Fractional Dynamics for Quantum Random Walkslbouck.github.io/research/siam_talk.pdf · Motivation: QRWs are essential tools for quantum computing Have applications in algorithm design

Fractional Dynamics for Quantum Random Walks

Lucas Bouck

George Mason University

September 30, 2017

Minisymposium 19: Recent Advances in Numerical PDEsSIAM Central States Section Meeting

Colorado State University Ft. Collins, CO

Partially supported by: NSF DMS-1407087, DMS-1521590

Collaborator: Harbir Antil (George Mason University)

L. Bouck (GMU) SIAM 1 / 22

Outline

1 An Introduction to Fractional Calculus

2 Background on QRWs and Our Fractional Model

3 A Numerical Method to Solve the Fractional QRW Problem

4 An Optimization Algorithm to Determine the Fractional Order in Time


Fractional Calculus: Fourier Approach

Fractional derivatives typically appear in the form of the fractionalLaplacian (−∆)s or the fractional time derivative ∂αt .

The Fractional Laplacian (−∆)s of order 0 < s < 1 is defined as:

(−∆)su = F−1(|ξ|2sF(u))

for u defined on Rn.While the fractional time derivative can be defined as

∂αt u = F−1((iω)αF(u))


From Fourier to Pointwise Definition of Caputo Derivative

Starting from∂αt u = F−1((iω)αF(u)),

we get

∂αt u = F−1(

Γ(1− α)Γ(1− α)(iω)

α−1(iω)F(u))

= F−1(F(

1

Γ(1− α)tα)F(∂tu)

)= F−1

(F(

1

Γ(1− α)tα ∗ ∂tu(t)))

where ∗ denotes convolution. We arrive at

∂αt u =1

Γ(1− α)

∫ t−∞

∂tu(y)

(t − y)α dy

which is the Marchaud fractional derivative. Setting u to be constant on(−∞, 0) recovers the Caputo fractional derivative of order 0 < α < 1

∂αt u(t) =1

Γ(1− α)

∫ t0

∂tu(y)

(t − y)α dy


Random Walk View of Fractional Derivatives

Fractional Laplacian:

(−∆)s comes from a long jump random walkIntuitively, this means that the fractional Laplacian is nonlocal inspace, i.e. is able to look farther around itself

Fractional Time Derivative:

∂αt with order 0 < α < 1 comes from a random walk with time delays

Time delays, τ , behave like αAαΓ(1−α)τ (1+α) , where Aα is a constant

depending on α

The fractional time derivative is then nonlocal in time. The derivativehas memory effects.


Our Application Area: Quantum Random Walks (QRW)

Motivation:

QRWs are essential tools for quantum computing

Have applications in algorithm design and can be a universal model ofcomputation (Venegas-Andraca 2012)

QRWs:

A quantum walk is described by a tensor product of two vectorsψc ⊗ ψp =

∑Ni=−N(aiw0 + biw1)⊗ vi

The basis vectors vi correspond to positions along a line

The probability of being at position i is P(i) = |ai |2 + |bi |2Coin and shift operators evolve the state

We are specifically studying a Hadamard walk, whose coin operator

is the matrix C = 1√2

(1 11 −1

)


Quantum Walk vs Classical Walk

100 50 0 50 100

Position

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Pro

babili

ty

Quantum Random Walk 100 Steps

Figure: Quantum Random Walk

100 50 0 50 1000.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Figure: Classical Random Walk


From Blanchard and Hongler (2004)

As t →∞, the following PDE describes the probability density, u

∂

∂tu(t, x) =

1

2

∂2

∂x2u(t, x)− ∂

∂x[tanh(x)u(t, x)]

u(0, x) =δ(x), limx→±∞

u(t, x) = 0

100 50 0 50 100

Position

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Pro

babili

ty

Quantum Walk and Continuous Model at t=50

Quantum Walk

Continuous Model The peaks of thecontinuous model movetoo quickly relative tothe QRW

A fractional model couldslow these peaks downand provide a better fitfor the QRW


Our Fractional Model

Why the Fractional Derivative Makes Sense:

Comes from the random walk view of fractional derivatives

Fractional derivative means time delays in the walk

Our Model

∂αt u(t, x) =1

2

∂2

∂x2u(t, x)− ∂

∂x[tanh(x)u(t, x)]

u(0, x) = δ(x), limx→±∞

u(t, x) = 0

∂αt is the Caputo fractional derivative of order 0 < α ≤ 1.


Numerical Method

Our numerical method does the following:

Solves the fractional PDE on the domain (0,T )× Ω withΩ = (−L2 , L2 ) and homogenous Dirichlet boundary conditions with Lsufficiently large

spectral method in space

L1 finite difference scheme in time

By taking the sine transform, we get the PDE in the frequency domain

∂αt Fs{u} = −π2ω2

2L2Fs{u}+

πω

LFc{tanh(x)u}

where

Fs denotes a sine transformFc denotes a cosine transformω is the frequency variable


L1-scheme for time discretization (Lin and Xu 2007)

We use the definition of the Caputo Fractional derivative

∂αt u(t, x) =1

Γ(1− α)

∫ t0

ut(y , x)

(t − y)α dy

By using a backwards difference approximation for ut(y , x), thediscretization of ∂αt u(x , t) at time tk = kτ with time step τ is

∂αt u(x , tk+1) ≈1

Γ(1− α)k∑`=0

∫ t`+1t`

u`+1 − u`τ(tk+1 − y)α

dy

≈ 1Γ(1− α)

k∑`=0

u`+1 − u`τ

∫ t`+1t`

1

(tk+1 − y)αdy

≈ 1Γ(2− α)

k∑`=0

u`+1 − u`τα

ak−`


Recall the PDE in the frequency domain:

∂αt ûk+1 = −π2ω2

2L2ûk+1 +

πω

LFc{tanh(x)u}

where ûk+1 denotes Fs{u} at t = (k + 1)τ . If we apply the L1 scheme tothe LHS we get a forward time marching scheme

ûk+1 = C2

[πω

LFc{tanh(x)uk+1}+ C1

(ûk −

k−1∑`=0

(û`+1 − û`)ak−`)]

C1 = (Γ(2− α)τα)−1 and C2 =(C1 +

π2ω2

2L2

)−1,

We cannot calculate Fc{tanh(x)u} directlyA nested fixed point iteration (FPI) can bypass this issue


Our Approximation Converges with Time and SpaceRefinements

We have an analytical solution when α = 1, below are L2 errors of ourmethod on the domain (0, 10)× (−200, 200).

101 102

Time Grid Points

10−1

L2(

(2,1

0)×

Ω)

Err

or

L2((2, 10)× Ω) Error with dx = 0.78125L2 Error

Slope=1

Slope=2

Figure: The convergence rate withrespect to space refinements is between1 and 2

103

Space Grid Points

10−1

L2(

(0,1

0)×

Ω)

Err

or

L2((0, 10)× Ω) Error with dt = 0.01L2 Error

Slope=1

Slope=2

Figure: The convergence rate withrespect to space refinements is between1 and 2


Optimization

The functional we are minimizing is

E [α] =1

2

∫ T0

∫Ω

(uα − q)2 + γ1

(1− α)α

uα is the cumulative probability distribution of the solution to ourPDE model with the derivative order α

q is the linear interpolant of the quantum random walk cumulativeprobability distribution

The far right term will prevent the optimization process from goingoutside the interval (0, 1), with γ ≤ 1


Optimization Method

Using a right hand rule Riemann sum approximation of the integral, weapproximate the gradient of our functional.

E ′[α] ≈ hτ∑i

∑j

[(uα(ti , xj)− q(ti , xj))

d

dαuα(ti , xj)

]+ γ

2α− 1(1− α)2α2

h is the spatial step size

τ is the time step sizeddαuα(ti , xj) will be a backward difference approximation

Using this approximation for the gradient, we’ll use a gradient descentmethod to find optimal α


Optimization Numerical Results

T 10 20 30 40 50 60 70

Optimal α 0.645 0.701 0.724 0.742 0.754 0.763 0.771

Table: Optimal α values at differing T values

−60 −40 −20 0 20 40 60x

0.0

0.2

0.4

0.6

0.8

1.0

CD

F

QRW

Fractional Model

Original Model

Figure: Comparison of CDFs from the QRW and the fractional model with theoptimal α value from when T = 30


Conclusion

We have done the following

Enriched Blanchard and Hongler’s (2004) model by introducing afractional derivative in time

Provided a numerical scheme to solve the fractional model andprovided a method to find optimal α

Future Work:

Analysis of our numerical scheme for our problem (error estimates andstability)

Analysis of the optimization problem to determine the fractional timeorder α


References

[1] E. Barkai, R. Metzler, and J. Klafter. From continuous time randomwalks to the fractional Fokker-Planck equation. Physical Review Letters61 no. 1. 2000.[2] Ph. Blanchard and M.-O. Hongler. QRWs and Piecewise DeterministicEvolution. Physical Review Letters 92 no. 12. 2004.[3] Y. Lin, C. Xu. Finite difference/spectral approximations for thetime-fractional diffusion equation. Journal of Computational Physics 225.2007.[4] E. Venegas-Andraca. Quantum walks: a comprehensive review.Quantum Information Processing. 2012.


An Introduction to Fractional CalculusBackground on QRWs and Our Fractional ModelA Numerical Method to Solve the Fractional QRW ProblemAn Optimization Algorithm to Determine the Fractional Order in Time

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Fractional Dynamics for Quantum Random Walkslbouck.github.io/research/siam_talk.pdf · Motivation: QRWs are essential tools for quantum computing Have applications in algorithm design