Continuous Limit of Discrete Quantum Walks

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by  Dheeraj M N

Continuous Limit of Discrete Quantum Walks
Dheeraj M N
∗
Department of Electrical Engineering, IIT Madras, Chennai, Tamil Nadu, India
Todd A. Brun
†
Communication Sciences Institute, University of Southern California, Los Angeles, California, USA
(Dated: January 29, 2015)
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time
quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in
which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the
discrete-time case, an additional “coin space” must be appended for the walk to have nontrivial
5
time evolution. Continuous-time quantum walks, however, have no such constraints. This means
1
that there is no completely straightforward way to treat a CTQW as a limit of DTQW, as can be
0
2 doneintheclassicalcase. Variousapproachestothisproblemhavebeentakeninthepast. Wegive
a construction for walks on d-regular, d-colorable graphs when the coin flip operator is Hermitian:
n
fromastandardDTQWweconstructafamilyofdiscrete-timewalkswithawell-definedcontinuous-
a
timelimitonarelatedgraph. Onecanthinkofthislimitasacoinedcontinuous-timewalk. Weshow
J
that these CTQWs share some properties with coined DTQWs. In particular, we look at spatial
7 searchbyaDTQWoverthe2-Dtorus(agridwithperiodicboundaryconditions)ofsize√N √N,
2 whereit wasshown that acoined DTQWcan search in timeO(√NlogN),butastandard C×TQW

takes Ω(N) time to search for a marked element. The continuous limit of the DTQW search over
]
h the 2-D torus exhibits the O(√NlogN) scaling, like the coined walk it is derived from. We also
p lookattheeffectsofgraphsymmetryonthelimitingwalk,andshowthatthepropertiesaresimilar
- tothose of theDTQW as shown in [3].
t
n
a
u I. INTRODUCTION does not arise in CTQWs. So CTQWs and DTQWs
q on the same graph have state spaces with different di-
[
Quantum walks are unitary analoguesof classicalran- mensions. This makes it difficult to define a sequence of

1 domwalks,andhavemanyapplicationsinquantumcom- DTQWshavingaCTQWasalimit,ascanbedonewith
v puting, as well as being interesting objects in their own classical random walks.
0
right. Quantum walksare defined separatelyfor discrete There have been a number of previous studies of this
5
time (DTQW) [4–9]andcontinuoustime (CTQW) cases problem. In[20]adiscrete-timewalkwithoutacoinisde-
9
6 [10–12]. fined by alternating unitaries. In [21], a correspondence
0 Algorithmsbasedonclassicalrandomwalkscansolvea is shown between the limiting behavior of CTQWs and
. varietyofclassicalcomputationalproblemsefficiently, as DTQWsontheinfinitelineandthe3Dsquarelatticeare
1
shown in [13]. The quantum analogues also have a wide shown, and this is extended to generalgraphs in [22]. In
0
5 variety of applications in quantum computation. They [23], a continuous-time limit is found for a limited class
1 can be used to solve the element distinctness problem of one-dimensional quantum walks.
v: [14]. A QW-based search algorithm over the hypercube In[24]Childs givesadiscussionofthe relationshipbe-
i [15] performs as efficiently as Grover’s algorithm ([16]), tweenCTQWsandDTQWs,andpresentsaconstruction
X i.e, in time O(√N) for a database of size N. More ap- to discretize in the time domain, transforming a CTQW
r plications of DTQWs are described in [17]. CTQWs can to a DTQW, and also shows how to recover the CTQW
a
solvethe (albeitsomewhatartificial)“glued-trees”prob- from the discretized DTQW by isometry mapping. In
lemexponentiallyfasterthanthebestclassicalalgorithm this paper,wearemainly interestedinstartingfromdis-
[12]. They give a polynomial speed-up in evaluating the cretetime andproducinga continuoustime limit. More-
NANDtree[18],whichhasbeengeneralizedtoevaluating over,ourapproachdiffersfrompreviouswork,inthatthe
any Boolean formula [19]. resulting CTQW is a coined continuous-time walk. Dif-
While algorithms have been found based on both ferences between coined DTQWs and uncoined CTQWs
DTQWs andCTQWs, these walkscannotnecessarilybe have been noted in the past; we will see that the behav-
used interchangeably,unlike the classical case. To main- ior of our coined CTQWs is more similar to that of the
tainbothunitarityandnontrivialdynamicsinaDTQW, coined DTQWs than to standard CTQWs.
the statespaceis expandedtohavea“coinspace.” Such
We consider one particular algorithm that exhibits a
walks are often called “coined” walks. This problems
differencebetweenCTQWandDTQWimplementations.
The DTQW search on a 2D grid of size √N √N
×
with periodic boundary conditions was studied in [1],
∗ [email protected] where it was shown that a probabilistic search can be
† [email protected] done in O(√NlogN). But in [2] it was shown that any
2
CTQWsearchalgorithmtakestimeofΩ(N)onthesame connected back to v. If G is d-colorable,then we can al-
graph. We show that by taking the continuous time wayschoosej(v,i)=i,sowalkingthethesamedirection
limit of the DTQW search from [1], the search scaling twicetakesonebacktothevertexwhereonestarted. We
of O(√NlogN) is recovered. will assume that later in our construction. In this case,
We also look at another property of DTQWs in the S is not only unitary but also Hermitian.
continuoustimelimit. In[3],itisshownthataDTQWon Strictly speaking, F needn’t have the tensor product
agraphwithcertainsymmetriescanbereducedtoawalk structure C⊗I. It could be of the form F = iCi⊗Pi
on a smaller graph (the “quotient graph”) for certain where Pi are projection operators such thatPiPi = I
unitary time operators and initial conditions. There are (identity over p) and the Ci are unitary. TPhis would
H
manyinterestingconsequencesofsuchsymmetry;forex- allow the coin to differ at different parts of the graph.
ample, infinite hitting times for certaininitial conditions Thistypeofcoinisusedinthesearchalgorithmpresented
asshownin[25]. Inthispaper,weshowthatthelimiting in this paper to “mark” the node to be found.
CTQW inherits the symmetries from the DTQW.
Definition 2. A CTQW over G is defined by the uni-
In the next section we give the standard definitions
tary tranformation U(t) = e iHt and the state vector at
of DTQWs and CTQWs, and compare their definitions −
any time t is Ψ(t) = U(t) Ψ(0) . Here, H = H is a
to the continuous-time limit of a classical random walk. †
| i | i
Hermitian operator such that for vertices i=j,
In Sec. III, we present the construction of a family of
6
DTQWs on a graph with a well-defined continuous-time
=0 if i and j share an edge,
limit. In Sec. IV we apply this construction to the Hij(cid:26)6=0 otherwise,
discrete-time walk-based search algorithm on the torus,
and show that the continuous time limit of this walk ex-
and H R.
hibitsthesamescalingwiththegridareaN. InSec.Vwe ii ∈
lookatthe effects ofgraphsymmetry,andshowthatifa
Thereis acanonicalchoiceofHamiltonianH that one
standardDTQWhasareductiontoaquotientgraphdue
might call a standard CTQW. Let A = [A ] be the ad-
ij
to graph symmetry, the family of DTQWs also has a re-
jacency matrix of the graph G, so a = 1 if i and j are
ij
duction to the quotient graph, including the continuous-
connected by an edge and a = 0 otherwise; then for
ij
time limit. In Sec. VI we conclude.
i = j, H = κa , and H = κd , where κ is an energy
ij ij ii i
6
scale (or rate) and d is the degree of vertex i.
i
TheCTQWsdefinedinthispaperarenotofthisstan-
II. DEFINITIONS OF QUANTUM WALKS
dard form; but they do satisfy the broader definition
above.
A. Discrete- and continuous-time quantum walks
We now define DTQWs and CTQWs. These defini- B. Comparison with classical random walks
tions are taken from [3] and apply to d-regular graphs,
but, the definitions can be extended to irregular graphs.
Discrete time classicalrandomwalks,which area spe-
Let Gbe a d-regulargraphonwhichthe walk is defined.
cialcaseofMarkovchains,admitacontinuoustimelimit.
TheHilbertspaceoftheDTQWis c p. Intheposi-
These are defined by linear difference equations with
tion space p, a basis vector v isHass⊗ocHiated with each
probabilities represented as vectors. The limiting pro-
H | i
ofthe verticesv,andinthe coinspace c,abasisvector
cess is well established and is found in most discussions
H
i is associated with each of the edges emanating from
of classical random walks (for example [26]).
| i
each vertex of G; i is a label of the different directions
LetaclassicalrandomwalkbedefinedoveragraphG.
one can walk. The basis vectors of the coin and vertex
Let p be a vector whose ith entry is the probability of
n
states together are i,v i v .
being at the vertex i at time step n. The time evolution
{| i≡| i⊗| i}
Definition 1. The time evolution of a state vector in a is given by
DTQW is Ψ =U Ψ , where U =SF, where
| n+1i | ni pn+1 =Mpn, (3)
S = j(i,v),v(i) i,v (1)
| ih | where M =[mij] is a stochastic matrix. The probability
Xv Xi m to gofromvertexi tovertexj is zerounless anedge
ji
connectsitoj. Inastandardundirectedrandomwalkon
and
G, from a vertex i there is an equal probability to walk
F =C I. (2) along any of the edges connected to i, so mji = 1/di
⊗ where d is the degree of vertex i.
i
In this definition, S is the shift operator and F is the As seen in [24], we can replace M by ǫM+(1 ǫ)I to
−
coin-flipoperator. Thesearebothunitary. Here,visany obtainafamilyofwalksparametrizedbyǫ. Thestandard
vertex of G; v(i) is the vertex connected to v along the discrete-time random walk corresponds to ǫ=1. Taking
direction i; and j(i,v) is the direction by which v(i) is the limit ǫ 0 while defining the time to be t =nǫ, we
n
→
3
obtain a differential equation for the probability vector A. Family of DTQWs and limit
p:
Constructing the family of DTQWs is based ona sim-
dp(t) ple property of operators that are both Hermitian and
=(M I)p(t). (4)
dt − unitary. If a finite dimensional operator A is Hermitian
and unitary, then A2 =I, and
This equation gives the continuous-time limit of the dis-
crete time classical random walk. e−iπ2(A−I) =A. (6)
Itisclearthatboththediscrete-timerandomwalkand By assumption, both S and F are Hermitian. Define a
its continuous-timelimit havethe samenumberofstates family of step operators
inthe Markovchain. But, inthe quantumcase,the con-
tinuoustimedynamicsisdefinedthroughtheSchr¨odinger U(s)=e−iπ2s(S−I)e−iπ2s(F−I). (7)
equation
where s is a parameter s [0,1].
∈
d Ψ(t) Lemma 1. U(s) is a local transformation for all s, and
| i = iH Ψ(t) , (5) U(1)=SF
dt − | i
Proof. Since S and F are both Hermitian and unitary,
where the state space is of the same dimension as the
S2 =F2 =I. This implies that
number of vertices in G. But, as described above, the
state spaceofthe DTQWalsoincludes the “coinspace,” e−iπ2s(S−I) =eiπ2s(cos(πs/2)I isin(πs/2)S),
−
and thus the dimensions of the state spaces of a DTQW
and similarly for F. Hence, U(s) = eiπs(cos2(πs/2)I
and a CTQW over the same graph G are different. We icos(πs/2)sin(πs/2)(S +F) sin2(πs/2)SF). Since I−,
will overcome this difficulty by retaining the coin space −
S,F andSF arealllocaltransformationsoverG,U(s)is
inthecontinuouslimitoftheDTQW.Wecanthenthink
alsoalocaltransformationoverG. Ifwe takes=1 then
of this as either a CTQW over a different (but related)
cos(πs/2)=0 and eiπs = 1, so we get U(1)=SF.
graph G′, or as a coined continuous-time walk. −
Let the family of local transformations U(s) be called
. SF . As s 0 , U(s) I iπs(S+F 2I)+
F ∈F → → − 2 −
O(s2). This shows that as s 0, the local transforma-
III. CONTINUOUS TIME LIMIT
→
tionsin behavelikethe CTQWoveranothergraphG
′
F
with Hamiltonian = S +F 2I (up to a time scal-
In this section, we show how to construct a family of ing of π). The coHnnectivity o−f G is described by the
quantumwalks,startingfromaDTQWdefinedonareg- 2 ′
entries of as seen in the definition of CTQW. Hence,
ular, d-colorable, undirected graph of degree d with a H
thisCTQWcanbeseenasthelimitofthefamilyoflocal
Hermitian coin flip operator F. This family of walks is
transformations as s 0. The continuous-time limit
parametrized by a real number s>0, and has as a limit F →
defined by =S+F 2I is equivalent to that defined
as s 0 a CTQW over a different but related graph. H −
→ by =S+F up to a global phase.
This walk can be considered a continuous-time coined H
walk. This construction is illustrated with an example,
and a few properties ofthe continuoustime limit are de-
B. Relationship between the original and new
rived. graph: the coined continuous-time walk
LetGbeanundirectedgraphwithaDTQWdefinedby
theshiftoperatorS andcoinflipoperatorF. S isdefined Henceforth in the paper, we denote with a prime the
by Eq. (1), with j(i,v)=i (which can always be done if graph over which the limiting CTQW of a DTQW is
G is regular and d-colorable). By the definition of S, it
defined; the original graph is unprimed. The number
isHermitian. WeassumealsothatF isHermitian;many of vertices in the limiting graph G is the same as the
′
widely studied coin operators (e.g., the Hadamard and
dimension of the state space of G. Hence, we can index
Grover coins) satisfy this assumption. In this standard the states associated with G with the same labels as
′
form of a DTQW, the shift operator S is a Hermitian
the states associated with G. Each vertex subspace of
permutation matrix of order 2 and the coin flip operator the original walk is mapped onto a collection of vertices
F maps a basis state of the coin space associated with a
in G. The edges among these vertices is given by the
′
vertex v to a superposition of such basis states. coin flip operator F, and each of them is connected to
a neighboring vertex of the original graph G. We can
Definition 3. Aunitarytransformation U actingon the groupthese collections of vertices together,and consider
state space of a graph G is a local transformation if it this limiting case to be a coined CTQW.
maps any vector associated with a vertex v to a super- Notethatifweallowselfloops,thentherearemultiple
position of vectors associated with v and those associated possible graphs G on which defines a CTQW. We
′
with vertices sharing an edge with v. thereforeconsideronlytheoneGH whichhasnoselfloops.
′
4
Example Consider the graph G to be the square Define λ =2sinφj. Then
j 2
shown in Fig. 1. Because this is a cycle (d = 2) with
an even number of vertices (n = 4), we can define a e i(S F)t = cosλ t j j isinλjt j j (S F) .
DTQW with a coin of dimension 2, and the graph G − − (cid:18) j | ih |− λ | ih | − (cid:19)
Xj j
is 2-colorable. The coin flip operator F could be any
2 2 matrix that is both Hermitian and unitary; the Whenφ issufficientlysmall,λ φ . Roughlyspeak-
Ha×damard is a commonly used choice. The CTQW is ing, this mj eans that the limitinjg≈waljk defined by the
defined on the graph G′ as shown in Fig. 2. Each ver- Hamiltonian =S F approximatestheDTQWpretty
tex of the original graph G is mapped to two vertices well, as far aHs time−evolution in the state space is con-
of G′, one for each coin state of each vertex; edges be- cerned.
tweencoinstatesofthe samevertexrepresent“coinflip”
transitions, while edges between coin states of different
vertices represent “shift” transitions. IV. SEARCH ALGORITHM
In the search algorithm as described in [1], S is the
shift operator over a 2D √N √N grid with periodic
×
boundary conditions, and the coin flip operator is
F =C I (C C ) x x , (8)
0 0 1
⊗ − − ⊗| ih |
where
1
C =2 S S I, S = i , C = I. (9)
0 c c c 1
| ih |− | i √d | i −
FIG. 1: graph G Xi
C iscalledtheGrovercoin. Thestate S istheuniform
0 c
| i
superposition of all coin states, and d is the dimension
of coin space. The state x is the marked vertex which
| i
is to be found. The idea behind the algorithm is that
we use the coin flip C on all vertices other than x, and
0
use the coin flip C on x. We start with a state which
1
is symmetric on all the basis states, and use this vertex-
dependentcointo“accumulate”probabilityinthestates
associated with x. Note that the number of coin states
associated with each vertex is d = 4, and S and F are
both Hermitian; hence (SF) 1 =FS.
−
Notation: We have already defined S as the uni-
c
| i
form superposition of all d = 4 coin basis states. We
FIG. 2: graph G
′ similarly define
1
S = v (10)
v
| i √N | i
Xv
C. Properties of continuous-time limit
as the uniform superposition of all vertex states. We
denote S ,S = S S .
c v c v
| i | i⊗| i
The evolution of the DTQW is the same if the coin F Consider the continuous limit of the algorithm with
is replaced with F except for a time-dependent global HamiltonianS F, as described in the previous section.
− −
phase. However, the family of maps may differ. The The probability of being at a vertex v in the continuous-
requirementforthistransformationwillbecomeapparent time walk is the probability of being in the subspace
inthesectionstofollow,duringtheanalysisofthesearch spanned by the vectors associatedwith v in the DTQW.
algorithm.
Theorem 3. The continuous limit of the DTQW search
algorithm described in [1] is a CTQW search algorithm
Lemma2. If j isaneigenvector ofSF witheigenvalue
eiφj, then |ji|isi also an eigenvector of (S − F)2 with (wtihtahtHisa,mOil(t√onNialnogSN−))F., with the same time complexity
eigenvalue 4sin2 φj. So if SF = eiφj j j then
2 j | ih |
P Proof. The initial state is Sc,Sv . Let x be the marked
| i
φ vertex. The theorem follows from five partial results:
(S F)2 = 4sin2 j j j .
− Xj 2 | ih | Claim 1. (S−F)|Sc,Svi= √2N |Sc,xi
5
Proof. S S ,S = S ,S , F S ,S = S ,S Claim 5. The magnitudes in Eq. (14) are:
c v c v c v c v
| i | i | i | i −
2 x S S ,x , and x S = 1/√N. Putting these
v c v
h | i| i h | i
together yields the result. 1
Φ (t) =O ,
k| rem1 ik (cid:18)logN(cid:19)
Claim 2. If j is an eigenvector of SF with eigenvalue
eiφj then | i
1
eiφj/2 Sc,Sv w α 1/2 =O .
|jihj |Sc,Svi=−i√Nsin(φ /2)|jihj |Sc,xi. kh | ± i∓p k (cid:18)logN(cid:19)
j
Proof. Taking the state at t=0,
Proof.
1
hj |Sc,Svi=e−iφjhj|FS|Sc,Svi |Ψ(0)i=|Sc,Svi= √2(|wαi−|w−αi)+|Φremi
=e−iφj j (Sc,Sv 2 x Sv Sc,x ).
h | | i− h | i| i = Sc,Sv w w + Sc,Sv w w +Φ (0) ,
α α α α rem1
h | i| i h | − i| − i | i(15)
Solving for j S ,S yields the result.
c v
h | i
Claim 3. which implies
|Ψ(t)i==eX−ji((cid:16)S−coFs)(tλ|jStc),+Svei−iφ2j sin(λjt)(cid:17)|jihj |Sc,S(v1i1.) |Φre(cid:16)mhiS=c,(cid:16)ShvS|cw,S−vαi|w+αpi−1/p2(cid:17)1|/w2−(cid:17)α|wi+αi|+Φrem1i. (16)
Since all three vectors on the RHS are orthogonal, their
Proof. Follows from Lemma 2, Claim 1 and Claim 2.
norms must each be less than or equal to the norm of
Φ on the LHS. Since Φ = Θ(1/logN), the
rem rem
| i k| ik
Claim4. Theprobability ofbeingatthenodexatatime result follows.
instant t is S ,x Ψ(t) 2.
c
|h | i|
Claim 6. At t=π/2θ ,
α
Proof. The initial state is symmetric with respect to all
directionsaboutthemarkednodex. TheunitariesS and 1
S ,x Ψ(t) =Ω .
F preservethisproperty. Thismeansthattheamplitude c
kh | ik (cid:18)√logN(cid:19)
of being at eachcoin state associatedwith x atany time
t is the same. The result follows from this.
Proof.
The initial state is Ψ(0) = Ψ = S ,S . In [1]
| i | 0i | c vi Ψ(π/2θ ) =e iα/2 w w S ,S
they show that there are eigenstates wα and w α of | α i − | αih α | c vi
U′ = SF with eigenvalues eiα and |e−iαi, resp|ec−tiviely, eiα/2 w α w α Sc,Sv + Φrem1(π/2θα) . (17)
such that − | − ih − | i | i
The result follows from Claim 5.
1
S ,S = (w w )+ Φ (12)
c v α α rem
| i √2 | i−| − i | i
With this last result we can prove the theorem.
and The probability of the particle being at the node x is
Ω(1/logN) at t = π/2θ where θ = Θ(1/√N). Re-
α α
1
S ,x w + S ,x w =Θ( ), (13) peatingthealgorithmO(logN)timesgivesusaconstant
c α c α
kh | i h | − ik √logN probability of finding the marked item. The time com-
plexity of the continuous time search algorithm is there-
where k|Φremik = Θ(log1N) and α = Θ(√1N). From fore O(√NlogN).
Lemma 2, it follows that
This result is in contrast to the proof in [2] that any
Ψ(t) =[f(α,t) wα wα Sc,Sv ] CTQWsearchalgorithmovera √N √N gridwithcir-
| i | ih | i
×
=[f( α,t) w α w α Sc,Sv ]+ Φrem1(t) . (14) cular boundary conditions takes Ω(N) time. But, by
− | − ih − | i | i
defining the CTQW search as the continuous time limit
Here, f(a,t) = cos(θat)+e−ia2 sin(θat), Φrem1(t) is a on the related graph, as described in Sec. III—that is,
| i
vector perpendicular to both w and w , and θ = a coined CTQW—the search becomes as efficient as the
α α a
2sin(a). | i | − i DTQW search algorithm.
2
6
V. EFFECT OF GRAPH SYMMETRIES DTQW. Hence, the CTQW given by the Hamiltonian
H
over G can be seen as the continuous limit of a DTQW
′H
In [3] it was shown that if a DTQW with an appro- over the quotient graph in the limit s 0.
→
priate unitary evolution is defined on a graph with sym-
metries,thenforcertainsymmetricinitialconditionsthe Proof. By the proof of Lemma 1, U(s) = a2I +ab(S +
walkcanbereducedtoawalkonitsquotientgraph. This F)+b2(SF). I, S+F andSF arelocaltransformations
is a smallergraphobtainedby identifying certaingroups over G . Hence U(s) is a local transformation over G
H H
of vertices and edges. We briefly review the reduction. s [0,1].
∀ ∈
Let H be a subgroup of the symmetry group over n
π
letters, where n is the dimension of the state-space of a U(s)=I i s(S+F 2I)+O(s2).
DTQWoverG. TheelementsofH aresuchthat h H, − 2 −
∀ ∈
[σ(h),S] = 0, where σ(h) is the matrix representation
Taking the limit s 0, as shown in Section III, we see
of the permutation and S is the shift operator. It is →
that =S+F is a Hamiltonian that defines a CTQW
shown in [3] that if U = SF (where F is the coin flip H
over G .
operator) and [U,σ(h)] = 0 for every h H, then there ′H
∈
exists a common set of eigenvectors A = , with
x
{ |O i}
eigenvalue 1 for all σ(h), such that
VI. CONCLUSION
A, U A. (18)
x x
∀|O i∈ |O i∈
In classical random walks, there is a straightforward
Locality of the graph is preserved, in the sense that two
sense inwhich continuoustime randomwalksarea limit
vectors and in A are connected if and only if
|Oxi |Oyi of discrete time random walks. Both can be defined on
every component i of is connected to some com-
| i |Oxi thesamegraph,withbehaviorsthatareoppositelimitsof
ponent j of in the graph G. Mathematically, this
| i |Oyi a continuous family of evolution rules. Because discrete-
means that S =0 if and only if for every state
hOy| |Oxi6 time and continuous-time quantum walks are defined on
i suchthat i =0thereexistsastate j suchthat
| i h |Oxi6 | i statespaceswith differentdimensions, constructingsuch
j =0 and j S i =0.
h |Oyi6 h | | i6 a correspondence is not simple. A small number of at-
The “quotient graph” G is the graph whose states
H tempts have been made to overcome this problem.
are the vectors in A and whose connectivity is defined
This paper presents a different approach. For a par-
as above. Two states and are associated with
|Oxi |Oyi ticular class of DTQWs with Hermitian coins and shift
thesamevertexinG if,foreverystate c ,v ofGsuch
H | 1 i operators, the continuous-time limit of a DTQW on a
that c ,v =0, there exists a coin state labeled by
h 1 |Oxi6 graphG is a continuous-timewalk ona different, but re-
c such that c ,v =0.
2 h 2 |Oyi6 lated,graphG′. Thetwoevolutionrulescanbedefinedas
Wecaneasilyseethatif[U,σ(h)]=0and[S,σ(h)]=0,
opposite limits of a continuous family of evolution rules,
then [F,σ(h)] = 0. Let F be Hermitian. Then by the
justasinthe classicalcase. Wecanthink ofthis walkon
results ofSection III the continuous limitof the DTQW
the graphG asbeing acoined continuoustime quantum
′
is generated by the Hamiltonian = S + F over the
H walk.
related graph G. From the above, [σ(h), ]=0.
′ H Because the continuous-time limit is defined on the
Theorem 4. defines a CTQW over G whose basis samespaceastheDTQW,itsharesmanypropertieswith
states are the bHasis states of G . ′H the originalwalk. We have shown, for example, that the
H
continuous-timelimitoftheDTQWsearchalgorithmhas
Proof. The proof is similar to as given in [3]. Since the the same √N speed-up as the original algorithm; the
x are eigenvectors of σ(h) with eigenvalue 1, usual CTQW on the same graph has no speed-up.
{|O i}
Similarly, DTQWs on symmetric graphs can exhibit a
σ(h) = σ(h) = . (19)
x x x reductiontoaDTQWonasmallerquotientgraph. This
H|O i H |O i H|O i
propertyiscloselyconnectedtotheexistenceofquantum
This proves that is an operator on the space spanned
H speed-upsincertainquantum-walkbasedalgorithms[12].
by the vectorsin A. As in Section III,we candefine the
We have shown the the CTQW limit of this walk shares
graphG whoseverticescorrespondto the vectorsinA.
′H this reduction to a walk on the quotient graph.
Hence, defines a CTQW overG , or a coined CTQW
H ′H The ability to take such limits—and the existence of
over G .
H coined CTQWs—adds another tool to the arsenal of
It straightforwardly follows that this walk is the quantum walks, and one that deserves to be further ex-
continuous-time limit of a DTQW over the quotient plored. Moreover,theinterestingquestionofwhytheuse
graph G . ofcoinedwalkscansometimesproducespeed-upsalsode-
H
serves further study. In addition to their own beautiful
Theorem 5. The family of unitary operators U(s) = properties, quantum walks have proven to be a fertile
e−iπ2s(S−I)e−iπ2s(F−I), parametrized by s (0,1], define field for the study of quantum algorithms. We hope to
∈
a local transformation over GH, which for s = 1 is a illuminate these questions in future work.
7
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