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Transcript of 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
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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
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Information is physical!
Use the las of 'uantum mechanics
for the (asic components of aninformation processin) machine*
+uantum computin)
+uantum crypto)raphy +uantum information
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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
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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 =
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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 =
=
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Example
Superposition1
:easure1
1
!
1+=ψ
Measure12
2
)!⟩
)1⟩
)ψ⟩
33
9 D ψ
=
39
dD9
=
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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
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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∈$!%1&n α )⟩ )⟩
Measure
Second bit / !
)α) I ) γ ) 7α )!!⟩* β )!1⟩*γ )1!⟩* δ)11⟩
1!!!
γα
γα
+
+
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Quantum computer: n qubits
n 'u(its ⇔ tensor product ψ⟩ ∈ CG4,3Hn such that ψ⟩
3?
⇒ ψ⟩ ΣQ∈G4,3Hn αQ Q⟩ ith ΣQ αQ 3
:easure
#artial :easure
Unitary transformation ψ⟩ →Uψ⟩ ith U∈ UEnF
eQ1 OR
Measure
α
Σ∈$!%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⟩
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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 ⟩
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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
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implest Quantum lgorithm:
"eutsch#s $roblem
!nput1 function f1G4,3H→G4,3H Ein (lack (oQF
+uestion1 f constant EfE4FfE3FF or (alanced EfE4F≠fE3FF
+uantum (lack (oQ Ereversi(leF1
>l)orithm1 one 'ueryonly***
f )⟩),
⟩
)⟩),
⊕f7
⟩
f 4
4
4)!⟩)1⟩
Measure
)!⟩ <constant
)1⟩ <balanced
( )( ) ( ) ( )[ ]
( ) ( )( )( ) ( ) ( ) ( ) ( )1
11!
111!11!1
1
717117!7!!11!1!
11!
717!717!717!
f f f f
H f f
f H H f f f f
−−−+
−+− → −−+−
=−+− → −+ → ⊗
/! if f balanced /! if f constant
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%niversal computation
Classical circuit model1
+uantum circuit model1
= ealuates boolean functions
= can be constructed from universal local gates e> N#N@% A:+B7
!1
!
C
1
bits∧
¬
∨ !
!
= unitary transformations !
"u bits
)!⟩
)!⟩
)1⟩
)1⟩
)!⟩
)!⟩
Measure
-
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Quantum circuits
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Quantum &ircuits
+uantum circuits can simulate classicalcircuits eTciently Eith polynomialoverheadF
Classical circuits can (e eTcientlysimulated (y classical reversi(le circuits
universal reversi(le )ate V e?)? 5oWoli$)ate 5oWoli$)ate can (e )enerated ith local
unitary )ates on a 'uantum computer
$X Classical circuits ⊆ +uantum circuits
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Quantum algorithms
Deutsch$Jo6sa al)orithm EY8F1 determines if a functionE(lack (oQF is constant or $3 ith only one 'uery
Simon Ys al)orithm EY8/F1 period Mndin)
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Quantum algorithms
Deutsch$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)
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 [
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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)
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
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"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!
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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
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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
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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>
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)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$S>5
al)orithm (ased on randomalk -Sch^nin)Y88,Pofmeister, Sch^nin) &7atana(eY40F
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)andom 'al(s+++
Random alk on the line1 :iQin) timePittin)time OEnDF stationary dist?uniform
+uestions1 Stationary distri(ution Eer)odic VX
independent of initial stateF
:iQin) time
Pittin) time
:ethods1 spectral )ap, conductance, o)
So(olev, couplin),
1212
:n
7
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&lassical,quantum random wal(s
Classical
5ransition matriQ1
translationally invariant
DtEiF$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
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&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
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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
#
=
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&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
→ ←
→
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&lassical random wal(
G→,←H ⊗ GE→,4F,E←,4F,E→,3F,,E→,n$3F,E←,n$3FH
:SbC
5race out Ei)nore, avera)e overF the direction$space
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
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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
=
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Quantum random wal(
G→,←H ⊗ GE→,4F,E←,4F,E→,3F,,E→,n$3F,E←,n$3FH
:SbC
>fter t steps measure
5race out Ei)nore, avera)e overF the direction$space
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 ← = → ←
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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∑
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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∑
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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 = ∑
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-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⟩
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)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⟩
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)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
[
J@> #haronov%#> #mbainis% \>]>% ->^azirani<ST:A.!1
) ⟩)1⟩
)⟩
)n<1⟩
ε τ
4ψ
,d $
) ε
3E lo) lo)E FF# *n nε
ε =
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)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
J@> #haronov%#> #mbainis% \>]>% ->^azirani<ST:A.!1
, uu ,
- π ∈
= ∑ ,,
, u $ uu , $ ,
. p π ∈ ∉
= ∑ , G⊂
43D
min
,
,
, G , -
.
-< <≤
Φ =
D
DE3 F DD
λ Φ ≤ − ≤ Φ
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&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
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Quantum /itting Time on /ypercube
Space1
444
434
344
343
333433
G , , H↔ ⊗ ( EG3,???, H G 1 G4,3H HFnn ⊗ ∈
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Quantum /itting Time on /ypercube
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= − =
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Quantum /itting Time on /ypercube
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 =
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/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
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)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$O
D
n
n
π ⇒
333433
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)esults on hitting time.
Classical1 from v to opposite vY hittin)$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$O
D
n
n
π ⇒
( )3 5K n pKD nπ
⇒ Ω
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1"etails2
Use symmetry to calculate ei)envaluesei)envectors of un!easured alk U
\>ssymptotics] to calculate hittin) pro(a(ility at 5
⇒ one$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
tEhit v if measured at tF p α D
9nD 3 Elo) F* n nπ α = −
3t t t β α α −≥ −
( ) 5 5 5
3t4 t4 t4
3 3Ehit (efore 5F 3'
t t t t p * n' ' '
α β β α α −
≥ ≥ − = =
∑ ∑ ∑
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)obustness of initial condition
#olynomial hittin) time to opposite corner, holon) from other sites Eor to sites close to cornerF
\close] initial
states )ive similar polynomial (ehavior
Upper (ound1Re)ion around v of polynomial hittin) time to vY at
mostEotherise e could Mnd search
al)orithm that (eats the loer (ound for'uantum searchin) E@roverFF
444
434
344
343
333433
( )Dn
*
( )DnΩ
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-pen graphs
nG
C C
n<level binar, tree
;QampleL1
J#>Ahilds% L>arhi% S> Gutman% quant<ph2!1C
starthit
1
12
2
2
12
n
12
2
n*1
2
2
C
12
2
12
2
Reduces to assymetric alk on the line Eclassically and '
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-pen graphs
nG
C C
n<level binar, tree
;QampleL1
J#>Ahilds% L>arhi% S> Gutman% quant<ph2!1C
starthit
1
12
2
2
12
n
12
2
n*1
2
2
C
12
2
12
2
Reduces to assymetric alk on the line Eclassically and '
Classical1 :epn77
hittin time
+uantum1 numeric7 pol,n7
hittin time
N>Shenvi \>]>.!7
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-utloo(,-pen questions
!n )eneral hich )raphs have eQponential'uantumclassical )aps in hittin) times
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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-utloo(,-pen questions
@eneral 5P;OR***
Connection to classical:arkov chains
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&ollaborators and related wor(
Discrete-time walks
(Mixing Time !" (On the+ine!!"
#orit Aharono 4ebreD -niversit,7 A% Ambainis, -% .ach, A% *aya,
Andris Ambainis E#S% +rinceton7 A% 'ishanath, $% 1atrous$% &% F;E% :rsa,-A HerIele,7 (STOC’01
!mesh 'a(irani -A HerIele,7
JSTOC’017
:iQin) on the Pypercu(e1
#olynomial hittin) time on the Pypercu(e1
$% &% (su&)itte* #0!
hittin) time on other )raphs Enumerical & >nalytical studiesF1
*eil Sheni and $% &% (in preparation #0!