QRW-Cambr02 quantum computing

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Quantum Random WalksQuantum Random Walks

Combinatorial and Computational Aspects of Statistical Physics/

Random Graphs and Structures

Cambridge, September 5, 2002

Julia Kempe

Computer Science Division and

Department of Chemistry,

University of California, Berkeley

&

CNRS & R!, Universit" de #aris$Sud, %rance



Towards nanotechnology

Size of the components

Number of components

Speed

Gordon Moore 1965

prevent or use quantum effects ?

Theoretical limitations reached in !! """

#pparition of quantum phenomena



Information is physical!

Use the las of 'uantum mechanics

for the (asic components of aninformation processin) machine*

+uantum computin)

+uantum crypto)raphy +uantum information



Main applications

Crypto)raphy #rotocol of unconditionally secure secret key distri(ution

-Bennett, Brassard ./0

!mplementation 1 2 344 km

+uantum information 5eleportation -B, B, Cr"peau, Jo6sa, #eres, 7ooters 890

!mplementation -Boumeester, #an, :attle, ;i(l, 7einfurter, <eilin)er

8=0 >l)orithms

%actorin), discrete lo)arithm, ??? -Shor 8/0

Data(ase search -@rover 8A0

Num? of 'u(its 388 1 , 388. 1 9, 44 1 . -Chuan) E!B:F0 $ 34 -os

>lamos0



The qubit

Classical (it1 (∈G4,3H

#ro(a(ilistic (it1 pro(a(ility distri(ution d∈RI$!%1&

such that d3 3? ⇒ dEp,3$pF ith p

∈-4,30

+uantum (it1 ψ⟩ ∈ CG4,3H such that ψ⟩ D3?

⇒ |ψ⟩ = α |0 ⟩ + β |1 ⟩ ith α DI β D3

EDiracnotationF

=

=

=

'( %

1! 1 %

!1 ! ψ

( ) ( ) ( )L L L4 3,4 , 3 4,3 , ,

t

ψ α β ψ = = = =

4J , K 3Jα ψ β ψ =



Qubit evolution

:easure1 reads and modiMes

Measureα

β

α )!⟩ * β )1⟩)!⟩

)1⟩

⇒ Superposition → +robabilit, distribution

Unitary transformation1 U∈ C× such that UUId

)ψ⟩ - )ψ .⟩ / - )ψ⟩

unitary → reversi(le1

-)ψ⟩ -0 )ψ⟩

D

E F J

J J

p i i

i i

ψ

ψ ψ =

=



Example

Superposition1

:easure1

1

!

1+=ψ

Measure12

2

)!⟩

)1⟩

)ψ⟩

33

9 D ψ

=

39

dD9

=



Example

Superposition1

:easure1

Unitary transformations1

NO51 4⟩ ⇔ 3⟩

Padamard1

1

!

1+=ψ

Measure12

2

)!⟩

)1⟩

)ψ⟩

)ψ⟩ - )ψ .⟩ / - )ψ⟩

= !1

1!

xσ

−

=11

11

1 H

1

1!

3 +=

xσ

16

1

!6

1

1!

1!

1

3

−

+

+

=

−

+

+

= H

4



Quantum computer: n qubits

n 'u(its ⇔ tensor product ψ⟩ ∈ CG4,3Hn such that ψ⟩

D3?

⇒ ψ⟩ ΣQ∈G4,3Hn αQ Q⟩ ith ΣQ αQ D 3

:easure

#artial :easure

Measure

α

Σ∈$!%1&n α )⟩ )⟩

Measure

Second bit / !

)α) I ) γ ) 7α )!!⟩* β )!1⟩*γ )1!⟩* δ)11⟩

1!!!

γα

γα

+

+



Quantum computer: n qubits

n 'u(its ⇔ tensor product ψ⟩ ∈ CG4,3Hn such that ψ⟩

3?

⇒ ψ⟩ ΣQ∈G4,3Hn αQ Q⟩ ith ΣQ αQ 3

:easure

#artial :easure

Unitary transformation ψ⟩ →Uψ⟩ ith U∈ UEnF

eQ1 OR

Measure

α

Σ∈$!%1&n α )⟩ )⟩

Measure

Second bit / !

)α) I ) γ ) 7α )!!⟩* β )!1⟩*γ )1!⟩* δ)11⟩

1!!!

γα

γα

+

+

!1!!1!!!

!!1!

!!!1

)!!⟩ )!1⟩ )1!⟩ )11⟩

*

)i⟩)8⟩

)i⟩ ):;i%87⟩



Quantum computing a function

et f1 G4,3Hn → G4,3Hm

Q → fEQF

Reversi(le1

Rf 1G4,3HnIm → G4,3HnIm

EQ,yF → EQ,y⊕fEQFF

+uantum1

Uf ∈UEnImF1 CnIm → CnIm

Q⟩y⟩ → Q⟩y⊕fEQF ⟩



implest Quantum lgorithm:

"eutsch#s $roblem

!nput1 function f1G4,3H→G4,3H Ein (lack (oQF

+uestion1 f constant EfE4FfE3FF or (alanced EfE4F≠fE3FF

+uantum (lack (oQ Ereversi(leF1

>l)orithm1 one 'ueryonly***

f )⟩),

⟩

)⟩),

⊕f7

⟩

f 4

4

4)!⟩)1⟩

Measure

)!⟩ <constant

)1⟩ <balanced



implest Quantum lgorithm:

"eutsch#s $roblem

!nput1 function f1G4,3H→G4,3H Ein (lack (oQF

+uestion1 f constant EfE4FfE3FF or (alanced EfE4F≠fE3FF

+uantum (lack (oQ Ereversi(leF1

>l)orithm1 one 'ueryonly***

f )⟩),

⟩

)⟩),

⊕f7

⟩

f 4

4

4)!⟩)1⟩

Measure

)!⟩ <constant

)1⟩ <balanced

( )( ) ( ) ( )[ ]

( ) ( )( )( ) ( ) ( ) ( ) ( )1

11!

111!11!1

1

717117!7!!11!1!

11!

717!717!717!

f f f f

H f f

f H H f f f f

−−−+

−+− → −−+−

=−+− → −+ → ⊗

/! if f balanced /! if f constant



%niversal computation

Classical circuit model1

+uantum circuit model1

= ealuates boolean functions

= can be constructed from universal local gates e> N#N@% A:+B7

!1

!

C

1

bits∧

¬

∨ !

!

= unitary transformations !

"u bits

)!⟩

)!⟩

)1⟩

)1⟩

)!⟩

)!⟩

Measure

-



Quantum circuits



Quantum &ircuits

+uantum circuits can simulate classicalcircuits eTciently Eith polynomialoverheadF

Classical circuits can (e eTcientlysimulated (y classical reversi(le circuits

universal reversi(le )ate V e?)? 5oWoli$)ate 5oWoli$)ate can (e )enerated ith local

unitary )ates on a 'uantum computer

$X Classical circuits ⊆ +uantum circuits



Quantum algorithms

Deutsch$Jo6sa al)orithm EY8F1 determines if a functionE(lack (oQF is constant or $3 ith only one 'uery

Simon Ys al)orithm EY8/F1 period Mndin)



Quantum algorithms



Shor EY8F1 eTcient factorin)

)eneral pro(lem Efactorin), discrete lo)F hiddensu()roup1

Input: function f: G → G s.t. f(x)=f(x+H) where H< G

Output: H (generators)

ecient uantu! al)orithm if @ $ >(elian or Z special [



Quantum algorithms



Shor EY8F1 eTcient factorin)

)eneral pro(lem Efactorin), discrete lo)F hiddensu()roup1 Input: function f: G → G s.t. f(x)=f(x+H) where H< G

Output: H (generators)

ecient uantu! al)orithm if @ $ >(elian or Z special [

@rover EY8AF1 Search of one entry in a data(ase of si6e Nith 'ueries EClassical loer (ound is ΩENFF

E'uantum loer (oundF

E F"θ

E F"Ω



"iscrete Quantum 'al(s

Discrete-time walks on fnite graphs

(Mixing Time !"#orit Aharono 4ebreD -niversit,7

Andris Ambainis E#S% +rinceton7

$% &% F;E% :rsa,-A HerIele,7

!mesh 'a(irani -A HerIele,7

JSTOC’017

:iQin) on the Pypercu(e1

C% )oore and A% Russel (quant-ph’01

#olynomial hittin) time on the Pypercu(e1

$% &% ( ’0!"

hittin) time on other )raphs Enumerical & >nalytical studiesF1

*eil Sheni and $% &% (in preparation #0!



Mar(ov chains

:arkov chains for al)orithms1

!dea!dea1 construct a :arkov chain Esimple, local

transitions only, eTciently implementa(leF E3F hose stationary distri(ution )ives the

solution to the pro(lem ⇒ #ixing ti!e#ixing ti!e

or EF hich hits the desired solution ⇒ Hitting ti!eHitting ti!e

Z +uantum [ :arkov chains Z +uantum [ :arkov chains



Example: )andom wal( for *T

Enput Hoolean formula Φ con8unction of clauses of variables7 in 1% C % n

e> 7

Kuestion Es Φ satisfaisable?

e> BLS% T is satisf,in assinment7

#lorithm 17 initialise the variables u>a> random T< OtrueO% <OfalseO7

7 if all clauses satisfied P ST:+% otherDise

7 chose a non<satisfied clause% chose one of its tDo variables andflip its valueQ return to 7

76767676 111 $ $ $ $ $ $ $ $ ¬∨¬∧∨∧∨¬∧¬∨=Φ



Example: )andom wal( for *T

#lorithm 17 initialise the variables u>a> random T< OtrueO% <OfalseO7

7 if all clauses satisfied P ST:+% otherDise

7 chose a non<satisfied clause% chose one of its tDo variables and

flip its valueQ return to 7

T

TT TT

TTTTT

TT

!

1

ST:+R12

R12

R12 12

12

4ammin distance

;andom DalI on a line Dith n*1 vertices "

#fter t/n repetitions 4ittin time U7 the succes probabilit, is R12

if Φ satisfiable7>



)andom 'al(s+++

9S>5 $ \(iased] random alk ith eQponentialhittin) time

in )eneral 1 local, simple :arkov chain oneQponential domain

! 1

R12

2

R12

22

R12

V

R12

2

5

2

R12

ST:+

Efastest knon 9$S>5

al)orithm (ased on randomalk -Sch^nin)Y88,Pofmeister, Sch^nin) &7atana(eY40F



)andom 'al(s+++

Random alk on the line1 :iQin) timePittin)time OEnDF stationary dist?uniform

+uestions1 Stationary distri(ution Eer)odic VX

independent of initial stateF

:iQin) time

Pittin) time

:ethods1 spectral )ap, conductance, o)

So(olev, couplin),

1212

:n

7



&lassical,quantum random wal(s

Classical

5ransition matriQ1

translationally invariant

DtEiF$distri(ution after time t

stationary distri(ution

measure of \closeness]1 total variation distance

miQin) time τ $ time until ∆_const.

1212

:n7

76 %i pro& ' i% →=

W@lim tt →∞→

t ∑Ω∈

−=−=∆i

t W6i76i7@

1W@ t



&lassical,quantum random wal(s

Classical +uantum

5ransition matriQ1

translationally invariant

DtEiF$distri(ution after time t

stationary distri(ution

measure of \closeness]1 total variation distance

miQin) time τ $ time until ∆_const.

1212

:n7

1 +

1

76 %i pro& ' i% →=

W@lim tt →∞→

t ∑Ω∈

−=−=∆i

t W6i76i7@

1W@ t

unitar,?

reversible?local

translationall, invariant



Quantum random wal(

\Classical] :arkov process1

+uantum Unitary :eyer -`8=01 >ll local, translationary invariant unitary

matrices are simple translations?

X;Y XFY

4 3 4 4 ??? 3

3 4 3 4 ??? 4

4 3 4 3 4 ???3

??? 4 3 4 ??? 4D

4 4 4 ??? ??? 33 4 ??? 4 3 4

#

=



&lassical random wal(

!ncorporate \coin$ip] into alk*

Classical alk in to steps1

G→,←H ⊗ GE→,4F,E←,4F,E→,3F,,E→,n$3F,E←,n$3FH

ip direction coin C

perform controlled shift S1 → ⇒ \R]← ⇒ \]

:SbC

5race out Ei)nore, avera)e overF the direction$space

X;Y XFY

3 3 3 43 c c c c

3 3 4 3D

→ ←

→



&lassical random wal(


:SbC


3 3

3 3

3 33

3 3C

???

3 3

3 3

=

ip direction coin perform controlled shift 1 → ⇒ \R]

← ⇒ \]

S /

3 4

4 4

=

4 4 4 3

=

C

C

→← →

←→←

C



Quantum random wal(

:eyer -`8=01 >ll local, translationary invariant unitarymatrices are simple translations?

\coined] alk in to steps1

G→⟩,←⟩H ⊗

]ip] direction coin E F

perform controlled shift 1 →⟩ ⇒ \R]

←⟩ ⇒ \]

( ) ( )3 3 D D

Η → = → + ← Η ← = → − ←

X;Y XFY

unitar, XDalIY -

U \collapses] to the classical random alk if emeasure directions or positions at every step*

4

43 33

3 $3D

=



Quantum random wal(


:SbC

>fter t steps measure


X;Y XFY

3 3

3 3

3 33

3 3C

???

3 3

3 3

− −=

−

ip direction coin perform controlled shift 1 → ⇒ \R]

← ⇒ \]

S /

3 4

4 4

=

4 4 4 3

=

C

C

)!⟩⟩( ) ( )

3 3Η → = → + ← Η ← = → ←



Quantum random wal(s

#xample" start in

induces pro(a(ility$dist? #tEiF on the sites Eafter measurementF

$on%ergence&$on%ergence&

NO* U is unitary ⇒ reversi(le* Eno stationary distri(?F

De' ? \avera)ed distri(ution] +t ECesaro limitF1

Theorem" QTheorem" Qtt con%erges to a stationar distri)ution*con%erges to a stationar distri)ution*

)!⟩)1⟩

)⟩

)n<1⟩

4→ ⊗

( ) ( )3 3 D 4 4 DH shift → → + ← ⊗ + → − ← ⊗ − → → ⊗ + ← ⊗ + → ⊗ − ← ⊗ −

( ) 4 3 3H shift → → + ← ⊗ → → ⊗ + ← ⊗ −

( ) ( )D 4 D ???DH shift → → + ← ⊗ + → ⊗ − → − ← ⊗ − →

- 4t

→ ⊗

t

t s

s !

1K 6v7 +

tE F$

=

= ∑

( ) ( ) D D

Η → = → + ← Η ← = → − ←

)!⟩) ⟩

t1K 6 7 + E F∑



tationary distribution

Theorem" QTheorem" Qtt con%erges to a stationarcon%erges to a stationar

distri)ution*distri)ution*

)!⟩)1⟩

)⟩

)n<1⟩t s

s !

K 6v7 +t

E F$ =

= ∑

Calculate ei)envectorsei)envalues of U ;Qpand initial state1

State at time t1

Stationary distri(ution1

if

E , Fi iφ λ

43

"

i ii

a φ =

Ψ = ∑

3

" t t i i i

i

a λ φ =

Ψ = ∑D L L

G , H , , 3

E F , E F , ,"

ss s i % i % i %

d i % d

& $ d $ aa d $ d $ ψ λ λ φ φ ∈ → ← =

= =∑ ∑

t

t st t

s !

1limK 6v7 lim +

tE F E F$ $ π

→∞ →∞ =

= = ∑t

i

i

s !i

61 6

t

L

L

L

3 F DF 4

3

t %s

% t

% i %

t

λ λ λ λ

λ λ λ λ →∞

=

−= ≤ →

− −∑i %λ λ ≠

)!⟩) 1⟩

t1K 6 7 + E F∑



tationary distribution

Theorem" QTheorem" Qtt con%erges to a stationarcon%erges to a stationar

distri)ution*distri)ution*

)!⟩)1⟩

)⟩

)n<1⟩t s

s !

K 6v7 +t

E F$ =

= ∑

Stationary distri(ution1

uniform if @ non$de)enerate E F1

!f @ also a(elian $X stationary distri(ution uniform1

characters of the a(elian )roup Eunit normF

D L L

G , H , , 3

E F , E F , ,"

ss s i % i % i %

d i % d

& $ d $ aa d $ d $ ψ λ λ φ φ ∈ → ← =

= =∑ ∑

i

tt

d%i%8?

limK 6v7 / LE F , J J ,

%

i % i %$ aa d $ d $

λ λ

π φ φ →∞

≠

= ∑

i %λ λ ≠

d%i

DD

E F , Ji i$ a d $ π φ = ∑i i i

wφ χ = ⊗3

E Fi i

$

$ $ n

χ χ = ∑



-bservations

Classically1 real ei)envalues +uantum1 compleQ ei)envalues

Classically1 \(ehavior] depends E F on secondlar)est ei)envalue

+uantum1 all ei)envalues e'ually important

;Q1 miQin) time determined (y conver)ence of

i?e? (y

3 3 ??? 3"λ λ λ = ≥ ≥ ≥ ≥ − L 3i i iλ λ λ = =

≈λ

ti

i

s ! i

1 t

L

L

L

3 F DF

3

t %s

%

% i %t

λ λ λ λ

λ λ λ λ =

−= ≤

− −∑

, 1min

i %

i %i %λ λ

λ λ ≠

∆ = −

minimum ap7

)!⟩

) 1⟩



)esults on mixing time.Cycle1

'uantum alk conver)es toards uniformdistri(ution

:iQin) time1 classical1 ε θEN lo)E3εFF

'uantum1 ε OEN lo) N ε9F

5otal variation distance1

Similar results in hi)her dimensions, for Cayley)raphs, )raphs on a(elian )roups, alks ithdiWerent coins,

Z t Z[ s>t> Z t [ 4 ,d$ ψ ∆ ≤ ∀ ≥ ∀ =

[[

J@> #haronov%#> #mbainis% \>]>% ->^azirani<ST:A.!1

D

, 1

3E?F E?F Di %

' ii % i %

( a' λ λ

π λ λ ≠

− ≤−

∑

ElnE DF 3FE?F E?F'

nd(

'

π π

+− ≤∆

) ⟩)1⟩

)⟩

)n<1⟩

)!⟩) 1⟩



)esults on mixing time.Cycle1

'uantum1 ε OEN lo) N ε

9

F

Z 7armstart [ to )et lo)arithmic ε$dependence1 !nitiali6e in

Run 'uantum alk for steps $X measure Enode vF Restart ne alk in Ed$randomF Repeat k$times

Resultin) distri(ution is $close to the stationarydistri(ution

Eorks if stationary distri(ution is independent ofinitial stateF

[


) ⟩)1⟩

)⟩

)n<1⟩

ε τ

4ψ

,d $

) ε

3E lo) lo)E FF# *n nε

ε =



)esults on mixing time.

Conductance$type loer (ound for miQin)time of any 'uantum alk on oundeddegree )raph1

capacitance o

conductance1

5heorem EJerrum,SinclairY.8F1

3E F

d

τ = ΩΦ

( ) ( )D3 O 3τ Ω Φ ≤ ≤ ΦClassical1

+uantum1 d$maQ?de)re


, uu ,

- π ∈

= ∑ ,,

, u $ uu , $ ,

. p π ∈ ∉

= ∑ , G⊂

43D

min

,

,

, G , -

.

-< <≤

Φ =

D

DE3 F DD

λ Φ ≤ − ≤ Φ



&onductance

+uantum1 d$maQ?de)ree

Cut E,F of @, (oundary

!dea1 start ith state concentrated in and sho thatat each time step \leaka)e] into is (ounded (y

? 5hen after steps

>nd hence

3E Fdτ = Ω Φ

G 1 v RH , / $ , edge= ∈ ∃ →

3

D

min d ,

, G

/

, ≤

Φ = Φ ≤ Φ

, /

,

E3 F , / ,

0 , ε τ ε ≥ −ε

τ

3

eε

τ = Ω Φ



Quantum /itting Time on /ypercube

Space1

444

434

344

343

333433

G , , H↔ ⊗ ( EG3,???, H G 1 G4,3H HFnn ⊗ ∈




Space1

7alk1

Conditional Shift Coin C Erespects permutational symmetry of

hypercu(eF

444

434

344

343

333433

G , , H↔ ⊗ ( EG3,???, H G 1 G4,3H HFnn ⊗ ∈

1i

2 i 1 i 1 e⊗ → ⊗ ⊕G44??43 44??4ii

e =

???

a + +

+ a + +-

+ + a

=

K

: O :

3a n n= − =




Space1

7alk1 Conditional Shift

Coin C Erespects permutational symmetry ofhypercu(eF

!nitial state1

444

434

344

343

333433

G , , H↔ ⊗ ( EG3,???, H G 1 G4,3H HFnn ⊗ ∈

1i

2 i 1 i 1 e⊗ → ⊗ ⊕G44??43 44??4ii

e =

???

a + +

+ a + +-

+ + a

=

K

: O :

3a n n= − =

3

3E4F 44???4

n

i

in

ψ =

= ⊗∑S,mmetric superposition over all directions

:iQin) time1 classical1 'uantum1

Ecoupon collectorF E:oore&RusselY43F

OE lo) Fn nτ = ? /inst ) nπ τ =( )9

* nε τ ε =



/itting time0

Dilemma1 constant measurement of position illcollapse U to the classical alk

5o options1

One$shot '$hittin)$time E5,pF1 :easure only at time 5

\Pits] desired tar)et$state Q ith pro(a(ility Xp

Concurrent '$hittin)$time E5,pF1 #artial measurement E\>m ! at Q>m ! not at Q]F at alltimes

Stop alk if Q is hit? #ro(a(ility Xp to hit Q (efore time 5

333433



)esults on hitting time.

Classical1 from v to opposite vY hittin)$time

+uantum1 One$shot hittin)$time from v to vY E5,pF

J\>]>.!

OEeQpE FF 3 $ n π =

5 E F, E F

D D

n *n n *nβ β π π ∈ − +

3$D

lo) n p3$O

n β

⇒ 3Dβ <T<n7 even%

444

434

344

343

433

and9lo)

5K n pK3$O

D

n

n

π ⇒

333433



)esults on hitting time.

Classical1 from v to opposite vY hittin)$time

+uantum1 One$shot hittin)$time from v to vY E5,pF

Need to kno ith accuracy hen to measure, success≈3 in linear time*

Concurrent hittin)$time from v to vY E5,pF

No information on hen to measure needed, ith

ampliMcation success ≈3 in 5OEnF*J\>]>.!

OEeQpE FF 3 $ n π =

5 E F, E F

D D

n *n n *nβ β π π ∈ − +

3$D

lo) n p3$O

n β

⇒ 3Dβ <T<n7 even%

444

434

344

343

433

( )O ng

and9lo)

5K n pK3$O

D

n

n

π ⇒

( )3 5K n pKD nπ

⇒ Ω



1"etails2

Use symmetry to calculate ei)envaluesei)envectors of un!easured alk U

\>ssymptotics] to calculate hittin) pro(a(ility at 5

⇒ one$shot hittin) time E5,pF

%or concurrent hittin) time )ive a loer (ound onhittin) pro(a(ility in terms of unmeasured alk U1

emma1

D

tEhit at tJ not stopped (efore tFK p β

D

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Classical1 :epn77

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-utloo(,-pen questions

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Po ro(ust is this )ap ?r?t? initialpositiondistri(ution

:iQin) times for non$a(elian alks

:iQin) times for alks on non$(ounded de)ree

)raphs%or de)enrate or non$a(elian )roups stationarydistri(ution depends on initial state$al)orithmic use

>l)orithmic use



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Connection to classical:arkov chains



&ollaborators and related wor(

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hittin) time on other )raphs Enumerical & >nalytical studiesF1

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