Anyonic quantum walks:
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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 ):

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Gavin Brennen Lauri Lehman Zhenghan Wang Valcav Zatloukal JKP

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Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

Anyonic quantum walks:The Drunken Slalom

Gavin Brennen

Lauri Lehman

Zhenghan Wang

Valcav Zatloukal

JKP

Ubergurgl, June 2010


Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

Anyonic Walks: Motivation

  • 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)

  • Quantum simulation


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    Bosons

    Fermions

    Anyons

    Anyons

    • Two dimensional systems

    • Dynamically trivial (H=0). Only statistics.

    3D

    2D

    View anyon as vortex with flux and charge.


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    Define particles:

    Define their fusion:

    Define their braiding:

    Fusion Hilbert space:

    Ising Anyon Properties


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    Assume we can:

    Create identifiable anyons

    pair creation

    Braid anyons

    Statistical evolution:

    braid representation B

    Fuse anyons

    Ising Anyon Properties

    time


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    Approximating Jones Polynomials

    “trace”

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

    [Markov, Alexander theorems]


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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)]


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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)]


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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:


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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):...


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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):...


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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)]


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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)]


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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 ...?


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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.


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    Ising anyons QW

    Evolve in time e.g. 5 steps

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


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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.


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    Trace & Kauffman’s brackets

    Trace

    (in pictures)

    TIME


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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).


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    Ising QW Variance

    ~T2

    Variance

    ~T

    step, T

    The variance appears to be close to the classical RW.


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    local vs non-local

    step, T

    step, T

    Ising QW Variance

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


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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


    Gavin brennen lauri lehman zhenghan wang valcav zatloukal jkp

    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!


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