Quantum Random Walks - Penn Math

Introduction to QRWDefinition of QRW

Finding a generating functionApplying mvGF theory to QRW

Quantum Random Walks

Robin Pemantle + Yuliy Baryshnikov, Wil Brady, AndyBressler, Torin Greenwood, Marko Petkovsek, Mark Wilson

Cornell Probability Summer School, July, 2009

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Why QRW?

Quantum random walks were first defined in the 1990’s as possibleelements of quantum computing machines.

QRW on a graph is a generalization of the Grover walk, that“finds” a distinguished node in a n-vertex graph in time of order√

n.

For a probabilist (and physicist), their most notable characteristicis ballistic behavior E|Xn| = Θ(n), as opposed to the diffusivebehavior E|Xn| = Θ(

√n) of classical random walks.

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Some pictorial examples

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The figure shows the probability profile of the “Hadamard” QRWin one dimension at time 100

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A two-dimensional example

This is an intensity plot of a two-dimensional QRW at time 100.As in the one-dimensional case, it is ballistic, and has interferencepatterns on a scale of Θ(1).

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More two-dimensional examples

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A three-dimensional example

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Overview of this lecture

QRW has been studied primarily in one dimension. This talkdescribes a body of work that includes Wil Brady’s Masters thesis,Andy Bressler’s Ph.D. thesis, and work of an undergraduatestudent, Torin Greenwood.

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Methodology

We use multivariate generating function theory to compute limitlaws for QRW on integer lattices. The generating functions aresmooth, which you will recall means that the QRW should exhibitOrnstein-Zernike behavior. In fact this is true. The challenge is toexplain the pictures. We explain them as projections of certaind-tori embedded in the (d + 1)-torus.

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Classical Random Walk

A classical random walk on Zd is a sequence of random variablesX0,X1,X2, . . . modeling a particle making a series of random steps.The increments Xn+1 − Xn are independent and all have the samedistribution.

For example, in a simple random walk on Z1, we take

P(Xn+1 − Xn = 1) = 1/2 = P(Xn+1 − Xn = −1)

and all other probabilities are zero.

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Semigroup construction

We may think of a classical random walk as powers of an operator.Let p(x) : x ∈ Zd denote a probability law on Zd . We associatewith p the operator µ 7→ ν on L1(Zd) defined by

ν(y) =∑x

µ(x)p(y − x) .

In (infinite) matrix notation,

ν = µp

where pi ,j := p(j − i).

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Example: classical SRW

Classical simple random walk on Z 1 is the operator

M :=

...

. . . 12 0 1

2 0 0 . . .. . . 0 1

2 0 12 0 . . .

. . . 0 0 12 0 1

2 . . .

...

.

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The Quantum World

In the quantum world, to model a particle located on the latticeZd , the state space is taken to be L2(Zd). In other words, ageneric state is ψ :=

∑axδx where ax are complex numbers and∑

x |ax |2 = 1. The quantity ax is called the amplitude of findingthe particle in state x . If we choose to measure the location, theprobability we will find the particle in state x is the squaredmodulus |ax |2 of the amplitude.

The only meaningful states are vectors of unit norm in L2(Zd).Evolution operators must preserve the property that

∑x |ax |2 = 1,

hence evolution operators are unitary.

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Sad Fact

There are no translation-invariant unitary operators except fortrivial ones such as “always move one step to the right”.

Try it for yourself and see. The classical RW operator M is notunitary, but neither is this one:

Q :=

...

. . .√

12 0

√12 0 0 . . .

. . . 0√

12 0

√12 0 . . .

. . . 0 0√

12 0

√12 . . .

...

.

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Chiralities

To overcome this problem, we usea device usually attributed toAharonov, Davidovich and Zagury (1993). We suppose eachparticle to have not only a location but a chirality which is ahidden state taking values in a finite set 1, . . . , k. We then fixany k vectors v(1), . . . , v(k) ∈ Zd which will be the possible stepsof the random walk, and we take the increment of the randomwalk to be v(j) when the particle has chirality j .

Every step of the QRW is really two steps: first the chirality isupdated, then the walk moves according to the new chirality.

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Formal definition

The state space is

Ω := L2(Zd × 1, . . . , k

).

Among the states ψ ∈ Ω, there are elementary states er,j , definedto be 1 at (r, j) and zero elsewhere. We interpret this state as aparticle known to be at location r and chirality j .

j = k

j = 3j = 2j = 1

location: chirality−3 −1 0 1 2 3 4−2

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The chiralities may update according to any k × k unitary matrixU. Let I ⊗ U denote the infinite matrix (operator) which fixes thelocation and operates on the chirality by U. This is a blockdiagonal matrix

...

. . . U 0 0 . . .

. . . 0 U 0 . . .

. . . 0 0 U . . .

...

.

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Success!

Let T be the operator on Ω defined by T (er,j) = er+v(j),j . Thisdoes what I said: it moves according to the chirality. The QRWoperator is defined by

S := T · (I ⊗ U) .

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Define the spacetime generating function

If we know that a particle began at the origin in chirality i , thenthe state of a QRW at time n is given by Sne0,i . The probabilityprofile of the QRW at time n is determined by the presentamplitudes which we may denote by

anr,i ,j .

For fixed chiralities i , j , the spacetime generating function for thequantities an

r,i ,j is the power series

Fi ,j(x1, . . . , xd , y) :=∑r,n

anr,i ,jx

r11 · · · x

rdd yn .

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For example, if we let d = 1 and we suppress the starting andending chiralities from the notation, things become a bit cleaner:

F (x , y) =∑n,r

anr x

ryn

where anr is the amplitude of finding the particle r steps to the

right of its initial location at time n.

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Solve using the transfer matrix method

Let F denote the k × k matrix whose (i , j)-entry is the generatingfunction Fi ,j . An easy application of the transfer matrix methodgives an exact expression for F:

F(x, y) = (I − yM(x)U)−1 .

Here, M(x) is the diagonal matrix whose (j , j)-entry is the

monomial xv(j):= x

v(j)1

1 · · · xv(j)d

d .

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Fi ,j are rational

The inverse of a matrix of polynomials is a matrix of rationalfunctions with a common denominator. Indeed, the (i , j)-entry isthe quotient of the (i , j)-cofactor by the determinant. Therefore,

Fi ,j(x, y) =Ci ,j(x, y)

Q(x, y)

where Ci ,j is a polynomial and Q(x, y) := det(I − yM(x)U) is alsoa polynomial.

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Example

The one-dimensional QRW with two steps (0 and 1) and unitarymatrix

U :=

√12

√12√

12 −

√12

is called the Hadamard QRW. Its spacetime amplitude generatingfunction is given by

F1,1(x , y) =1 +

√12xy

1− 1−x√2

y − xy2.

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We now recall theorems about asymptotics of multivariategenerating functions and apply them to the generating functionsfor QRW.

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Some definitions

Recall thatV := (x , y) : Q(x , y) = 0

denotes the pole variety. Let n : V → Rd denote the logarithmicGauss map

n(Z) := 5(Q exp)(log(Z)) .

In this notation, the critical points in direction r are the points Zfor which n(Z) ‖ r.

Recall that Z(r) denotes the solution in V to n(Z) = r if it isunique and lies on the boundary of the domain of convergence.Extend this to the non-unique case by letting Z(r) denote amultivalued function

Z(r) := n−1(r) ∩ V ∩ ∂D .

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Basic formula, extended

Recall that in the case where Z(r) is a singleton smooth point andcertain quantities do not vanish, then

ar ∼ C (r)|r|−(d−1)/2Z(r)−r .

The basic formula extends easily to the multi-valued case, as longas the cardinality is finite, to yield

ar ∼∑

Z∈Z(r)

C (Z)|r|−(d−1)/2Z−r .

Here, as before, C (Z) is the Hessian of a map parametrizing logV.

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The constant C (Z)

The definition of C (r) involved a Hessian determinant:

C (Z) = (2π)−d/2 Cij

Zd∂Q/∂Zd

∣∣∣∣Z

H−1/2

where H is the determinant of the Hessian matrix of a mapparametrizing logV near log Z by the first d coordinates.

There is a more coordinate-free formula, namely

C (Z) =Cij(Z)

| 5 (Q exp)(log Z)|κ(Z)−1/2

where κ(Z) is the Gaussian curvature of logV evaluated at log Z.

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The feasible region

LetV1 := V ∩ T 2

be the intersection of V with the unit torus

T 2 := (x , y) : |x | = |y | = 1 , .

Then the (closed) feasible region, which we will denote by G , issimply the image of V1 under the gauss map: G := n[V1].

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Theorem

Let F = Fi ,j = PQ =

∑r,n ar,nxryn be the spacetime amplitude

generating function for any QRW. Suppose that V1 is smooth.Then as r →∞ with r/n → r, uniformly as r varies over compactsubsets of G c , we have

limn→∞

n−1 log |ar,n| → β(r) < 0 if r /∈ G .

Conversely, there is a subset G0 ⊆ G whose complement haspositive co-dimension, such that if r varies over compact subsets ofG0, then

ar,n ∼ ψ(r, n)χ(r)n−d/2

where ψ(r) =∑

Z∈Z(r) c(Z) exp(inω(Z)) is a sum of complexvectors with rotating phases.

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Proof

Proof of second statement (formula when r ∈ G0):

I Check that Z(r) is a finite set of smooth points wheneverr ∈ Interior (G ).

I Check that κ(Z) 6= 0 except on an algebraic set of positiveco-dimension.

I Apply the (extended) basic mvGF theorem.

I The points of Z(r) are on the unit torus, hence, C (r)Z−r

becomes c(Z)e inω(Z).

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Proof







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Proof







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Proof







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Proof

Proof of exponential decay when r /∈ G :

I If −r · x is minimized somewhere on B other than at theorigin, then the smooth point asymptotics give β = −r · x < 0as the rate of exponential decay.

I If the origin is the minimizing point but r /∈ G , then there isno critical point on the unit torus. Use Morse theory todeform the contour further, establishing exponential decay.

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Proof

Proof of exponential decay when r /∈ G :

I If −r · x is minimized somewhere on B other than at theorigin, then the smooth point asymptotics give β = −r · x < 0as the rate of exponential decay.

I If the origin is the minimizing point but r /∈ G , then there isno critical point on the unit torus. Use Morse theory todeform the contour further, establishing exponential decay.

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Pictorial interpretation - one dimension

The manifold V1 is the union of topogical circles embedded in the2-torus. On this manifold, the logarithmic Gauss map is real andbounded (the velocity of the particle is bounded by the maximumstep size). Therefore, as one traces each circle, the Gauss mapmust meander back and forth.

Where it turns around, its derivative vanishes. The derivative ofthe Gauss map is the curvature, hence these are exactly the pointswhere H = 0. Values of r where H = 0 are where C (r) blows up.Thus we expect the limiting probability profile to have infinitepeaks: there should be an even number of these, and they shouldinclude the endpoints of the feasible region.

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1-D Examples

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Do we see peaks at the endpoints?

Do we see peaks wherever (and only wherever) κ = 0?

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The vertical lines show where κ = 0. Some of these peaks seemawfully small, and what is that extra peak doing at around 800?

We tried increasing n by a factor of 10. The extra peak was stillthere!

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It turns out that the constant is simply very large there: the Gaussmap slows to a crawl but does not stop. On closer look, comparingthe sizes of the peak at n = 1000 and n = 10000, it goes down bya factor of 10−1/2, as it should if it is not really a peak. Bycontrast, the dinky peak at the right endpoint goes down only by afactor of 10−1/3. For large enough n, the dinky peak willoutperform the peak at 800.

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Pictorial interpretation - two dimensions

In two dimensions, again, the feasible region (the silhouette of thepicture) is the image of one or more topological 2-tori under asmooth map to a bounded region of R2. Moreover, the interiorstructure can be explained by this map.

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Imagine that V1 is a soap bubble film, evenly tinted with blue dye,and that n projects it to your plane of vision. Wherevever n foldsover, its Jacobian vanishes, hence one sees a dark crease in theapparent intensity of the dye.

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Of course, one must see a dark crease around the outer boundaryof the feasible region, but there are also creases internally, aroundthe holes in the tori, and around places where the Gauss maphappens to fold over itself when it is not near the boundary.

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A funny story

Before we had proved any of this stuff, we had the prediction forthe feasible region. A promising graduate student, Wil Brady, waslured away by Microsoft, so he wanted to write a Masters thesisand leave. One of the things he did was to plot the image of theGauss map for various QRW’s and match them against probabilityplots of quantum random walks in one and two dimensions. Theresults are shown on the next slide.

[By the way, he also proved early version of some of thesetheorems. Really, his thesis was very close to a Ph.D.-level thesis.Does this matter to anyone who oversees software engineers?]

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Comparison

Wil was trying to verify that the regions were the same. Indeed,the rounded diamonds in the two figures match. His left handfigure did not attempt to plot χ with the region G – it was just abunch of dots that were supposed to fill the region.

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Mystery!

In the right-hand figure, darker regions correspond to greatervalues of χ. Why did the same pattern of dark and lightmysteriously appear in the left-hand figure? That is, why did he,by accident, plot the intensity rather than the indicator function?

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Only one explanation:

Recall that V1 is a topological 2-torus embedded in T 3. Theregion on the left was produced by parametrizing logV1 by twovariables (u, v), embedding a grid of dots (u, v) ∈ m−1Z2, andmapping these dots via n into R2 ... Ergo, J−1.

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Overtime

I am guessing we’re out of time, but in case not, there’s more goodstuff! The Gauss map takes a torus and smooshes it onto a regionin the plane. At the boundary of this region, the map folds over,meaning its Jacobian vanishes. In other words, the Gaussiancurvature becomes zero at the boundary of G . But the equationsfor vanishing logarithmic Gaussian curvature of a rational functionare polynomial equations. Thus ∂G is an algebraic curve.

Computing this can be hairy, but last year, Marko Petkovsekhelped us proved that the boundary of G was defined by thevanishing of the following polynomial in r and s.

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132019r16 + 2763072s2r20− 513216s2r22− 6505200s2r18 + 256s2r2 + 8790436s2r16−10639416s10r8 + 39759700s12r4 − 12711677s10r4 + 4140257 ∗ s12r2 − 513216s22r2 −7492584s2r14 + 2503464s10r6 − 62208s22 + 16s6 + 141048r20 + 8790436s16r2 +

2763072s20r2 − 6505200s18r2 − 40374720s18r6 + 64689624s16r4 − 33614784s18r4 +

14725472s10r10 + 121508208s16r8 − 1543s10 − 23060s2r6 + 100227200s10r12 +

7363872s20r4 − 176524r18 + 121508208s8r16 − 197271552s8r14 − 13374107s8r6 +

1647627s8r4 + 18664050s8r8 − 227481984s10r14 − 19343s4r4 + 279234496s12r12 −67173440s14r4 − 7492584s14r2 + 4140257s2r12 + 291173s2r8 − 1449662s2r10 +

7363872s4r20 − 227481984s14r10 + 132019s16 − 197271552s14r8 − 59209r14 −1449662s10r2 + 100227200s12r10 − 1543r10 − 153035200s14r6 − 13374107s6r8 +

3183044s6r6 + 39759700s4r12 − 176524s18 + 72718s6r4 + 1647627s4r8 − 62208r22 +

141048s20 − 1472s4r2 + 11664s24 − 33614784s4r18 + 128187648s16r6 − 1472s2r4 −67173440s4r14 + 291173s8r2 + 64689624s4r16 − 10639416s8r10 − 59209s14 +

72718s4r6 + 92321584s8r12 − 56r8 + 92321584s12r8 − 153035200s6r14 − 23060s6r2 +

128187648s6r16 − 40374720s6r18 + 72282208s12r6 + 14793r12 + 11664r24 +

14793s12 + 16r6 + 2503464s6r10 − 56s8 − 12711677s4r10 + 72282208s6r12

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Open questions

Many open questions remain:

I What do these look like in dimensions three and higher?

I In one dimension, can we construct a double peak, where thecurvature vanishes to order 2, yielding n−1/4 scaling and aquartic-Airy limit in a scaling window?

I Can we classify limit behavior near various double zeros of κin dimension two?

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Open questions





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Open questions





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