quantum random walks combinatorial and computational aspects of statistical physics/ random graphs...
Post on 21-Dec-2015
215 views
TRANSCRIPT
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 & LRI, Université de Paris-Sud, France
Towards nanotechnology
Size of the components
Number of components
Speed
Gordon Moore 1965
prevent or use quantum effects ?
Theoretical limitations reached in 2020 !!!
Apparition of quantum phenomena
Information is physical!
Use the laws of quantum mechanics for the basic components of an
information processing machine!Quantum computingQuantum cryptographyQuantum information …
Main applications
Cryptography Protocol of unconditionally secure secret key distribution
[Bennett, Brassard 84]Implementation : ~ 100 km
Quantum information Teleportation [B, B, Crépeau, Jozsa, Peres, Wooters 93] Implementation [Bouwmeester, Pan, Mattle, Eibl, Weinfurter,
Zeilinger 97]
Algorithms Factoring, discrete logarithm, ... [Shor 94] Database search [Grover 96]Num. of qubits ? 1995 : 2, 1998 : 3, 2002 : 8 [Chuang (IBM)] - 10 [Los
Alamos]
The qubit
Classical bit: b{0,1}Probabilistic bit: probability distribution dR+
{0,1} such that ||d||1 =1. d=(p,1-p) with p [0,1]
Quantum bit: | C{0,1} such that || | ||2=1.
| = |0 + |1 with | | 2+ | | 2=1
(Dirac notation)
β
α ,
1
0 1 ,
0
1 0 ψ
* * *0 1,0 , 1 0,1 , ,t
0| , = 1|
Qubit evolution
Measure: reads and modifies
Measure| | 2
| | 2
|0 + |1|0
|1
Superposition Probability distribution
Unitary transformation: U C22 such that UU†=Id
| U |’ = U |
unitary reversible:
U| U† |
2( ) |
| |
p i i
i i
Example
Superposition:
Measure:
Unitary transformations:
NOT: |0 |1
Hadamard:
13
20
3
1
Measure1/3
2/3
|0
|1|
| U |’ = U |
01
10
x
11
11
2
1H
13
10
3
2ψ x
16
210
6
21
2
10
3
2
2
10
3
1ψ
H
H
Quantum computer: n qubits n qubits tensor product | C{0,1}n such that || | ||
2=1.
| = x{0,1}n x |x with x |x |2 =1 Measure
Partial Measure
Measure|x | 2
x{0,1}n x |x |x
Measure
Second bit = 0 (||2 + | |2 )
|00+ |01+ |10+ |11 22
1000
γα
γα
Quantum computer: n qubits n qubits tensor product | C{0,1}n such that || | ||
2=1.
| = x{0,1}n x |x with x |x |2 =1 Measure
Partial Measure
Unitary transformation | U| with U U(2n)
ex: XOR=
Measure|x | 2
x{0,1}n x |x |x
Measure
Second bit = 0 (||2 + | |2 )
|00+ |01+ |10+ |11 22
1000
γα
γα
0100
1000
0010
0001|00 |01 |10 |11
+
|i|j
|i |XOR(i,j)
Quantum computing a function
Let f: {0,1}n {0,1}m
x f(x)
Reversible: Rf :{0,1}n+m {0,1}n+m
(x,y) (x,yf(x))
Quantum:Uf U(2n+m): Cn+m Cn+m
|x|y |x|yf(x)
Simplest Quantum Algorithm:Deutsch’s Problem
Input: function f:{0,1}{0,1} (in black box) Question: f constant (f(0)=f(1)) or balanced (f(0)f(1)) ?
Quantum black box (reversible):
Algorithm: one query only!!!
f|x|y
|x|yf(x)
fHH
H|0|1
Measure|0 -constant|1 -balanced
Simplest Quantum Algorithm:Deutsch’s Problem
Input: function f:{0,1}{0,1} (in black box) Question: f constant (f(0)=f(1)) or balanced (f(0)f(1)) ?
Quantum black box (reversible):
Algorithm: one query only!!!
f|x|y
|x|yf(x)
fHH
H|0|1
Measure|0 -constant|1 -balanced
12
110
2
11101101
2
1
)1()1(1)0()0(02
11010
2
110
)1()0()1()0()1()0(
ffffHff
fHH ffff
=0 if f balanced =0 if f constant
Universal computation
Classical circuit model:
Quantum circuit model:
• evaluates boolean functions• can be constructed from universal local gates (ex.: NAND, COPY)
010…1
bits
0
0
• unitary transformations U
qubits
|0
|0
|1
|1
|0
|0
Measure
U
Quantum CircuitsQuantum circuits can simulate classical
circuits efficiently (with polynomial overhead)
Classical circuits can be efficiently simulated by classical reversible circuits; universal reversible gate – e.g. Toffoli-gate
Toffoli-gate can be generated with local unitary gates on a quantum computer
-> Classical circuits Quantum circuits
Quantum algorithms
Deutsch-Jozsa algorithm (’92): determines if a function (black box) is constant or 2-1 with only one query
Simon ’s algorithm (’94): period finding
Quantum algorithms
Deutsch-Jozsa algorithm (’92): determines if a function (black box) is constant or 2-1 with only one query
Simon ’s algorithm (’94): period finding Shor (’95): efficient factoring general problem (factoring, discrete log) = hidden
subgroup: Input: function f: G G s.t. f(x)=f(x+H) where H G Output: H (generators) efficient quantum algorithm if G - Abelian or « special »
Quantum algorithms
Deutsch-Jozsa algorithm (’92): determines if a function (black box) is constant or 2-1 with only one query
Simon ’s algorithm (’94): period finding Shor (’95): efficient factoring general problem (factoring, discrete log) = hidden
subgroup: Input: function f: G G s.t. f(x)=f(x+H) where H G Output: H (generators) efficient quantum algorithm if G - Abelian or « special »
Grover (’96): Search of one entry in a database of size N with queries (Classical lower bound is (N))
(quantum lower bound)
( )N( )N
Discrete Quantum Walks
Discrete-time walks on finite graphs (Mixing Time) *:
Dorit Aharonov (Hebrew University) Andris Ambainis (IAS, Princeton) J. K. (LRI, Orsay&UC Berkeley) Umesh Vazirani (UC Berkeley)
(*STOC’01)
Mixing on the Hypercube:
C. Moore and A. Russel (quant-ph’01)
Polynomial hitting time on the Hypercube:
J. K. ( ’02+)
hitting time on other graphs (numerical & Analytical studies):
Neil Shenvi and J. K. (in preparation ‘02)
Markov chains
Markov chains for algorithms:
IdeaIdea: construct a Markov chain (simple, local transitions only, efficiently implementable) (1) whose stationary distribution gives the
solution to the problem Mixing timeMixing time or (2) which hits the desired solution
Hitting timeHitting time
« Quantum » Markov chains ?« Quantum » Markov chains ?
Example: Random walk for 2SAT
Input: Boolean formula (conjunction of clauses of 2 variables) in X1, … , Xn
(ex. )
Question: Is satisfaisable?
(ex. YES, FFT is satisfying assignment)
Algorithm: 1) initialise the variables u.a. random (T- ¨true¨, F-¨false¨) 2) if all clauses satisfied – STOP, otherwise: 3) chose a non-satisfied clause, chose one of its two variables and flip its value; return to 2)
)()()()( 31323121 XXXXXXXX
Example: Random walk for 2SAT
Algorithm: 1) initialise the variables u.a. random (T- ¨true¨, F-¨false¨) 2) if all clauses satisfied – STOP, otherwise: 3) chose a non-satisfied clause, chose one of its two variables and flip its value; return to 2)
FFT
TFT FFFFTT
FTFTFFTTT
TTF
0
1
2
3
STOP>1/2
>1/2
>1/2 <1/2
<1/2Hamming distance
Random walk on a line with n+1 vertices !After t=2n2 repetitions (« Hitting time ») the succes probability is >1/2
(if satisfiable).
Random Walks...3SAT - “biased” random walk with
exponential hitting time
in general : local, simple Markov chain on exponential domain
0 1
>1/3
<2/3
2
>1/3
<2/3<2/3
>1/3
3 4
>1/3
<2/3
5
<2/3
>1/3
STOP
(fastest known 3-SAT algorithm based on random walk [Schöning’99, Hofmeister, Schöning & Watanabe’02])
Random Walks... Random walk on the line: Mixing time=Hitting
time =O(n2) stationary dist.=uniform
Questions: Stationary distribution? (ergodic –>
independent of initial state?) Mixing time? Hitting time?
Methods: spectral gap, conductance, Log Sobolev, coupling, ………
1/21/2
O(n2)
Classical/quantum random walksClassical
Transition matrix:
translationally invariant
Dt(i)-distribution after time t
stationary distribution
measure of “closeness”: total variation distance
mixing time - time until <const.
1/21/2
O(n2)
)( jiprobM ij
πDlim tt
t
i
t π(i)(i)D2
1πD t
Classical/quantum random walksClassical Quantum
Transition matrix:
translationally invariant
Dt(i)-distribution after time t
stationary distribution
measure of “closeness”: total variation distance
mixing time - time until <const.
1/21/2
O(n2)
21 +
21
?
)( jiprobM ij
πDlim tt
t
i
t π(i)(i)D2
1πD t
unitary?reversible?
localtranslationally invariant
Quantum random walk “Classical” Markov process:
Quantum??? Unitary??? Meyer [‘97]: All local, translationary invariant unitary
matrices are simple translations.
“R” “L”
0 1 0 0 ... 1
1 0 1 0 ... 0
0 1 0 1 0 ...1... 0 1 0 ... 020 0 0 ... ... 1
1 0 ... 0 1 0
M
Classical random walk Incorporate “coin-flip” into walk!
Classical walk in two steps:{,} ={(,0),(,0),(,1),…,(,n-1),(,n-1)}
flip direction coin C=
perform controlled shift S: “R”
“L” M=S•C Trace out (ignore, average over) the direction-
space
“R” “L”
1 1 1 01 " "= " "=
1 1 0 12
Classical random walk
{,} ={(,0),(,0),(,1),…,(,n-1),(,n-1)}
M=S•C Trace out (ignore, average over) the direction-space
1 1
1 1
1 11
1 1C 2
...
1 1
1 1
flip direction coin perform controlled shift : “R”
“L”
S =
1 0
0 0
0 0
0 1
……
…
Quantum random walk Meyer [‘97]: All local, translationary invariant unitary
matrices are simple translations.
“coined” walk in two steps:
{|,|}
”flip” direction coin ( )
perform controlled shift : | “R”
| “L”
1 1
2 2
“R” “L”
unitary “walk” U
U “collapses” to the classical random walk if we measure directions or positions at every step!
HH
1 11
1 -12
Quantum random walk
{,} ={(,0),(,0),(,1),…,(,n-1),(,n-1)}
M=S•C After t steps measure Trace out (ignore, average over) the direction-space
“R” “L”
1 1
1 1
1 11
1 1C 2
...
1 1
1 1
flip direction coin perform controlled shift : “R”
“L”
S =
1 0
0 0
0 0
0 1
……
Quantum random walksExample: start in
induces probability-dist. Pt(i) on the sites (after measurement)
Convergence?Convergence?
NO! U is unitary reversible! (no stationary distrib.)
Def. “averaged distribution” Qt (Cesaro limit):
Theorem: QTheorem: Qtt converges to a stationary distribution. converges to a stationary distribution.
|0|1
|2
|n-1
0
1 1 2 0 0 2H shift 0 1 1H shift
2 0 2 ...2 H shift
U 0 t
t
t ss 0
1Q (v) P
t( )v
1 1
2 2
Stationary distributionTheorem: QTheorem: Qtt converges to a stationary converges to a stationary
distribution.distribution.
|0|1
|2
|n-1t
t ss 0
1Q (v) P
t( )v
Calculate eigenvectors/eigenvalues of U Expand initial state:
State at time t:
Stationary distribution:
if
( , )i i
01
N
i ii
a
1
Nt
t i i ii
a
2 * *
{ , } , , 1
( ) , ( ) , ,N
ss s i j i j i j
d i j d
P v d v aa d v d v
t
t st t
s 0
1limQ (v) lim P
t( ) ( )v v
t
i
is 0 i
(1 (t
*
*
*
1 ) 2) 0
1
tjs
j tj i jt
i j
Stationary distributionTheorem: QTheorem: Qtt converges to a stationary converges to a stationary
distribution.distribution.
|0|1
|2
|n-1t
t ss 0
1Q (v) P
t( )v
Stationary distribution:
uniform if G non-degenerate ( ):
If G also abelian -> stationary distribution uniform:
characters of the abelian group (unit norm)
2 * *
{ , } , , 1
( ) , ( ) , ,N
ss s i j i j i j
d i j d
P v d v aa d v d v
i
tt
d,i, j:
limQ (v) = *( ) , | | ,
j
i j i jv aa d v d v
i j
d,i
22( ) , |i iv a d v
i i iw 1( )i i
v
v vn
Observations
Classically: real eigenvalues Quantum: complex eigenvalues
Classically: “behavior” depends ( ) on second largest eigenvalue
Quantum: all eigenvalues equally important
Ex: mixing time determined by convergence of
i.e. by
1 21 ... 1N 2 * 1i i i
2
ti
is 0 i
(1 (t
*
*
*
1 ) 2)
1
tjs
j
j i jt
, :mini j
i ji j
(minimum gap)
Results on mixing time*Cycle:
quantum walk converges towards uniform distribution
Mixing time:classical: = (N2 log(1/))quantum: =O(N log N / 3)
Total variation distance:
Similar results in higher dimensions, for Cayley graphs, graphs on abelian groups, walks with different coins,…
ε t ετ s.t. ε t τ 0 ,d v
ττ
*D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC’01
2
, :
1(.) (.) 2
i j
T ii j i j
Q aT
(ln( /2) 1)(.) (.)T
ndQ
T
|0|1
|2
|n-1
Results on mixing time*Cycle:
quantum: =O(N log N / 3)
« Warmstart » to get logarithmic -dependence: Initialize in Run quantum walk for steps -> measure (node v) Restart new walk in (d-random) Repeat k-times
Resulting distribution is -close to the stationary distribution
(works if stationary distribution is independent of initial state)
τ
*D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC’01
|0|1
|2
|n-1
0
,d v
k
1( log log( ))M On n
Results on mixing time*
Conductance-type lower bound for mixing time of any quantum walk on bounded degree graph:
capacitance flow
conductance:
Theorem (Jerrum,Sinclair’89):
1( )d
21 O1
Classical:
Quantum: d-max.degree
*D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC’01
X uu X
C
,,
X uv uu X v X
F p
X G
012
min
X
X
X GXC
FC
2
2(1 ) 22
ConductanceQuantum: d-max.degree
Cut (X,X) of G, boundary
Idea: start with state concentrated in X and show that at each time step “leakage” into X is bounded by .
Then after steps
And hence
1( )d
{ : v X}XB v X edge
12
' min ' dX
X G
B
X
XB
X
(1 )XB XVX
1'
Quantum Hitting Time on Hypercube
Space:
000
010
100
101
111011
{ , , } ({1,..., } { : {0,1} })nn z z
Quantum Hitting Time on Hypercube
Space:
Walk: Conditional Shift Coin C (respects permutational symmetry of
hypercube)
000
010
100
101
111011
{ , , } ({1,..., } { : {0,1} })nn z z
: iS i z i z e 00..01 00..0ii
e
...
a b b
b a b bC
b b a
2 21 a bn n
Quantum Hitting Time on Hypercube
Space:
Walk: Conditional Shift Coin C (respects permutational symmetry of
hypercube)
Initial state:
000
010
100
101
111011
{ , , } ({1,..., } { : {0,1} })nn z z
: iS i z i z e 00..01 00..0ii
e
...
a b b
b a b bC
b b a
2 21 a bn n
1
1(0) 00...0
n
i
in
Symmetric superposition over all directions
Mixing time: classical: quantum: (coupon collector) (Moore&Russel’01)
O( log )n n . 4instk n 3O n
Hitting time?Dilemma: constant measurement of position will
collapse U to the classical walk…
Two options:One-shot q-hitting-time (T,p):
Measure only at time T “Hits” desired target-state x with probability >p
Concurrent q-hitting-time (T,p): Partial measurement (“Am I at x/Am I not at x?”) at
all times Stop walk if x is hit. Probability >p to hit x before
time T
Results on hitting time*
Classical: from v to opposite v’ hitting-time
Quantum: One-shot hitting-time from v to v’ (T,p)
*J.K.’02
O(exp( )) 1 vn
T ( ), ( )2 2
n On n On 1-2
log n p=1-O
n
12 (T-n) even,
000
010
100
101
111011
and3log
T= n p=1-O2
nn
Results on hitting time*
Classical: from v to opposite v’ hitting-time
Quantum: One-shot hitting-time from v to v’ (T,p)
Need to know with accuracy when to measure, success 1 in linear time!
Concurrent hitting-time from v to v’ (T,p)
No information on when to measure needed, with amplification success 1 in T=O(n2)!
*J.K.’02
O(exp( )) 1 vn
T ( ), ( )2 2
n On n On 1-2
log n p=1-O
n
12 (T-n) even,
000
010
100
101
111011
O n
and3log
T= n p=1-O2
nn
1T= n p=2 n
“Details” Use symmetry to calculate eigenvalues/eigenvectors
of unmeasured walk U “Assymptotics” to calculate hitting probability at T
one-shot hitting time (T,p)
For concurrent hitting time give a lower bound on hitting probability in terms of unmeasured walk U:
Lemma:
2t(hit at t| not stopped before t)=p
2t(hit v' if measured at t)=p
2 3n2 1 (log )O n n
1t t t
22 2T T T
2
1t=0 t=0 t=0
1 1(hit before T)= 1T
t t t tp O nT T T
Robustness of initial condition
Polynomial hitting time to opposite corner, how long from other sites (or to sites close to corner)?
“close” initialstates give similar polynomial behavior
Upper bound:Region around v of polynomial hitting time to v’ at
most(otherwise we could find search
algorithm that beats the lower bound for quantum searching (Grover))
000
010
100
101
111011
2nO
2n
Open graphs
nG
… …
n-level binary tree
Example*:
*A.Childs, E.Farhi, S. Gutman, quant-ph/01…
starthit
1
1/3
2/3
2
2/3
1/3
n
1/3
2/3
n+1
2/3
2/3
…
1/3
2/3
1/3
2/3
Reduces to assymetric walk on the line (classically and quantum).
Open graphs
nG
… …
n-level binary tree
Example*:
*A.Childs, E.Farhi, S. Gutman, quant-ph/01…
starthit
1
1/3
2/3
2
2/3
1/3
n
1/3
2/3
n+1
2/3
2/3
…
1/3
2/3
1/3
2/3
Reduces to assymetric walk on the line (classically and quantum).
Classical: O(exp(n)) hitting time
Quantum: (numeric) poly(n) hitting time(N.Shenvi & J.K.’02)
Outlook/Open questions
In general which graphs have exponential quantum/classical gaps in hitting times ?
How robust is this gap w.r.t. initial position/distribution?
Mixing times for non-abelian walks ?Mixing times for walks on non-bounded
degree graphs?For degenrate or non-abelian groups
stationary distribution depends on initial state -algorithmic use?
Algorithmic use?
Collaborators and related work
Discrete-time walks (Mixing Time) *: (On the Line)**:
Dorit Aharonov (Hebrew University) A. Ambainis, E. Bach, A. Nayak,Andris Ambainis (IAS, Princeton) A. Vishwanath, J. WatrousJ. K. (LRI, Orsay&UC Berkeley) (**STOC’01)Umesh Vazirani (UC Berkeley)
(*STOC’01)
Mixing on the Hypercube:
C. Moore and A. Russel (quant-ph’01)
Polynomial hitting time on the Hypercube:
J. K. (submitted ‘02)
hitting time on other graphs (numerical & Analytical studies):
Neil Shenvi and J. K. (in preparation ‘02)