<s>
Quantum	B-Algorithm
walks	I-Algorithm
are	O
quantum	O
analogues	O
of	O
classical	O
random	O
walks	O
.	O
</s>
<s>
In	O
contrast	O
to	O
the	O
classical	O
random	O
walk	O
,	O
where	O
the	O
walker	O
occupies	O
definite	O
states	O
and	O
the	O
randomness	O
arises	O
due	O
to	O
stochastic	O
transitions	O
between	O
states	O
,	O
in	O
quantum	B-Algorithm
walks	I-Algorithm
randomness	O
arises	O
through	O
:	O
(	O
1	O
)	O
quantum	O
superposition	O
of	O
states	O
,	O
(	O
2	O
)	O
non-random	O
,	O
reversible	O
unitary	O
evolution	O
and	O
(	O
3	O
)	O
collapse	O
of	O
the	O
wave	O
function	O
due	O
to	O
state	O
measurements	O
.	O
</s>
<s>
As	O
with	O
classical	O
random	O
walks	O
,	O
quantum	B-Algorithm
walks	I-Algorithm
admit	O
formulations	O
in	O
both	O
discrete	O
time	O
and	O
continuous	O
time	O
.	O
</s>
<s>
Quantum	B-Algorithm
walks	I-Algorithm
are	O
motivated	O
by	O
the	O
widespread	O
use	O
of	O
classical	O
random	O
walks	O
in	O
the	O
design	O
of	O
randomized	O
algorithms	O
,	O
and	O
are	O
part	O
of	O
several	O
quantum	B-Device
algorithms	I-Device
.	O
</s>
<s>
For	O
some	O
oracular	O
problems	O
,	O
quantum	B-Algorithm
walks	I-Algorithm
provide	O
an	O
exponential	O
speedup	O
over	O
any	O
classical	O
algorithm	O
.	O
</s>
<s>
Quantum	B-Algorithm
walks	I-Algorithm
also	O
give	O
polynomial	O
speedups	O
over	O
classical	O
algorithms	O
for	O
many	O
practical	O
problems	O
,	O
such	O
as	O
the	O
element	O
distinctness	O
problem	O
,	O
the	O
triangle	O
finding	O
problem	O
,	O
and	O
evaluating	O
NAND	O
trees	O
.	O
</s>
<s>
The	O
well-known	O
Grover	B-Algorithm
search	I-Algorithm
algorithm	I-Algorithm
can	O
also	O
be	O
viewed	O
as	O
a	O
quantum	B-Algorithm
walk	I-Algorithm
algorithm	O
.	O
</s>
<s>
Quantum	B-Algorithm
walks	I-Algorithm
exhibit	O
very	O
different	O
features	O
from	O
classical	O
random	O
walks	O
.	O
</s>
<s>
Continuous-time	O
quantum	B-Algorithm
walks	I-Algorithm
arise	O
when	O
one	O
replaces	O
the	O
continuum	O
spatial	O
domain	O
in	O
the	O
Schrödinger	O
equation	O
with	O
a	O
discrete	O
set	O
.	O
</s>
<s>
Under	O
particular	O
conditions	O
,	O
continuous-time	O
quantum	B-Algorithm
walks	I-Algorithm
can	O
provide	O
a	O
model	O
for	O
universal	O
quantum	B-Architecture
computation	I-Architecture
.	O
</s>
<s>
This	O
construction	O
naturally	O
generalizes	O
to	O
the	O
case	O
that	O
the	O
discretized	O
spatial	O
domain	O
is	O
an	O
arbitrary	O
graph	O
and	O
the	O
discrete	O
laplacian	O
is	O
replaced	O
by	O
the	O
graph	O
Laplacian	O
where	O
and	O
are	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
and	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
respectively	O
.	O
</s>
<s>
Common	O
choices	O
of	O
graphs	O
that	O
show	O
up	O
in	O
the	O
study	O
of	O
continuous	O
time	O
quantum	B-Algorithm
walks	I-Algorithm
are	O
the	O
d-dimensional	O
lattices	O
,	O
cycle	O
graphs	O
,	O
d-dimensional	O
discrete	O
tori	O
,	O
the	O
d-dimensional	O
hypercube	O
and	O
random	O
graphs	O
.	O
</s>
<s>
The	O
evolution	O
of	O
a	O
quantum	B-Algorithm
walk	I-Algorithm
in	O
discrete	O
time	O
is	O
specified	O
by	O
the	O
product	O
of	O
two	O
unitary	O
operators	O
:	O
(	O
1	O
)	O
a	O
"	O
coin	O
flip	O
"	O
operator	O
and	O
(	O
2	O
)	O
a	O
conditional	O
shift	O
operator	O
,	O
which	O
are	O
applied	O
repeatedly	O
.	O
</s>
<s>
When	O
this	O
choice	O
is	O
made	O
for	O
the	O
coin	O
flip	O
operator	O
,	O
the	O
operator	O
itself	O
is	O
called	O
the	O
"	O
Hadamard	O
coin	O
"	O
and	O
the	O
resulting	O
quantum	B-Algorithm
walk	I-Algorithm
is	O
called	O
the	O
"	O
Hadamard	O
walk	O
"	O
.	O
</s>
<s>
A	O
quantum	B-Algorithm
walk	I-Algorithm
corresponds	O
to	O
iterating	O
the	O
shift	O
and	O
coin	O
operators	O
repeatedly	O
.	O
</s>
