Quantum Random Walks

1
Quantum 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
2
Towards nanotechnology
Gordon Moore 1965
Size of the components Number of
components Speed
Theoretical limitations reached in 2020 !!!
Apparition of quantum phenomena
prevent or use quantum effects ?
3
Information is physical!

• Use the laws of quantum mechanics for the basic
  components of an information processing machine!
• Quantum computing
• Quantum cryptography
• Quantum information

4
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

5
The qubit

• Classical bit b?0,1
• Probabilistic bit probability distribution
  d?R0,1 such that d1 1. ? d(p,1-p)
  with p ?0,1
• Quantum bit ?? ? C0,1 such that ?? 21.
  
• ? ?? ? 0? ? 1? with ? 2 ? 21
• (Dirac notation)

6
Qubit evolution

• Measure reads and modifies

0?
? 2
Measure
? 0? ? 1?
1?
? 2
? Superposition ? Probability distribution

• Unitary transformation U? C2?2 such that UUId

U
?? U ??
??
unitary ? reversible
U
??
U??
7
Example

• Superposition
• Measure

0?
1/3
Measure
??
1?
2/3
8
Example

• Superposition
• Measure
• Unitary transformations
• NOT 0? ? 1?
• Hadamard

0?
1/3
Measure
??
1?
2/3
U
?? U ??
??
H
9
Quantum computer n qubits

• n qubits ? tensor product ?? ? C0,1n such that
  ?? 21.
• ? ?? ?x?0,1n ?x x? with ?x ?x 2 1
• Measure
• Partial Measure

?x 2
?x?0,1n ?x x?
x?
Measure
Second bit 0 (?2 ? 2 )
Measure
? 00? ? 01?? 10? ?11?
10
Quantum computer n qubits

• n qubits ? tensor product ?? ? C0,1n such that
  ?? 21.
• ? ?? ?x?0,1n ?x x? with ?x ?x 2 1
• Measure
• Partial Measure
• Unitary transformation ?? ?U?? with U? U(2n)
• ex XOR

?x 2
?x?0,1n ?x x?
x?
Measure
Second bit 0 (?2 ? 2 )
Measure
? 00? ? 01?? 10? ?11?
00? 01? 10? 11?
i?
i?
XOR(i,j)?

j?
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Quantum computing a function

• Let f 0,1n ? 0,1m
• x ? f(x)
• Reversible
• Rf 0,1nm ? 0,1nm
• (x,y) ? (x,y?f(x))
• Quantum
• Uf ?U(2nm) Cnm ? Cnm
• x?y? ? x?y?f(x) ?

12
Simplest Quantum AlgorithmDeutschs Problem

• Input function f0,1?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!!!

x?
x?
f
y?
y?f(x)?
H
H
0?
Measure
f
1?
H
0? -constant 1? -balanced
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Simplest Quantum AlgorithmDeutschs Problem

• Input function f0,1?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!!!

x?
x?
f
y?
y?f(x)?
H
H
0?
Measure
f
1?
H
0? -constant 1? -balanced
0 if f constant
0 if f balanced
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Universal computation

• Classical circuit model
• Quantum circuit model

• evaluates boolean functions
• can be constructed from universal local gates
  (ex. NAND, COPY)

?
0 0
0 1 0 1
?
bits
?

• unitary transformations U

0? 0? 1?
1? 0? 0?
U
qubits
Measure
15
Quantum circuits
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Quantum Circuits

• Quantum 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
• -gt Classical circuits ? Quantum circuits

17
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

18
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(xH) where
  H? G
• Output H (generators)
• efficient quantum algorithm if G - Abelian or
   special 

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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(xH) 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)

20
Discrete Quantum Walks
Discrete-time walks on finite graphs (Mixing
Time) Dorit Aharonov
(Hebrew University) Andris Ambainis (IAS,
Princeton) J. K. (LRI, OrsayUC
Berkeley) Umesh Vazirani (UC
Berkeley) (STOC01)
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)
Mixing on the Hypercube C. Moore and A. Russel
(quant-ph01)
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Markov chains

• Markov chains for algorithms
• Idea construct a Markov chain (simple, local
  transitions only, efficiently implementable)
• (1) whose stationary distribution gives the
  solution to the problem ? Mixing time
• or (2) which hits the desired solution ? Hitting
  time
•  Quantum  Markov chains ?

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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)
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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
0
STOP
gt1/2
TFT
FFF
FTT
Hamming distance
1
lt1/2
gt1/2
FTF
TFF
TTT
2
lt1/2
gt1/2
TTF
3
Random walk on a line with n1 vertices ! After
t2n2 repetitions ( Hitting time ) the succes
probability is gt1/2 (if ? satisfiable).
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Random Walks...

• 3SAT - biased random walk with exponential
  hitting time
• in general local, simple Markov chain on
  exponential domain

lt2/3
lt2/3
lt2/3
lt2/3
lt2/3
(fastest known 3-SAT algorithm based on random
walk Schöning99, Hofmeister, Schöning
Watanabe02)
STOP
0
1
2
3
4
5
gt1/3
gt1/3
gt1/3
gt1/3
gt1/3
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Random Walks...

• Random walk on the line Mixing timeHitting time
  O(n2) stationary dist.uniform
• Questions
• Stationary distribution? (ergodic gt
  independent of initial state?)
• Mixing time?
• Hitting time?
• Methods spectral gap, conductance, Log Sobolev,
  coupling,

1/2
1/2
O(n2)
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Classical/quantum random walks

• Classical
• Transition matrix
• translationally invariant
• Dt(i)-distribution after time t
• stationary distribution
• measure of closeness total variation distance
• mixing time ? - time until ?ltconst.

1/2
1/2
O(n2)
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Classical/quantum random walks

• Classical Quantum
• Transition matrix
• translationally invariant
• Dt(i)-distribution after time t
• stationary distribution
• measure of closeness total variation distance
• mixing time ? - time until ?ltconst.

1/2
1/2
O(n2)
?
unitary? reversible? local translationally
invariant
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Quantum random walk

• Classical Markov process
• Quantum??? Unitary???
• Meyer 97 All local, translationary invariant
  unitary matrices are simple translations.

R
L
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Classical random walk
R
L

• 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   
• MSC
• Trace out (ignore, average over) the
  direction-space

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Classical random walk
? ?
? ?

? ?

• ?,? ? (?,0),(?,0),(?,1),,(?,n-1),
  (?,n-1)
•  
• MSC
• Trace out (ignore, average over) the
  direction-space

• perform controlled shift ? ? R   
• ? ? L   
• S

• flip direction coin

31
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   

R
L
H
H
unitary walk U
U collapses to the classical random walk if we
measure directions or positions at every step!
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Quantum random walk
R
L

• ?,? ? (?,0),(?,0),(?,1),,(?,n-1),
  (?,n-1)
•  
• MSC
• After t steps measure
• Trace out (ignore, average over) the
  direction-space

• perform controlled shift ? ? R   
• ? ? L   
• S

• flip direction coin

33
Quantum random walks
0?
n-1?
1?
2?

• Example start in
• induces probability-dist. Pt(i) on the sites
  (after measurement)
• Convergence?
• NO! U is unitary ? reversible! (no stationary
  distrib.)
• Def. averaged distribution Qt (Cesaro limit)
• Theorem Qt converges to a stationary
  distribution.

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Stationary distribution
0?
n-1?
1?
2?

• Theorem Qt converges to a stationary
  distribution.

• Calculate eigenvectors/eigenvalues
  of U
• Expand initial state
• State at time t
• Stationary distribution
• if

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Stationary distribution
0?
n-1?
1?
2?

• Theorem Qt converges to a stationary
  distribution.

• Stationary distribution
• uniform if G non-degenerate ( )
• If G also abelian -gt stationary distribution
  uniform
• characters of the abelian group (unit norm)

36
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

(minimum gap)
37
Results on mixing time
0?
n-1?
1?
2?

• 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,

D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC01
38
Results on mixing time
0?
n-1?
1?
2?

• Cycle
• quantum ? O(N log N / ?3) 
•  Warmstart  to get logarithmic ?-dependence
• Initialize in
• Run quantum walk for steps -gt 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)

D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC01
39
Results on mixing time

• Conductance-type lower bound for mixing time of
  any quantum walk on bounded degree graph
• capacitance flow
• conductance
• Theorem (Jerrum,Sinclair89)

Classical
Quantum d-max.degree
D. Aharonov,A. Ambainis, J.K., U.Vazirani-STOC01
40
Conductance

• Quantum 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

41
Quantum Hitting Time on Hypercube

• Space

42
Quantum Hitting Time on Hypercube

• Space
• Walk
• Conditional Shift
• Coin C (respects permutational symmetry of
  hypercube)

43
Quantum Hitting Time on Hypercube

• Space
• Walk
• Conditional Shift
• Coin C (respects permutational symmetry of
  hypercube)
• Initial state

Symmetric superposition over all directions
Mixing time classical quantum (coupon
collector) (MooreRussel01)
44
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 gtp
• 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 gtp to hit x
  before time T

45
Results on hitting time

• Classical from v to opposite v hitting-time
• Quantum
• One-shot hitting-time from v to v (T,p)

and
(T-n) even,
J.K.02
46
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 TO(n2)!

and
(T-n) even,
J.K.02
47
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

48
Robustness of initial condition

• Polynomial hitting time to opposite corner, how
  long from other sites (or to sites close to
  corner)?
• close initial
• states 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))

49
Open graphs
Example


start
hit
n-level binary tree
Reduces to assymetric walk on the line
(classically and quantum).
2/3
2/3
2/3
2/3
1/3
1/3

1
2
n
n1
1/3
1/3
1/3
2/3
2/3
2/3
A.Childs, E.Farhi, S. Gutman, quant-ph/01
50
Open graphs
Example
Classical O(exp(n)) hitting time


start
hit
Quantum (numeric) poly(n) hitting
time (N.Shenvi J.K.02)
n-level binary tree
Reduces to assymetric walk on the line
(classically and quantum).
2/3
2/3
2/3
2/3
1/3
1/3

1
2
n
n1
1/3
1/3
1/3
2/3
2/3
2/3
A.Childs, E.Farhi, S. Gutman, quant-ph/01
51
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?

52
Outlook/Open questions

• General THEORY???!!!
• Connection to classical
• Markov chains?

53
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. Watrous J. K. (LRI,
OrsayUC Berkeley) (STOC01) Umesh Vazirani
(UC Berkeley) (STOC01)
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)
Mixing on the Hypercube C. Moore and A. Russel
(quant-ph01)