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Gavin Brennen Lauri Lehman Zhenghan Wang Valcav Zatloukal JKP

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Anyonic quantum walks:The Drunken Slalom

Gavin Brennen

Lauri Lehman

Zhenghan Wang

Valcav Zatloukal

JKP

Ubergurgl, June 2010

Anyonic Walks: Motivation

- Random evolutions of topological structures arise in:
- Statistical physics (e.g. Potts model):
- Entropy of ensembles of extended object

- Statistical physics (e.g. Potts model):
- Plasma physics and superconductors:
- Vortex dynamics

- Diffusion of polymer chains

- DNA folding

Quantum simulation

Bosons

Fermions

Anyons

Anyons

- Two dimensional systems
- Dynamically trivial (H=0). Only statistics.

3D

2D

View anyon as vortex with flux and charge.

Define particles:

Define their fusion:

Define their braiding:

Fusion Hilbert space:

Ising Anyon Properties

Assume we can:

Create identifiable anyons

pair creation

Braid anyons

Statistical evolution:

braid representation B

Fuse anyons

Ising Anyon Properties

time

Approximating Jones Polynomials

“trace”

Knots (and links) are equivalent to braids with a “trace”.

[Markov, Alexander theorems]

Approximating Jones Polynomials

“trace”

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)]

Approximating Jones Polynomials

“trace”

Take “Trace”

With QC polynomially easy to approximate:

Simulate the knot with anyonic braiding

[Freedman, Kitaev, Wang (2002); Aharonov, Jones, Landau (2005);

et al. Glaser (2009)]

Classical Random Walk on a line

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

- Recipe:
- Start at the origin
- Toss a fair coin: Heads or Tails
- Move: Right for Heads or Left for Tails
- Repeat steps (2,3) T times
- Measure position of walker
- Repeat steps (1-5) many times

- Probability distribution P(x,T): binomial
- Standard deviation:

QW on a line

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

- Recipe:
- Start at the origin
- Toss a quantum coin (qubit):
- Move left and right:
- Repeat steps (2,3) T times
- Measure position of walker
- Repeat steps (1-5) many times

- Probability distribution P(x,T):...

QW on a line

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

- Recipe:
- Start at the origin
- Toss a quantum coin (qubit):
- Move left and right:
- Repeat steps (2,3) T times
- Measure position of walker
- Repeat steps (1-5) many times

- Probability distribution P(x,T):...

CRW vs QW

CRW

QW

P(x,T)

Quantum spread ~T2, classical spread~T

[Nayak, Vishwanath, quant-ph/0010117;

Ambainis, Bach, Nayak, Vishwanath, Watrous, STOC (2001)]

QW with more coins

dim=2

dim=4

Variance =kT2

More (or larger) coins dilute the effect of interference (smaller k)

New coin at each step destroys speedup (also decoherence)

Variance =kT

New coin every two steps?

[Brun, Carteret, Ambainis, PRL (2003)]

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 ~ kTafor 1<a<2?

Anyonic quantum walks are promising due to their non-local character.

QW vs RW vs ...?

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.

Ising anyons QW

Evolve in time e.g. 5 steps

What is the probability to find the walker at position x after T steps?

Ising anyons QW

Hilbert space:

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.

Trace & Kauffman’s brackets

Trace

(in pictures)

TIME

Ising anyons QW

A link is proper if the linking between the walk and any other link is even.

Non-proper links Kauffman(Ising)=0

Evaluate Kauffman bracket.

Repeat for each path of the walk.

Walker probability distribution depends on the distribution of links (exponentially many).

Locality and Non-Locality

Position distribution, P(x,T):

- z(L): sum of successive pairs of right steps
- τ(L): sum of Borromean rings

Very local

characteristic

Very non-local

characteristic

Ising QW Variance

~T2

Variance

~T

step, T

The variance appears to be close to the classical RW.

local vs non-local

step, T

step, T

Ising QW Variance

Assume z(L) and τ(L) are uncorrelated variables.

Anyonic QW & SU(2)k

probability

P(x,T=10)

index k

position, x

The probability distribution P(x,T=10) for various k.

k=2 (Ising anyons) appears classical

k=∞ (fermions) it is quantum

k seems to interpolate between these distributions

Conclusions

- 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?

Thank you for your attention!