Gavin Brennen Lauri Lehman Zhenghan Wang Valcav Zatloukal JKP
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Transcript of Gavin Brennen Lauri Lehman Zhenghan Wang Valcav Zatloukal JKP
Gavin BrennenLauri Lehman
Zhenghan Wang Valcav Zatloukal
JKP
Ubergurgl, June 2010
Anyonic quantum walks:Anyonic quantum walks:The Drunken SlalomThe Drunken Slalom
Random evolutions of topological structures arise in:•Statistical physics (e.g. Potts model):
Entropy of ensembles of extended object •Plasma physics and superconductors:
Vortex dynamics•Polymer physics:
Diffusion of polymer chains•Molecular biology:
DNA folding•Cosmic strings•Kinematic Golden Chain (ladder)
Anyonic Walks: Motivation
Quantum simulation
Anyons
2ie
Bosons
Fermions
U
ei 2
Anyons
3D
2D
View anyon as vortex with flux and charge.
•Two dimensional systems •Dynamically trivial (H=0). Only statistics.
Ising Anyon Properties• Define particles:
• Define their fusion:
• Define their braiding:
,,1
1
11
B
,,1,Fusion Hilbert space:
Ising Anyon Properties• Assume we can:
– Create identifiable anyons pair creation
– Braid anyons Statistical evolution: braid representation B
– Fuse anyons
time
,1,B
1
1
1
B
1
1
Approximating Jones Polynomials
Knots (and links) are equivalent to braids with a “trace”.
[Markov, Alexander theorems]
“trace”
Approximating Jones Polynomials
Is it possible to check if two knots are equivalent or not? The Jones polynomial is a topological invariant: if it differs, knots are not equivalent. Exponentially hard to evaluate classically –in general.Applications: DNA reconstruction, statistical physics…
[Jones (1985)]
“trace”
Approximating Jones Polynomials
4
1t
4 t
[Freedman, Kitaev, Wang (2002); Aharonov, Jones, Landau (2005);et al. Glaser (2009)]
With QC polynomially easy to approximate: Simulate the knot with anyonic braiding
tt
1
Take “Trace”
“trace”
Classical Random Walk on a line
Recipe:1)Start at the origin2)Toss a fair coin: Heads or Tails3)Move: Right for Heads or Left for Tails4)Repeat steps (2,3) T times5)Measure position of walker6)Repeat steps (1-5) many times
Probability distribution P(x,T): binomial Standard deviation:
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
T~x2
QW on a line
Recipe:1)Start at the origin2)Toss a quantum coin (qubit):
3)Move left and right:4)Repeat steps (2,3) T times5)Measure position of walker6)Repeat steps (1-5) many times
Probability distribution P(x,T):...
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
2)/10(1H2)/10(0H
1,1xx,1S,1,0xx,0S
1111
21H
QW on a line
Recipe:1)Start at the origin2)Toss a quantum coin (qubit):
3)Move left and right:4)Repeat steps (2,3) T times5)Measure position of walker6)Repeat steps (1-5) many times
Probability distribution P(x,T):...
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
1,1xx,1S,1,0xx,0S
2)/10(1H2)/10(0H
1111
21H
CRW vs QW
QWCRW
Quantum spread ~T2, classical spread~T [Nayak, Vishwanath, quant-ph/0010117;
Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC (2001)]
22 xx
P(x,
T)
QW with more coins
Variance =kT2
More (or larger) coins dilute the effect of interference (smaller k)New coin at each step destroys speedup (also decoherence) Variance =kT[Brun, Carteret, Ambainis, PRL (2003)]New coin every two steps?
dim=2
dim=4
QW vs RW vs ...?• If walk is time/position independent then
it is either: classical (variance ~ kT) or quantum (variance ~ kT2)
• Decoherence, coin dimension, etc. give no richer structure...
• Is it possible to have time/position independent walk with variance ~ kTa for 1<a<2?
• Anyonic quantum walks are promising due to their non-local character.
Ising anyons QW
QW of an anyon with a coin by braiding it with other anyons of the same type fixed on a line.
Evolve with quantum coin to braid with left or right anyon.
sb1s1 2 s 1s 1n n
1sb
Ising anyons QWEvolve in time e.g. 5 steps
What is the probability to find the walker at position x after T steps?
Hilbert space:
n~2~2
(n)H(n)HHH(n)
n
positionanyonsqubit
Ising anyons QW
P(x,T) involves tracing the coin and anyonic degrees of freedom:
add Kauffman’s bracket of each resulting link (Jones polynomial)P(x,T), is given in terms of such Kauffman’s brackets: exponentially hard to calculate! large number of paths.
Markov122001 )B(B)BΨΨtr(B
TIM
E
Trace (in pictures)
Markov122001 )B(B)BΨΨtr(B
0
1B
2B
0
Trace & Kauffman’s brackets
Ising anyons QWEvaluate Kauffman bracket.
Repeat for each path of the walk.
Walker probability distribution depends on the distribution of links (exponentially many).
A link is proper if the linking between the walk and any other link is even.
Non-proper links Kauffman(Ising)=0
B
1
1
Locality and Non-LocalityPosition distribution, P(x,T):
L
τ(L)z(L)
T propernonisLif0properisLif1)(
21T)P(x,
•z(L): sum of successive pairs of right steps•τ(L): sum of Borromean rings
Very localcharacteristic
Very non-localcharacteristic
Ising QW Variance
The variance appears to be close to the classical RW.
step, T
Varia
nce
~T
~T2
Ising QW Variance
Assume z(L) and τ(L) are uncorrelated variables.
local vs non-local
T)(x,rT)(x,rNNT)(x,δPT)(x,PT)(x,P oddτevenτ
total
properQWRWAQW
step, T step, T
Anyonic QW & SU(2)k
The probability distribution P(x,T=10) for various k. k=2 (Ising anyons) appears classical k=∞ (fermions) it is quantumk seems to interpolate between these distributions
position, x
index k
prob
abilit
yP(
x,T=
10)
2a1a
•Possible: quant simulations with FQHE, p-wave sc, topological insulators...?
•Asymptotics: Variance ~ kTa 1<a<2 Anyons: first possible example
•Spreading speed (Grover’s algorithm) is taken over by •Evaluation of Kauffman’s brackets (BQP-complete problem)
•Simulation of decoherence?
Conclusions
Thank you for your attention!