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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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

  • Plasma physics and superconductors:

    • Vortex dynamics

  • Polymer physics:

    • Diffusion of polymer chains

  • Molecular biology:

    • DNA folding

  • Cosmic strings

  • Kinematic Golden Chain (ladder)

  • 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!


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