHamiltonian
FIG.4. (Coloronline)Searchon12×12cellgraphenelattice
hasatailintotherestofthelattice;thisleadstoalossof
withatriple-bondperturbation,usingstartingstate|s(cid:105)from
probabilitytobefoundattheneighboringverticesbelow
Equation 17. For tori with m = n the dynamics at each
100%.
neighboring site is the same so only one is shown.
It is worth emphasising that the marked vertex plays
no role in the search - it is removed from the lattice.
avoided crossing Rather, we force the system to localise on the nearest-
neighbors,essentiallymakingthemarkedvertexconspic-
 
(cid:114) 0 0 e−iµo uous by its absence. This feature differentiates our algo-
6
H˜ =  0 0 e−iνo (14) rithm from previous searches, which have generally con-
N
eiµo eiνo 0 centrated on localising directly on the marked vertex.
An approach considering the neighboring vertices was
with µo = 23π (αo+2βo) and νo = 23π (2αo+βo). detailed in [27], however, the focus was here still on lo-
This reduced Hamiltonian has eigenvalues E˜ = calisingonasinglesitebutwithclassicalpost-processing
±
±2(cid:113)3,E˜ =0, and eigenvectors steps considering the tail of the localised state extending
N 0 into the neighborhood around the marked vertex.
(cid:12) (cid:69) 1(cid:16) √ (cid:17) At this point, there are a number of issues which need
(cid:12)(cid:12)ψ˜± = 2 e−iµo|K(cid:105)A+e−iνo|K(cid:48)(cid:105)A± 2|(cid:96)(cid:105) (15) to be mentioned. The first is that our starting state in
Eq.(17)containsinformationaboutthemarkedvertexin
(cid:12) (cid:69) 1 (cid:16) (cid:17)
(cid:12)ψ˜ = √ e−iµo|K(cid:105)A−e−iνo|K(cid:48)(cid:105)A . (16) the form of the relative phase between the Dirac states.
(cid:12) 0
2
However, this phase can only take three different values.
UsingtheeigenvectorsofthereducedHamiltonian,we Therefore, we havethree possibleoptimal startingstates
can construct a search starting state which is a superpo- for an A-type perturbation and the same number for B-
sition of Dirac states type perturbations; there are thus in total six possible
optimal states. As not all of these states are orthogonal,
weonlyfindanincreaseofnecessaryrunsforasuccessful
1 (cid:16)(cid:12) (cid:69) (cid:12) (cid:69)(cid:17)
|s(cid:105)= √ (cid:12)ψ˜ +(cid:12)ψ˜ (17) search by a factor of 4; this additional overhead is inde-
(cid:12) + (cid:12) −
2 pendent of N and, therefore, does not affect the overall
= e−√iµo (cid:16)|K(cid:105)A+e−i23π(αo−βo)(cid:12)(cid:12)K(cid:48)(cid:11)A(cid:17) . ttiamtieoncoomf tphleexDityiraocfstthaetesseainrcEh.q.T(h1e0)pwaratsiccuhloasrenrepinressuecnh-
2
a way as to make the calculations and conceptualisation
Allowing our search starting state |s(cid:105) to evolve under of the system dynamics easier. In reality, constructing
the reduced Hamiltonian we find startingstateswhichexistononesublatticeonly,experi-
mentallyisextremelydifficulty;rather,inanexperiment,
one is likely to excite a superposition of Dirac states, re-
|ψ(t)(cid:105)=e−iH˜t|s(cid:105) ducing the success probability, on average, by a factor of
1 (cid:16) (cid:12) (cid:69) (cid:12) (cid:69)(cid:17) 1/4.
= √ e−iE˜+t(cid:12)ψ˜ +e−iE˜−t(cid:12)ψ˜ (18)
2 (cid:12) + (cid:12) − Our search is based on the conic dispersion relation in
(cid:16) √ (cid:17)
=cos(cid:16)E˜ t(cid:17)|s(cid:105)−isin(cid:16)E˜ t(cid:17)|(cid:96)(cid:105) , thespectrumandtheO 1/ N scalingbetweensucces-
+ +
sive energy levels in the linear regime, giving rise to the
(cid:16)√ (cid:17)
so that our system rotates from our Dirac superposition O N search time found in our reduced model. How-
|s(cid:105) to a state which is localised on the neighbors of the ever, previous searches in continuous-time using a mod-
(cid:113)
perturbed vertex |(cid:96)(cid:105) in a time t= π N. Thus, we find ified Dirac Hamiltonian [9] or in discrete-time [7] have
4 3
(cid:16)√ (cid:17) found logarithmic corrections to the search time. As we
a O N search time, a polynomial improvement over have seen in Eq. (18) the search time is related to the
the classical search time. energy gap at the avoided crossing. In the inset of Fig-
We plot in Figure 4 a numerically calculated search ure 3 we have numerically calculated the scaling of the
for a lattice with N = 288 sites. The system has been gap in the spectrum of the full search Hamiltonian and
6
weindeedfindevidenceforalogarithmiccorrection. The of H which are at the same time eigenstates of −A with
lack of a logarithmic term in the search time for our re- the same energy.
duced model is due to the neglecting contributions from Thiscanbeseeninthefollowingway. Wefirstconsider
the rest of the spectrum. an unperturbed eigenstate |ψo(cid:105) such that −A|ψo(cid:105) =
a a
In what follows we establish a more accurate estimate E |ψo(cid:105). Let us assume that there is an eigenvector
a a
of the running time and success probability by working |ψ (cid:105) of the search Hamiltonian with the same eigenen-
a
with the full Hamiltonian. We also justify the findings ergy E , that is, H|ψ (cid:105) = E |ψ (cid:105). Considering the
a a a a
of our reduced model: while this model is insufficient to matrixelement(cid:104)ψo|H|ψ (cid:105),wefind(cid:104)ψo|(cid:96)(cid:105)(cid:104)α ,β |ψ (cid:105)+
a a a o o a
estimate the finer details, such as logarithmic correction (cid:104)ψo|α ,β (cid:105)(cid:104)(cid:96)|ψ (cid:105) = 0. As |α ,β (cid:105) is an eigenvector of
a o o a o o
terms, the accurate calculations show that our search al- the search Hamiltonian we know that (cid:104)α ,β |ψ (cid:105) = 0.
o o a
gorithm indeed takes place mainly in a two-dimensional This leaves us with (cid:104)ψo|α ,β (cid:105)(cid:104)(cid:96)|ψ (cid:105) = 0. As the un-
a o o a
subspace spanned by the Dirac states and a state lo- perturbedeigenstate|ψo(cid:105)issimplyaBlochstateweknow
a √
calised on the neighbors of the marked vertex. (cid:104)ψo|α ,β (cid:105)=(cid:54) 0. Thus, we obtain (cid:104)(cid:96)|ψ (cid:105)= R =0. It
a o o a a
is then clear that eigenstates of the search Hamiltonian
whose eigenenergies remain in the spectrum of −A do
V. QUANTUM SEARCH ON GRAPHENE - not play a role in the time-evolution of the search.
DETAILED ANALYSIS We now derive an eigenvalue condition for those per-
turbedeigenvalueswhichareinthespectrumofH. Using
We follow here closely our derivation as presented in theorthogonalityofeigenstates(cid:104)αo,βo|ψa(cid:105)=0,Eq.(22)
[10] which builds on ideas given in [8] for a similar a leads to
similar calculation for regular rectangular lattices. We (cid:112)3R (cid:104)α ,β |(E +A)−1|α ,β (cid:105)=0. (23)
focus first on the search success amplitude, that is, a o o a o o
(cid:104)(cid:96)|e−iHT |start(cid:105)= (cid:88)(cid:104)(cid:96)|ψ (cid:105)(cid:104)ψ |start(cid:105)e−iEaT, (19) By expressing (cid:104)αo,βo| in terms of the eigenstates of the
a a unperturbed walk Hamiltonian −A, we may write this
|ψa(cid:105) as a quantization condition
where T is the search time, that is, the time our search
F (E )=0 (24)
a
probability reaches a maximum, and our search Hamil-
tonian H is given by Eq. (13) with γ =1, that is, with
√ √ √
(cid:20) (cid:21)
H=−A+ 3|(cid:96)(cid:105)(cid:104)α ,β |+ 3|α ,β (cid:105)(cid:104)(cid:96)| ; (20) 3(cid:88) 1 1
o o o o F (E)= + , (25)
N E−(cid:15)(k) E+(cid:15)(k)
here,|ψ (cid:105),E aretheeigenstatesandeigenenergiesofH. k
a a
We assume, without loss of generality, a specific starting
wereN isthetotalnumberofsitesinthelattice. Eq.(25)
state|start(cid:105)wherethemarkedvertexischosensuchthat
iswrittenastwosummationstoincorporatethefactthat
ei23π(αo+2βo) =1 leading to the spectrum of −A as well as H is symmetric around
E =0.
|start(cid:105)= √1 (cid:0)|K(cid:105)+(cid:12)(cid:12)K(cid:48)(cid:11)(cid:1) . (21) Choosing |ψa(cid:105) to be normalised (cid:104)ψa|ψa(cid:105)=1, Eq. (22)
2 also implies
In the following, we suppress the sublattice superscript 3R (cid:104)α ,β |(E +A)−2|α ,β (cid:105)=1, (26)
a o o a o o
astheanalysisisthesameregardlessonwhichsublattice
the perturbation lives. We also denote (cid:15)(k) the positive which allows R to be rewritten as
a
eigenenergies of −A from Eq. (6) and set (cid:15) =0.
D
1
Foreigenstates|ψ (cid:105)witheigenenergiesE thatarenot
a a R = √ . (27)
a
intheunperturbedgraphenespectrum(E (cid:54)=(cid:15)(k)forall 3|F(cid:48)(E )|
a a
points k in the dual lattice), we may rewrite Eq. (20) in
the form We may now rewrite the amplitude in Eq. (19) in the
form
(cid:112)
|ψ (cid:105)= 3R (E +A)−1|α ,β (cid:105) , (22)
a a a o o (cid:88) e−iEaT
where√Ra =(cid:104)(cid:96)|ψa(cid:105)andthephaseof|ψa(cid:105)ischosensuch (cid:104)(cid:96)|e−iHT |start(cid:105)=(cid:104)αo,βo|start(cid:105) a Ea|F(cid:48)(Ea)|,
that (cid:104)(cid:96)|ψ (cid:105)≥0. (28)
a
At this point, we can remove several states from the where we have used the adjoint of Eq. (22). Note again
summation in Eq. (19). Note that the basis state associ- that eigenstates of H, which are also in the spectrum
ated with the marked vertex, |α ,β (cid:105), is itself an eigen- of −A, do not contribute to the time-evolution of the
o o
state of the search Hamiltonian with H|α ,β (cid:105)=0, and search; it may be seen from the definition of F (E) in
o o
also, (cid:104)(cid:96)|α ,β (cid:105) = 0 so that |α ,β (cid:105) does not contribute Eq.(25)that|F(cid:48)(E )|→∞,whereE isinthespectrum
o o o o a a
to the time-evolution. We may also remove eigenstates of −A.
7
As we saw in the previous section, the avoided cross- ConvergenceoftheEpsteinzetafunctioniswell-known
ing is formed by two perturbers approaching symmetri- for k ≥2 [21], and leads to the bounds
cally either side of the Dirac states with energy (cid:15) = 0.
Thus, we concentrate on evaluating the contributDion to lim I2k =4√3(cid:0)Z (S ,k)+Z (S ,k)(cid:1) (34)
thetime-evolutionfromtheseperturbedeigenstatesofH N→∞Nk−1 2 K 2 K(cid:48)
either side of the Dirac point and we label these states
for k ≥2. A sharp estimate for
|ψ (cid:105). In order to evaluate these contributions we cal-
±
culate the perturbed eigenenergy E+ and also derive a I2 =O(lnN) . (35)
leading-order expression for F(cid:48)(E ) (since the spectrum
+
is symmetric, we have E =−E >0). isgiveninAppendixA.Inordertocalculateanestimate
+ −
Using the definition of F (E) in Eq. (25), we esti- for E , we truncate the Taylor expansion of F (E) in
+
mate F (E ) by separating out the Dirac points, where Eq. (29) at the I term (the first term in the summa-
+ 2
(cid:15)(K)=(cid:15)(cid:0)K(cid:48)(cid:1)=0, from the summations and then Tay- tion), and apply the eigenvalue condition F (E ) = 0.
√ +
lor expanding the remaining terms at E =0, to find That is, we solve 4 3 −I E = 0 and obtain the ap-
√ NE√+ 2 +
F (E+)= N4E3 −(cid:88)∞ I2nE+2n−1, (29) pesrtoixmimataetiFon(cid:48)(EE++2)≈≈4N−I232I.2.This solution also leads to the
+
n=1 We consider next whether our solution for E lies in-
+
where the sums I are given by side the radius of convergence of the Taylor expansion.
n (cid:0) (cid:1)
We note that each term in Z S ,k is smaller than
√ 2 K
(cid:20) (cid:21) (cid:0) (cid:1)
3 (cid:88) 1 1 the corresponding term in Z S ,2 for k > 2, and so
I = + . (30) 2 K
n N [(cid:15)(k)]n [−(cid:15)(k)]n it follows that Z2(SK,k) < Z2(SK,2) for k > 2. This
k(cid:54)=K,K(cid:48) property of the Epstein zeta function and the estimate
from Eq. (34) imply
AstheunperturbedspectrumissymmetriconlythoseI
n
withevennarenon-zero;thusfromnowonwefocusonly
∞ ∞
onWI2ekstwahteedreekar≥lie1r.thatthespectrumofgrapheneiswell- (cid:88)I2nE+2n−1 < NCE+ (cid:88)(NE+2)n
n=2 n=2
approximated by a conic dispersion relation around the
Dirac points. Thus, it is from around these points that i.e√. the infinite sum in Eq. (29) converges for E+ <
the major contributions to the I sums arise. By ap- 1/ N. Thus, for large N, our solution lies within the
2k
plying the linear approximation from Eq. (7) and the radiusofconvergenceofourTaylorexpansioninEq.(29).
momenta quantum numbers from Eq. (9) we may ap- In order to show that the dominant contributions to
proximate the I summations as thesearchcomefrom|ψ (cid:105),weneedtoestablishthelead-
2k ±
  ing order error term. All the I2k sums are positive so
I2k =4√3Nk−1 (cid:88) Z2(cid:0)SK,2(cid:1)+ (cid:88) Z2(cid:0)SK(cid:48),2(cid:1) tphaants,iogniviennEtqh.e(2si9g)n, tohfeatlrlutehveatlueremosf Ein t>he0Thaayslotroebxe-
+
(p,q)∈L (p,q)∈L(cid:48) smaller than the estimate we have obtained. Thus, we
+O(1). (31) write the true value of E as
+
√
Here Z2(S,x) is the Epstein zeta-function [21] 4 3
E2 = −∆>0, (36)
Z (S,x)= 1 (cid:88) (cid:0)S p2+2S pq+S q2(cid:1)−x , + NI2
2 2 11 12 22
with∆>0. OnemayrewriteF(E )=0, usingthetrue
(p,q)∈Z2\(0,0) +
(32) value of E+, to give
forarealpositivedefiniterealsymmetric2×2matrixS.
∞
For our purposes we use I ∆= (cid:88)I E2n. (37)
2 2n +
(cid:18) 2 −1(cid:19) n=2
S =S =4π2 , (33)
K K(cid:48) −1 2 Wefollowthesameargumentsasusedforthecalculation
of the radius of convergence to obtain an upper bound
which describes the spectrum close to the Dirac points.
for the summation. This together with the already es-
The linear approximation around both Dirac points K
tablishedfactthatE isinsidethisradiusforsufficiently
and K(cid:48) is the same and, thus, the matrices S and S +
K K(cid:48) large N, we get the following inequality:
are equal.
Our summation in Eq. (31) is over the rectangular re- (cid:88)∞
gions L and L(cid:48) of the lattice Z2. Both are√centered on 0<NI2∆<C (NE+2)n
(0,0)andhavesidelengthsproportionalto N,however n=2
the center, (0,0), corresponding to the relevant Dirac = CN2E+4 =O(I−2).
point, is omitted from the summation. 1−NE2 2
+
8
So ∆ = O(I−3N−1) = O(cid:0)(lnN)−3N−1(cid:1). Thus, we ob- WeusethesameunperturbedwalkHamiltonianasbe-
2
tain fore,thatis,H =−Aandthesametypeofperturbation
0
√ matrix as in Eq. (11), but now at two different sites in
E2 = 4 3(cid:0)1+O((lnN)−2)(cid:1) . (38) the lattice. Explicitly, our communication Hamiltonian
+ NI is
2
It also follows that H=−A+W +W , (45)
s t
(cid:18) (cid:19)
1 where the perturbation matrices W mark the source
F(cid:48)(E )=−2I +O . (39) s/t
± 2 lnN and target sites, respectively.
As prescribed in [17], the communication protocol op-
This shows that our perturbed eigenstates |ψ (cid:105) have erates by preparing a state localised on the source per-
±
an O(1)-overlap with the starting state and are, there- turbation,inourcaseontheneighborsofthesourcesite,
fore, the relevant states to be considered in the time- and allowing the system to evolve under our communi-
evolution of the algorithm. Using the definitions of |ψ (cid:105) cation Hamiltonian. The system then localises on the
a
andR inEqs.(22), (27), theinnerproductofthestart- neighboring vertices of the target site. The effect of the
a
ingstateandtheperturbedeigenvectorscanbeexpressed two perturbation matrices W amounts to disconnect-
s/t
as ing the two vertices from the underlying lattice.
As noted before, our search algorithm has several dif-
(cid:115) √
1 3 ferent optimal starting states depending on the position
(cid:104)start|ψ (cid:105)= (cid:104)start|α ,β (cid:105) , (40)
a E |F(cid:48)(E )| o o ofthemarkedvertices. Therefore,wedivideourcommu-
a a
nication analysis into three different cases: the two sites
i) are on the same sublattice and have the same optimal
where (cid:104)start|α ,β (cid:105) is the overlap of the starting state
o o
searchstartingstate;ii)areonthesamesublatticebutdo
with the marked vertex state. Applying our previous
approximations for E and F(cid:48)(E ), that is, for our per- not share the same optimal search starting state; iii) are
+ +
on different sublattices. Both of the first two cases can
turbed eigenstates closest to the Dirac point, we find
be treated by applying the reduced Hamiltonian method
1 (cid:18) 1 (cid:19) demonstratedearlier,butcommunicationbetweendiffer-
|(cid:104)start|ψ (cid:105)|= √ +O . (41)
± 2 ln2N entsublatticesisnottractablebythismethodandsowe
focus on numerics in this case.
Thus, our starting state is indeed a superposition of the For signal transfer between two sites on the same sub-
perturbed eigenstates, |ψ (cid:105) as assumed in the previous lattice, we will assume, without loss of generality, that
±
section. Thismakesitpossibletoinvestigatetherunning our communication takes place between two sites on the
time and success amplitude of the algorithm in more de- A-sublattice, (cid:0)α ,β (cid:1)A; here, the subscript s or t de-
s/t s/t
tail, that is, notesthesourceortargetverticesrespectively. Usingar-
gumentsasemployedinSec.IV,wecanreducethenum-
(cid:12)(cid:12)(cid:104)(cid:96)|e−iHt|start(cid:105)(cid:12)(cid:12) (42) ber of relevant states. Ultimately, the search dynamics
≈(cid:12)(cid:12)(cid:12)(cid:12)√12(cid:0)e−iE+t(cid:104)(cid:96)|ψ+(cid:105)−eiE+t(cid:104)(cid:96)|ψ−(cid:105)(cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (43) (cid:12)(cid:12)tKak(cid:48)e(cid:11)sAp,ltahcee iAn-ttyhpeesuDbisrpaaccsetaspteasn,naenddb|y(cid:96)st(cid:105)h,e|(cid:96)bt(cid:105)a,sitsh|eKu(cid:105)nAi-,
1 form superpositions over the neighbors of the source and
= |sin(E t)| . (44)
314I212 + tnaiargnet vertices. This basis leads to the reduced Hamilto-
It is clear from our earlier results for E and I that
± 2
our algorithm localises on the neighbor state |(cid:96)(cid:105) in time  0 0 eiµs eiµt
(cid:16)√ (cid:17) (cid:114)
T (cid:16)= √2Eπ+ =(cid:17)O NlnN with probability amplitude H˜ = N6 e−0iµs e−0iνs ei0νs ei0νt , (46)
O 1/ lnN . This confirms the logarithmic correction e−iµt e−iνt 0 0
observednumericallyanddisplayedintheinsetofFig.3.
where µ ≡ 2π (cid:0)α +2β (cid:1) and ν ≡
s/t 3 s/t s/t s/t
2π (cid:0)2α +β (cid:1).
3 s/t s/t
VI. COMMUNICATION In the first case considered we assume that there
are two perturbations on the graphene lattice located
It has been demonstrated in [17] that discrete-time at the points (αs,βs)A and (αt,βt)A, chosen such that
search algorithms can be modified to create a commu- ei23π(αs+2βs) = ei23π(αt+2βt). This implies that a search
nication protocol. We show here that a communica- for either vertex, using our search algorithm from
tionsetupcanbeestablishedalsointhecontinuous-time Secs.IV&V,wouldusethesameoptimalstartingstate.
searchalgorithmwithminorchangesduetothesubtleties Asthephasesareequal,wedropthesubscriptontheper-
of our search. turbationcoordinatesinwhatfollows. Fixingourphases
9
a single perturbation located at vertex (α ,β )A until it
0.4 Source s s
reachesmaximumsuccessprobability,andthenapplythe
y Target
bilit second perturbation to the vertex (αt,βt)A. The figure
a confirmsthebehaviourexpectedfromthereducedmodel
b0.2
o calculation.
r
P
The communication mechanism essentially works in
0 the same way as the quantum search algorithm, as it
0 20 40 60 80 can be viewed as one marked vertex ‘finding’ another.
Time
Theinitiallocalisedsourcestatedecaysbacktowardsthe
searchstartingstate,andthesystemthensearchesforthe
FIG. 5. (Color online) Numerically calculated signal transfer
target state.
ona12×12cellgraphenelatticebetweenequivalentvertices,
Inoursecondcaseofsignaltransferweanalysethebe-
using the communication Hamiltonian in Eq. (45). The sys-
haviour of a communication system where we have two
temisinitialisedin|(cid:96) (cid:105)andlocaliseson|(cid:96) (cid:105). Onlythesumof
s t
probabilities to be found on the neighbor vertices is shown. perturbed vertices on the same sublattice, but with the
restriction that ei23π(αs+2βs) (cid:54)=ei23π(αt+2βt). As such, the
two marked sites cannot interact via the same search
anddiagonalisingthereducedHamiltonian,wefindithas starting state. However, the optimal search starting
(cid:113)
eigenvalues λ± =±2 6 and λ1,2 =0 with eigenvectors states are not orthogonal so that signal transfer is still
2 N 0 possible, but the resulting interference effects make the
analysis slightly more complicated.
(cid:12)(cid:12)(cid:12)ψ˜±2(cid:69)= 21(cid:16)eiµ|K(cid:105)A+eiν(cid:12)(cid:12)K(cid:48)(cid:11)A±|(cid:96)s(cid:105)±|(cid:96)t(cid:105)(cid:17) (47) theWseourercwer,itαe t=heαco+orxdinaantdesβof=thβe +taryg.etAignaitne,rmfixsinogf
t s t s
(cid:12)(cid:12)(cid:12)ψ˜01(cid:69)= √12(cid:16)eiµ|K(cid:105)A−eiν(cid:12)(cid:12)K(cid:48)(cid:11)A(cid:17) (48) oEuqr.p(4h6a)s,eswaenfidnddiaitgohnaasleisiignegnvtahleureesdλu±√ced=H±am√i3lt(cid:113)on6iananind
(cid:12)(cid:12)(cid:12)ψ˜02(cid:69)= √12(|(cid:96)s(cid:105)−|(cid:96)t(cid:105)) . (49) λ±1 =±(cid:113)N6 with eigenvectors 3 N
Ubosrinsgtattheesaeseigenstates,wemayrewritethesourceneigh- (cid:12)(cid:12)(cid:12)ψ˜±√3(cid:69)= 2e√iµs3(cid:16)ei23π(x+2y)−1(cid:17)|K(cid:105)A (54)
|(cid:96)s(cid:105)= 21(cid:16)(cid:12)(cid:12)(cid:12)ψ˜2(cid:69)−(cid:12)(cid:12)(cid:12)ψ˜−2(cid:69)(cid:17)+ √12(cid:12)(cid:12)(cid:12)ψ˜02(cid:69) . (50) + 2e√iνs3(cid:16)e−i23π(x+2y)−1(cid:17)(cid:12)(cid:12)K(cid:48)(cid:11)A∓ 21|(cid:96)s(cid:105)± 12|(cid:96)t(cid:105)
Placing the system in the source state |(cid:96)s(cid:105) and allowing (cid:12)(cid:12)(cid:12)ψ˜±1(cid:69)=∓ei2µs (cid:16)ei23π(x+2y)+1(cid:17)|K(cid:105)A (55)
thesystemtoevolveunderthereducedHamiltonian,one
finds ± eiνsei232π(x+2y) (cid:12)(cid:12)K(cid:48)(cid:11)A− 12|(cid:96)s(cid:105)− 12|(cid:96)t(cid:105) .
|ψ(t)(cid:105)=e−iH˜t|(cid:96) (cid:105) (51)
s
Using these eigenstates, one may write the source per-
1(cid:16) (cid:12) (cid:69) (cid:12) (cid:69)(cid:17) 1 (cid:12) (cid:69)
= 2 e−iλ+2t(cid:12)(cid:12)ψ˜2 −e−iλ−2t(cid:12)(cid:12)ψ˜−2 + √2(cid:12)(cid:12)ψ˜02 turbation as
1(cid:16) (cid:12) (cid:69) (cid:12) (cid:69) (cid:12) (cid:69) (cid:12) (cid:69)(cid:17)
(52) |(cid:96) (cid:105)= −(cid:12)ψ˜√ +(cid:12)ψ˜ √ −(cid:12)ψ˜ −(cid:12)ψ˜ . (56)
s 2 (cid:12) 3 (cid:12) − 3 (cid:12) 1 (cid:12) −1
= −2ieiµsin(cid:0)λ+2t(cid:1)(cid:16)|K(cid:105)A+ei23π(α−β)(cid:12)(cid:12)K(cid:48)(cid:11)A(cid:17) The full expression for the time-evolution is rather cum-
+ 1(cid:0)cos(cid:0)λ+t(cid:1)+1(cid:1)|(cid:96) (cid:105)+ 1(cid:0)cos(cid:0)λ+t(cid:1)−1(cid:1)|(cid:96) (cid:105) . bersome; therefore, we only show the terms and prefac-
2 2 s 2 2 t tors we are interested in, namely |(cid:96)s(cid:105) and |(cid:96)t(cid:105)
(53)
We note that, in the last line above, the term in the |ψ(t)(cid:105)=21(cid:104)cos(cid:16)λ+√3t(cid:17)+cos(cid:0)λ+1t(cid:1)(cid:105)|(cid:96)s(cid:105) (57)
bopraticmkeatlsseinavrochlvsintagrtoinnlgysttahteeDfoirrabcotshtapteesrtiusrbaecdtuvaelrlytictehse, − 21(cid:104)cos(cid:16)λ+√3t(cid:17)−cos(cid:0)λ+1t(cid:1)(cid:105)|(cid:96)t(cid:105) .
defined in Eq. (17). The system thus oscillates between
We can see here that the prefactors do not depend upon
the states localised on the neighbors of the perturbed
(cid:113) the coordinates of either the source or the target sites;
vertices, |(cid:96)s(cid:105) and |(cid:96)t(cid:105), in a time T = π2 N6, via their thetransportsignalbetweensitesonthesamesublattice
optimal search starting state. but with different optimal search starting state is thus
Figure5showsthesystemevolvingunderthefullcom- independent of the position of the source or target site.
municationHamiltonian,Eq.(45). Theinitialstateused In Figure 6 we show the system evolved under the full
for the time evolution shown in Figure 5 is the true lo- communication Hamiltonian. Again, the initial state is
calised state, that is, we run the quantum search with the true localised state on the nearest-neighbors of the
10
0.4 Source 0.4 Source
y Target y Target
bilit bilit
a a
b0.2 b0.2
o o
Pr Pr
00 20 40 60 80 100 120 00 500 1000 1500 2000
Time Time
0.4 Source 0.4 Source
y Target y Target
bilit bilit
ba0.2 ba0.2
o o
Pr Pr
00 20 40 60 80 00 200 400 600 800 1000
Time Time
FIG. 6. (Color online) Upper: Numerically calculated sig- FIG. 7. (Color online) Numerically calculated signal transfer
nal transfer on a 12×12 cell graphene lattice between non- on a 12×12 cell graphene lattice between vertices on dif-
equivalent vertices, using the communication Hamiltonian in ferent sublattices, using the communication Hamiltonian in
Eq.(45). Thesystemisinitialisedin|(cid:96)s(cid:105)andlocaliseson|(cid:96)t(cid:105). Equation45. Thesystemisinitialisedin|(cid:96)s(cid:105)andlocaliseson
Onlythesumofprobabilitiestobefoundontheneighborver- |(cid:96)t(cid:105). Only the sum of probabilities to be found on the neigh-
tices is shown. Lower: Analytically calculated behaviour for bor vertices is shown. The two figures have the same source
thesamesystem,usingthereducedHamiltonianmethodand position but different positions of the target site.
Eq. (57).
source vertex, obtained by running the search algorithm
using one marked vertex until it reaches its peak success
been demonstrated previously that perturbations to one
probability. The time-evolution is radically different to
sublatticedonotinteractwiththeDiracstateswhichlive
the previous communication case; the behaviour here is
on the other sublattice. Therefore, when attempting to
erratic with uneven peaks of probability at the two per-
reduce the communication Hamiltonian as done before,
turbations involved in the protocol. However, there are
wemerelyfindthatitdecouplesintotwonon-interacting
still significant probability revivals.
reduced Hamiltonians, each describing a search proto-
The transport behaviour from Eq. (57), calculated us-
col on one sublattice. The interaction between the two
ing the reduced Hamiltonian, is also shown in Figure 6.
sublattices is here facilitated due to interactions via the
The probability at time t = 0 has been scaled to match
bulk of the spectrum. In what follows, we focus on nu-
that shown in Figure 6. Our calculated behaviour has
merics and inspect the behaviour for systems involving
the same signal pattern as the numerically calculated
the same source perturber but different targets. The nu-
behaviour from the full Hamiltonian, although over a
merics show a finite number of different signal patterns,
shorterperiodoftime. Asourreducedmodelonlymakes
two examples are shown in Figure 7. We can see from
use of the Dirac states and the perturber states, we lose
theseexamplesthatthecommunicationmechanismtakes
the contribution to the time-evolution from the rest of
place over a much longer timescale with a superimposed
thespectrumgivingrisetologarithmiccorrectionsasdis-
oscillatory dynamics.
cussed in Sec. V. This leads to differences in the overall
time scales for the reduced model and the full Hamilto-
nian. Thisalsosupportsourfindingsthatsignaltransfer Theincreaseintimescalecanbeattributedtotheweak
between all non-equivalent vertices on the same sublat- nature of the interaction between the two perturbations
tice is the same. The communication protocol is again due the fact that the localised states live mainly on one
set up by a search mechanism in reverse, where one ver- sublattice. ThesignalpatterninFigure7thusdecouples
tex finds another. The slightly erratic behaviour that into a fast oscillation between one site, say the source
emerges here is due to interference between the two sep- site, and the delocalised lattice state and a slow time
arate search mechanisms which interact due to the non- scale on which a small amount of probability amplitude
zero inner product of the three possible optimal search escapesintotheothersublatticeduetotheweakinterac-
starting states for vertices on the A-sublattice. tion. This process continues until the recurrence proba-
Inourfinalcaseofsignaltransfer,weconsidercommu- bilityatthetargetperturbationreachesthesamepeakas
nication between sites on different sublattices which we the initial localised source state, and then the behaviour
can not treat in the reduced Hamiltonian model. It has reverses.