1. Quantum
walks:
Test
case
for
Quantumness
Pavan
Iyengar
Project
Fellow
Prof.TS
Mahesh
NMR-‐QIP
Lab
IISER
Pune
Earlier
at
RRI,
Bengaluru
as
a
VSP
2. Discussion
structure
• What
are
1
dimensional
quantum
(not
random!)
walks
(in
relaPon
to
Classical
random
walks)
?
• Noise
(models)
model
a
quantum
system
into
one
with
a
classical
signature
(read
Gaussian,
apart
from
coherent
states!)
.
• Why
a
quantumness
index
!
That’s
so
non-‐
classical!
3. What
are
1
dimensional
quantum
walks?
• First
quesPon,
What
is
a
walk
-‐
A
(Classical)
random
walk
is
the
simulaPon
of
the
random
movement
of
a
parPcle
on
a
line.
• A
quantum
walk
is
analogous
–
but
with
simultaneous
right
AND
leY
steps.
So
(Classical)
Random
walk
with
simultaneous
superposiPon
of
the
leY
and
the
right
walk
gives
the
quantum
walk
distribuPon.
The
difference
is
in
the
quantum
‘gunta’
…
i.e.
entanglement
• (FYI!)
Random
walks
are
a
useful
model
for
developing
classical
algorithms;
quantum
walks
provide
a
new
way
of
developing
quantum
algorithms
– which
is
parPcularly
important
because
producing
new
quantum
algorithms
is
so
hard
4. Classical
random
walk
on
the
line
Consider a walk on the following simple infinite graph:
• When
the
walker
has
equal
probability
to
move
leY
or
right,
it s
well-‐known
that
the
average
distance
from
the
start
posiPon
aYer
Pme
n
is
sqrt(n)
• But
we
can
define
a
quantum
walk
on
the
same
graph
with
different
behaviour:
an
average
distance
of
n
.
This
speedup
is
due
to
the
leY
and
right
step
contribuPng
in
both
direcPon
travel
simultaneously
!
(itni
bhi
kya
jaldi
….)
5. Quantum
walk
on
the
line
• We
have
two
quantum
registers:
a
coin
register
holding
|L〉
or
|R〉,
and
a
posiPon
register
|x〉
• Our
walk
operaPon
is
a
coin
flip
followed
by
a
shiY
(discrete)
– coin
flip:
– shiY:
send
|L〉
→
|L〉
+
i|R〉,
|R〉
→|L〉
-‐
i
|R〉
send
|L〉|x〉
→
|L〉|x-‐1〉
|R〉|x〉
→
|R〉|x+1〉
• These
are
both
unitary
operaPons,
and
hence
their
combinaPon
is
too
– so,
together,
they
provide
a
way
of
defining
a
quantum
walk
on
the
line
– there
are
other
ways
–
e.g.
the
con-nuous-‐-me
formulaPon
of
quantum
walks
6. A
few
iteraPons
of
the
walk
on
the
line
1.
start
→
|R〉|0〉
2.
coin
→
(|L〉
+
i|R〉)|0〉
shiY
→
i|L〉|-‐1〉
+
|R〉|1〉
3.
coin
→
(i|L〉
-‐
|R〉)|-‐1〉
+
(i|L〉
+
|R〉)|1〉
shiY
→
i|L〉|-‐2〉
-‐
|R〉|0〉
+
i|L〉|0〉
+
|R〉|2〉
4.
coin
→
(i|L〉
-‐
|R〉)|-‐2〉
+
(i|L〉
+
|R〉)|2〉
shiY
→
i|L〉|-‐3〉
-‐
|R〉|-‐1〉
+
i|L〉|1〉
+
|R〉|3〉
Equal
probability
to
be
at
|-‐3〉,
|-‐1〉,
|1〉
or
|3〉
-‐
whereas
classical
random
walk
favours
|-‐1〉,
|1〉
7. Classical
vs.
quantum
walk
on
the
line
Running a
classical
walk on the
line
results in a
probability
distribution
like the left
graph
Whereas running this quantum walk for the
same number of steps gives the right graph
The peaks and troughs in the right graph are caused by
quantum interference.
9. Goal
was
(and
is)
-‐
• To
find
whether
Photosynthesis
uses
quantum
mechanics
(holy
grail
–
increase
efficiency
of
Organic
Dye
sensiPsed
solar
cells)
• ConPnuous
quantum
walks
have
been
proposed
as
models
for
Energy
Transport
(through
Dipole
int.)
in
the
FMO
complex
in
Photosynthesis
.
(Mohseni
et.al.)
• So
the
present
study
on
discrete
quantum
walks!
17. Conclusions
and
further
reading
• Density
matrices
of
Discrete
Quantum
Walks
at
slightly
different
Pmes
(t,t-‐2)
don’t
commute
at
low
noise.
A
rela've
measure
of
quantumness.
• Our
scheme
usefull
experimentally
when
we
have
tomographic
data
.
For
interference
experiments
Fazio
et.al.
propose
AnP
Commutators
to
measure
quatumness.
• Working
on
NMR
expt.
Design
and
feasibility
of
our
work
for
Quantum
Dots
(cyclic
array
cont.qw),
Quantum
Photosynthesis
...
• Further
reading
(on
www.arxiv.org):
–
Quantum
random
walks
–
an
introductory
overview ,
J.
Kempe,
quant-‐ph/0303081
2 books on Quantum
Walks NOT written by me !
18. Prof.R.Srikanth (PPISR,Visiting
Prof. @ RRI ) (summer internships!)
Thanks
to
my
Collaborators
,
mentors
and
guides
(IISER
Pune,UoP
Pune,
IISC,RRI,IUCAA
Sci-‐pop)
and
last
but
not
the
least
…
Family
and
Friends.
GN.Chandan , Grad
Student at PPISR
20. QuesPons
(shooPng
round
2)
…
Catch
me
at
iyengarpavan@gmail.com
AdverPsement-‐
A
lot
of
usefull
and
impacpull
Info
for
B.Sc.,
M.Sc.
Students
on
pavaniyengar.weebly.com
(including
this
work
and
towards
Quantum
Biology)
