Table Of ContentStrongly trapped two-dimensional quantum walks
B. Koll´ar,1 T. Kiss,1 and I. Jex2
1Wigner RCP, SZFKI, Konkoly-Thege M. u. 29-33, H-1121 Budapest, Hungary
2Department of Physics, Faculty of Nuclear Sciences and Physical Engineering,
Czech Technical University in Prague, Bˇrehov´a 7, 115 19 Praha 1 - Star´e Mˇesto, Czech Republic
(Dated: January 28, 2015)
Discrete time quantum walks (DTQWs) are nontrivial generalizations of random walks with a
broad scope of applications. In particular, they can be used as computational primitives, and they
are suitable tools for simulating other quantum systems. DTQWs usually spread ballistically due
to their quantumness. In some cases, however, they can remain localized at their initial state
(trapping). The trapping and other fundamental properties of DTQWs are determined by the
5
choiceofthecoinoperator. Weintroduceandanalyzeanuptonowunchartedtypeofwalksdriven
1
0 by a coin class leading to strong trapping, complementing the known list of walks. This class of
2 walksexhibitanumberofexcitingpropertieswiththepossibleapplicationsrangingfromlightpulse
trappingin a medium to topological effects and quantumsearch.
n
a
PACSnumbers: 03.67.Ac,05.40.Fb
J
7
2 Quantumwalks are analogsofclassicalrandomwalks, coins,andpointoutimportantphysicalconsequencesfor
gainingconsiderableattentionsincetheirintroductionin the corresponding DTQWs. In particular, we report the
]
h the nineties [1, 2]. They did prove themselves as inter- neweffectof“strongtrapping”. Inadditiontocontribut-
p esting constructs which find their applications gradually ing to the understanding and classification of quantum
- [3–13] (for review see [14]). Recent experiments demon- walks, we comment on the implications — possible ap-
t
n strated their basic properties [15], and stimulated fur- plication for the trapping of light, its relation to spatial
a ther theoretical studies. While the early experiments quantum search algorithms, appearance of topological
u
have been limited to DTQWs on a line, recently, two- phases, and its spectral similarity to the so-called split
q
dimensional quantum walks have been successfully real- step QWs.
[
ized[16]. Although atwo-dimensionallattice is stillcon- Let us begin with the formal definition of two-
2 sidered simple in physics, two-dimensional DTQWs [17] dimensional DTQWs, briefly. The composite Hilbert
v
displaya richvarietyofinteresting effects. In particular, space of a DTQW is given as: = . In the
4 H HP ⊗HC
they exhibit a partial quantum speedup in search [3]. present paper the position space corresponds to a
4 HP
8 The fundamental behavior of DTQWs is governed by two-dimensional Cartesian lattice. We denote position
5 the choice of the unitary operator acting on the inter- Hilbertvectorswith x P,wherex=(x,y)representsthe
| i
0 nal structure of the particle — the so-called coin opera- positiononthelattice. Thecoinspace C isspannedby
H
1. tor. Similarlytorandomwalks,DTQWscanbeclassified vectorscorrespondingtothedirections: |Li,|Di,|Ui,|Ri.
0 based on their spreading properties. One of the classifi- Throughout the present paper we expand operators on
5 cations is based on the definition of the Po´lya number the coin space using this basis. We note that this coin
1 [18], dividing walks into recurrent and transient walks. Hilbertspacecanbe viewedasatwo-qubitHilbertspace
v: Classically,this property depends only on the dimension with L = 00 , D = 01 , U = 10 , R = 11 . The
| i | i | i | i | i | i | i | i
i of the underlying lattice. However, the quantum exten- time evolution is unitary: U = S (IP C). Here, S is
X · ⊗
sions of the Po´lya number [19–21] exhibit significantdif- thestepoperatorresponsiblefortheconditionaldisplace-
ar flaertetinccee,s.theApclaarstsififrcoamtiotnhealdsoimdeenpseionndsofonthteheuncdheoriclyeinogf cmoernrets:pSo|nxdisPt⊗o|tchie=di|rxe⊕ctcioiPns⊗a|ncid,xwhecredcen∈ot{eLs,tDhe,Un,eRar}-
the coin operator and the internal state of an initially estneighborofpositionxinthedire⊕ctionc. ByC U(4)
∈
localized walker. In the family of two-dimensional re- we denote the coin operator acting on the internal coin
current quantum walks a rather interesting class can be degreeoffreedom. Due to the translationalinvarianceof
found, namely that exhibiting the effect of trapping (lo- theDTQWswestudy,theglobalphaseoftheU(4)coins
calization) [17]. In this class the probability of finding is ignored. The time evolution in the quasi-momentum
the particle at its initial location throughout the whole picture is given by
evolution is non-vanishing. For leaving walks, i.e. when U˜ =D˜ C, (1)
the walker is forced to leave its actual position during ·
a single step, this effect does not have a natural classi- where D˜ is the Fourier-transformedstep operator S, i.e.
calcounterpart,thus,itisapurelyquantuminterference D˜ =Diag(e ik,e il,eil,eik). Byk,lwedenotethequasi-
− −
phenomenon. Onecanpostthe naturalquestion: Which momenta of the walk. The phase ω of a given λ = eiω
coins exhibit trapping? First examples of such a coin eigenvalue of U˜ is frequently termed as quasi-energy.
were found in [22, 23]. Undoubtedly, the most notorious two-dimensional
In this paper we construct a large class of trapping DTQW is the one driven by the Grover diffusion opera-
2
always has a constant eigenvalue µ = 1 with the multi-
plicityoftwo. Weapplyaunitarytransformationkeeping
the constant eigenvalues:
2 W 2W =W(U U )WW(U U )W . (5)
2 1 1 2
U ≡ U ⊗ ⊗
FIG. 1. Quantum circuit representing the coin class de-
C
fined in Eq. (8). The abstract two-qubit space of ψ(1) Here,W istheunitarySWAPgateonatwo-qubitsystem
| in(out)i
and ψ(2) isspannedbybasisvectors 00 , 01 , 10 , 11 (coin space): W L L + D U + U D + R R .
corre|spino(noudt)tio the four dimensional coi{n| spiac|e iof|thie|twio}- Note that W = W≡−|1ih=|W†|, ainhd|we|alsioh in|ser|tedihth|e
dimensional quantumwalk with L = 00 , D = 01 , U = identity I = WW into Eq. (5). Performing the SWAP
| i | i | i | i | i
10 , R = 11 . operation on the first part of Eq. (5), and taking a
| i | i | i
square root results in = (U U )W. We substi-
1 2
U ⊗
tute the arbitrary SU(2) operators U with U =
1(2) 1(2)
tor: Diag(e i(k+( )l)/2,ei(k+( )l)/2) C , where C
− − − 1(2) k= 1,2
SU(2), to arrive at the formula:· { } ∈
1
C(G) = δ , (2)
i,j 2 − i,j
=D˜(C C )W =U˜, (6)
1 2
U ⊗
where δ is Kronecker’s delta symbol. Apart from its
i,j
use in quantum walk based search algorithms [3], the
whichis a time evolutionoperationofa two-dimensional
Grover walk is well-known for its trapping (localization)
DTQW [cf. Eq. (1)]. By its construction, has at least
property [17, 22]: The probability of finding the walker U
two constant eigenvalues 1. These eigenvalues appear
at its initial position at any time is non-vanishing for ±
as flat bands at ω =0,π quasi-energies in the spectrum.
almost all walkers started at the origin. For the well
Thus, we constructed a trapping walk with the new coin
chosen localized initial state
class:
1
|ψiNinTit =|x0iP ⊗ 2(|Li−|Di−|Ui+|Ri) (3) =(C C )W. (7)
1 2
C ⊗
the trapping is avoided and the walker escapes. A one-
parameter generalization of the Grover walk exists [23], One can observe that any operation commuting with D˜
sharingthe abovelistedtraits: state-dependenttrapping and W can generalize the coin without losing its trap-
C
and a single localized escaping-state. The extensive re- pingproperty(constanteigenvalues—flatbands). Since
searchdedicatedtotheGrovertrappingalsoledtoother C C cover all local single-qubit rotations, a non-
1 2
⊗
interesting effects, namely dynamics exhibiting full re- separableoperationisdesirabletoavoidredundancy. We
vivals [24], a way to design recurrent quantum walks in found that the most suitable operation is the controlled
arbitrary dimensions [19], and trapping in walks on per- phase gate: P(ϕ) = Diag(1,1,1,eiϕ). Thus, the general
colation graphs [25]. form of the trapping coin class ( -class) is
C
In general, the trapping in DTQWs is caused by the
presence of flat bands in the quasi-energy spectrum of
=P(ϕ)(C C )P( ϕ)W , (8)
1 2
the walk [22], which naturally depends on the choice of C ⊗ −
the coin operator. It was shown that the spectra of all
trapping two-dimensional DTQWs contain at least two which contains 7 real parameters (+1 for the global
such flat bands with a π difference between their quasi- phase). This coin class can be realized as the two-qubit
energies[24]. FindingtrappingcoinsinvolvessolvingEq. quantumcircuitillustratedinFig. 1. Wenotethatthe -
C
(1) for a pair of constant eigenvalues. However, it turns classcontainstheGrovercoin[cf. Eq. (2)]anditsknown
out that the direct solution in the complete U(4) space one-parameter generalization [23] as special cases.
is very demanding both numerically and analytically. Let us show some basic properties of the -class. We
C
Letuspresentabroadclassoftrappingcoinoperators define the elements of C SU(2):
k= 1,2
using a constructive approach. To begin, we define two { } ∈
arbitrarySU(2) operatorsU andU , thus, det(U )=
1 2 1(2) e iαkcosδ e iβksinδ
U1. UItisshsatrraeitghhetfsoarmwaerpdatiorosefeeitgheantvtahlueepsr(otdhueyctasrUe1uUn2itaanrdy Ck =(cid:18) −eiβksinδkk − e−iαkcosδkk (cid:19) . (9)
2 1
equivalent), and since det(U U ) = det(U U )= 1 these
1 2 2 1
eigenvalues are complex conjugate pairs. Consequently, One can see that the time evolution operator U˜ of Eq.
the following operator (1)usingthe -classhastwoconstanteigenvalues: 1and
C
1. Solvingthe eigenvectorproblem,aninverseFourier-
2 −
=(U U ) (U U ), (4) transform reveals the stationary eigenstates correspond-
2 1 1 2
U ⊗ · ⊗
3
ing to the eigenvalues λ= 1:
±
ψ(x,y) =
λ= 1
| ± i
1
x,y ( e iβ1sinδ L e iα1cosδ D )
P − 1 − 1
2n| i ⊗ − | i− | i
x,y+1 ( e iβ2sinδ L e iα2cosδ U )
P − 2 − 2
±| i ⊗ − | i− | i
x+1,y (eiα2cosδ D ei(β2+ϕ)sinδ R )
P 2 2
±| i ⊗ | i− | i
+x+1,y+1 (eiα1cosδ U ei(β1+ϕ)sinδ R ) .
P 1 1
| i ⊗ | i− | i o
(10)
An equally weighted superposition of two such FIG. 2. (Color online) Minimum probability of finding the
eigenstates living on the same lattice sites particle at its initial position after 40 steps of a -class coin
1 ψ(x,y) ψ(x,y) gives full reviv- [see Eq. (8)] driven quantum walk. The probaCbilities are
√2{| λ=+1i±| λ=−1i} determined by numerically searching the initial states yield-
ing states. These states generalize the results of [24] for
ing the minimum probability. The coin is investigated in the
the -class.
ACn initial state having a nonzero overlap with any of δ1,δ2 parameter space with α1 = α2 = β1 = β2 = π2 and
ϕ=π. The red linescorrespond tothecondition of Eq. (11)
the eigenstates in Eq. (10) is trapped. We found that
where strong trapping disappears. On these lines the mini-
for most of the coins in the -class, all initially localized
mum probability is naturally zero due to the appearance of
C
states are trapped. We term this novel phenomena as the escaping state of Eq. (12). However, where strong trap-
“strong trapping”. However, when the the coin parame- pingis present,the minimum probability is larger than zero.
ters satisfy
cos2δ1 =cos2δ2, (11) contacted bulk regions using certain pairs of coins lead
to the appearance of edge states. The found edge states
similarly to the Grover walk, a single initially localized are non-degenerate, can propagate only in one direction
state will lead to escape. For δ2 = δ1 +k π, k N+, determinedbythelayoutofthetwobulkregions,andare
· ∈
this state has the form: alsotopologicallyprotectedfromthedecoherencecaused
1 by stationary (spatial) noise. The obtained topological
|ψiEInSitC =|x0iP ⊗ √2(cid:16)e−iβ1cosδ1|Li−e−iα1sinδ1|Di mapfortheC-classandaspectrumexhibitingedgestates
are shown in Fig. 3. We note that up to our knowledge
e−i(α2+β1−β2)sinδ1 U e−i(α1+α2−β2−ϕ)cosδ1 R . thisis thefirstcasewhentopologicaleffects arereported
− | i− | i(cid:17)
in simple two-dimensional DTQWs using a single coin.
(12)
It is interesting to note that the -class walks can be
C
We note that the strong trapping effect can be avoided linked with the so-called split-step walks. The concept
bystartingthewalkfromaninitiallynon-localizedstate. of split-step (alternate step) two-dimensional quantum
We studied the strength of the trapping effect numeri- walks open a way to simulate some DTQWs using only
cally and illustrated it on Fig. 2. a two-dimensionalcoin space [33]. The time evolutionof
The -classcoins wouldmake goodcandidates to trap such walks is given as
C
and manipulate wave packets — light pulses in the pho-
tonic implementation of experiments. The expected be- Usplit =S C2S C1, (13)
↔ l
haviorofDTQWdynamicsdrivenbyageneralcoinisthe
where C SU(2). S (S ) displaces the particle
propagation of the wavefunction over the lattice. How- on a twok-=d{i1m,2e}n∈sional latticel do↔wn/up (left/right) with
ever, changing the coin to a -class coin can trap (stop)
C respecttoitsactualcoin(qubit)state. Thisconstruction
a part of the spreading wave. This trapped pulse is held
implies that the particle cannot step backwards: Split-
in its place by the trapping coin, and can be released by
step walks live on the directed square lattice. Conse-
changing back to the original non-trapping coin.
quently, such walks can maintain their ballistic spread-
The symmetries of the quasi-energy spectra allow for
ingbetterundernoisyconditions[34]. Thesplit-stepQW
non-trivial topological phases in quantum walks [26–32].
also exhibits topological effects [27, 30–32].
The controlled opening and closing of gaps in the quasi-
Let us rewrite the time evolution (13) of the split-
energy spectrum can be a signature of the presence of
stepwalkintheFourier-picture,whilerelabelingthemo-
such topological phases. We found that the spectra of
menta. This results the following time evolution opera-
weakly localizing walks satisfying Eq. (11) are gapless,
tor:
whereasinthecaseofstrongtrapping,gapsopenaround
the flat bands at ω = 0,π quasi-energies. Thus, param- U˜ =Diag(e i(k l)/2,ei(k l)/2) C
split − − − 2
eters δ and δ tune the continuity of the spectrum. To · ×
mappo1ssiblet2opologicalphases,wenumericallysearched Diag(e−i(k+l)/2,ei(k+l)/2) C1.
·
this parameter space. We found that DTQWs with two (14)
4
✁✆✞✝
✁✆✞✁
✁✆✁✝
(cid:0)✁ ✂✁ ✄✁ ☎✁
FIG.4. (Coloronline)Probabilityoffindingamarkedvertex
on a 10 80 torus using the original spatial quantum search
×
algorithm(bluedashedcurve),andamodifiedone(redcurve)
employing the coin class defined in Eq. (8) with parame-
FIG.3. (Coloronline)Topologicalmap(leftplot)andquasi- ters δ =δ = 31π, α =βC = π, α =β = π and ϕ=π.
energyspectrum(rightplot)of -classcoins[seeEq. (8)]. The 1 2 90 1 2 2 2 1 −2
C Theverticalblacklinedenotesthemeasuringtimedefinedin
light brown and white areas correspond to different topolog-
the algorithm √2LM = 40. At that time the probabilities
ical phases, i.e. in the boundary between two topologically
of theoriginal and modified algorithms at themarked vertex
differentbulkregionswefoundatopologicallyprotectededge
are0.0616and0.0827respectively. Notethattheseprobabili-
state. The thick lines represent the gap closing condition
tiesareobtainedbeforethefinalamplitudeamplificationstep
(11),whichisalsotheconditiondescribingthedisappearance
concluding thealgorithm.
of strong trapping. We used the definitions (8) and (9) with
parameter values α = α = β = β = π and ϕ = π, in-
1 2 1 2 2
vokingarealvaluedsubclassof ,whichcontainstheGrover
coin for δ = δ = π (marked byC the black dot). The quasi- texwithprobabilityO(1/logN). Torealizeapractically
1 2 4 useful probability in order of O(1), the algorithmshould
energy ω spectrum as the function of momentum k on the
rightplotcorrespondstothetwo-dimensionalQWonatorus be repeatedforO(√logN)times andthroughamplitude
using two parallel bulk regions with δ = π,δ = 4π and amplification [37] the O(1) probability is achieved. In
1 10 2 10
δ = 4π,δ = π (denoted by circles on the topological map summary, the total runtime is O(√NlogN) steps.
1 10 2 10
on the left). The topologically protected edge states are em- This algorithm is designed to perform searches on
phasized by magenta and cyan shading. M M tori, and on a general L M torus it turns
× ×
out to be suboptimal. However, employing the -class
C
coins one can possibly optimize the algorithm for gen-
Note that the relabeling of the momenta has the walker eral tori. Our preliminary numerical studies show that
step in the diagonal and anti-diagonal directions on the fornon-squaretoriaproper -classcoinoutperformsthe
C
lattice. An investigation of the spectrum of this model originaltransformedGrovercoin. See Fig. 4. for details.
reveals a surprising fact: It is perfectly identical to the FindingcoinsfromtheU(4)groupfortwo-dimensional
non flat band spectrum of the regular two-dimensional DTQWs that fulfill a specific goal is definitely a hard
DTQW governedby the -class (8) with ϕ=0. task. In this paper, we presented a coin class which ex-
C
This result unveils a correspondence between the two hibits a trapping feature similar to the Grovercoin. The
models, which helps us to draw consequences on the ex- proposed many parameter, analytically given class is a
pected behavior. For example, the group velocities of considerablesteptowardsclassificationofcoinsavailable
the propagating peaks [35] and weak-limit distributions on regular lattices. DTQWs driven by the new coin ex-
[23, 36] shouldbe similar. Moreover,the analyticaltools hibitanovelphenomenonwhichwetermas“strongtrap-
proposedforthesplit-stepmodelsmightbeemployedfor ping”, allowing for the trapping of all localized initial
-class based walks. The spectral correspondence also states. Generalizationstohigherdimensionallatticesare
C
strengthens our observations on the topological effects. also possible allowing for constructing strongly trapping
Finally, let us briefly comment on the applicability of walks. The closedformof the coin matrix allowsfor fur-
the new class of coins for the quantum search. Two- thertheoreticalstudies. We alsodemonstratedthe topo-
dimensional DTQWs are suitable for searching marked logicalproperties of the class,revealingnon-trivialtopo-
elements on a M M torus [3]. By design they allow logicalphases,whichwerepreviouslyunobservedforsim-
×
forfinding asingle markedvertexinO(√NlogN)steps, ple single-coin two-dimensional DTQWs. The proposed
with M2 =N. The walk performing the search is a reg- system has a spectral connection with two-dimensional
ular two-dimensional DTQW on the torus employing a split-step quantum walks, giving a bridge between the
transformed4 4GrovercoinC(G)(σ σ ). Infact,this two models. We have revealed that the quantum walk
x x
× ⊗
transformed coin is also in the -class with parameters based search algorithm on a non-square torus benefits
δ = δ = π, α = β = π, α C= β = π and ϕ = π. from the proposed coin, possibly allowing for optimiza-
1 2 4 1 2 2 2 1 −2
Atthemarkedvertex—whichthe algorithmisdesigned tion.
to find — the coin is changed to σ σ . The initial The proposed coin class is potentially very useful.
x x
− ⊗
state ofthe systemis anequalsuperpositionofallstates These coins are promising choices for two-dimensional
in the natural basis, which is evolved for √2N steps. experiments [16], as all of the necessary elements are
≈
Following,apositionmeasurementyieldsthemarkedver- currentlyavailable. Suchexperimentswillallowtostudy
5
topologicaleffects,trapping,strongtrapping,search,and 13-33906S, RVO 68407700, the Hungarian Scientific Re-
other quantum walk related effects in a new setting. search Fund (OTKA) under Contract Nos. K83858,
We thank J. K. Asbo´th and M. Sˇtefanˇa´k for stimu- NN109651 and the Hungarian Academy of Sciences
lating discussions. We acknowledge support by GACR (Lendu¨let Program,LP2011-016).
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