Quantum walks on graphs

Andrew ChildsCalifornia Institute of Technology

Information is physical.

Outline1. Introduction: Quantum systems as information processing machines

2. Exponential speedup by quantum walk

3. Spatial search by quantum walk

4. Evaluating Boolean formulas

1. Introduction

•Prepare n qubits in the state•Apply a sequence of unitary operations acting on one or two qubits

at a time•Perform a measurement to get the result

The universal quantum computer(The ultimate quantum physics lab!)

|0!

|0!

|0!

|0! U1

U2

U3

U4

U5

U6

!!"

!!"

!!"

!!"

|00 . . . 0!

Note: Many equivalent models exist (Hamiltonian dynamics of coupled spins, braiding of nonabelian anyons, quantum cellular automata, ...).

Implementations of quantum computers

... and many others!

Trapped ions

!"##$!%&'$

("%&")*+),-!

&").

#/.$0*1$/(

***&)%$0)/# **("%&")/#

231&%*.%/%$. 231&%*.%/%$.

/%*0$.%

+),4!.5&)*/)%&5/0/##$#

.5&)*5/0/##$#

6&730$*8

(Monroe & Wineland)

Nuclear spins

(Chuang et al.)

Quantum dots

/u/divince/tex/revtex/mmm2000/divloss3

e ee

quantum wellheterostructure magnetized

e

or high-g layerback gates

e

(Loss & DiVincenzo)

Superconducting circuits

(Nakamura et al.)

Motivation

• Coherent control of an artificial two-level system in solid-state device

• Understanding the mechanism of decoherence

!

2 µm

The threshold theoremRealistic quantum systems are subject to noise. Is quantum computation still possible?

The threshold theorem

von Neumann 1956 (“Probabilistic logics and the synthesis of reliable organisms from unreliable components”): Provided the noise level is below some threshold value, any computation can be performed with success probability arbitrarily close to 1, by encoding the computation redundantly (incurring only logarithmic overhead).

Realistic quantum systems are subject to noise. Is quantum computation still possible?

The threshold theorem

von Neumann 1956 (“Probabilistic logics and the synthesis of reliable organisms from unreliable components”): Provided the noise level is below some threshold value, any computation can be performed with success probability arbitrarily close to 1, by encoding the computation redundantly (incurring only logarithmic overhead).

Realistic quantum systems are subject to noise. Is quantum computation still possible?

Shor 1996, Aharonov and Ben-Or 1997, Kitaev 1997, et al.: Similar result for quantum computers. Threshold estimates to .! 10!3 10!5

The threshold theorem

von Neumann 1956 (“Probabilistic logics and the synthesis of reliable organisms from unreliable components”): Provided the noise level is below some threshold value, any computation can be performed with success probability arbitrarily close to 1, by encoding the computation redundantly (incurring only logarithmic overhead).

Realistic quantum systems are subject to noise. Is quantum computation still possible?

Shor 1996, Aharonov and Ben-Or 1997, Kitaev 1997, et al.: Similar result for quantum computers. Threshold estimates to .! 10!3 10!5

In the rest of this talk, we assume a perfectly functioning quantum computer.

Interference as a resource

Interference as a resource

Given a black box for , determine .

Toy problem [Deutsch 1985]

f : {0, 1}! {0, 1} f(0)! f(1)

Interference as a resource

Classically: Two queries required.

Given a black box for , determine .

Toy problem [Deutsch 1985]

f : {0, 1}! {0, 1} f(0)! f(1)

Interference as a resource

Classically: Two queries required.

Quantumly: One query sufficient!

Given a black box for , determine .

Toy problem [Deutsch 1985]

f : {0, 1}! {0, 1} f(0)! f(1)

Interference as a resource

Classically: Two queries required.

Quantumly: One query sufficient!

Given a black box for , determine .

Toy problem [Deutsch 1985]

f : {0, 1}! {0, 1} f(0)! f(1)

|0!

|1!

H

H

H|x, y! "# |x, y $ f(x)!

Interference as a resource

Classically: Two queries required.

Quantumly: One query sufficient!

|0!+ |1!"2

# |0! $ |1!"2

Given a black box for , determine .

Toy problem [Deutsch 1985]

f : {0, 1}! {0, 1} f(0)! f(1)

|0!

|1!

H

H

H|x, y! "# |x, y $ f(x)!

Interference as a resource

Classically: Two queries required.

Quantumly: One query sufficient!

|0!+ |1!"2

# |0! $ |1!"2

(!1)f(0)|0"+ (!1)f(1)|1"#2

$ |0" ! |1"#2

Given a black box for , determine .

Toy problem [Deutsch 1985]

f : {0, 1}! {0, 1} f(0)! f(1)

|0!

|1!

H

H

H|x, y! "# |x, y $ f(x)!

Interference as a resource

Classically: Two queries required.

Quantumly: One query sufficient!

|0!+ |1!"2

# |0! $ |1!"2

(!1)f(0)|0"+ (!1)f(1)|1"#2

$ |0" ! |1"#2

|f(0)! f(1)"

(!1)f(0)!f(1) |0" ! |1"#2

Given a black box for , determine .

Toy problem [Deutsch 1985]

f : {0, 1}! {0, 1} f(0)! f(1)

|0!

|1!

H

H

H|x, y! "# |x, y $ f(x)!

Interference as a resource

Classically: Two queries required.

But don’t classical systems exhibit interference too?

Quantumly: One query sufficient!

|0!+ |1!"2

# |0! $ |1!"2

(!1)f(0)|0"+ (!1)f(1)|1"#2

$ |0" ! |1"#2

|f(0)! f(1)"

(!1)f(0)!f(1) |0" ! |1"#2

Given a black box for , determine .

Toy problem [Deutsch 1985]

f : {0, 1}! {0, 1} f(0)! f(1)

|0!

|1!

H

H

H|x, y! "# |x, y $ f(x)!

Hilbert space is a big place

n classical bits: 2n discrete values

Hilbert space is a big place

n classical bits: 2n discrete values

n quantum bits: 2n complex numbers |!! =2n!1!

x=0

"x|x!

Hilbert space is a big place

n classical bits: 2n discrete values

n random bits: 2n real numbers (probabilities)

n quantum bits: 2n complex numbers |!! =2n!1!

x=0

"x|x!

Hilbert space is a big place

n classical bits: 2n discrete values

n random bits: 2n real numbers (probabilities)

But probabilities do not exhibit interference!

n quantum bits: 2n complex numbers |!! =2n!1!

x=0

"x|x!

Hilbert space is a big place

n classical bits: 2n discrete values

How can we exploit the efficient representation of interference phenomena to perform fast computations?

n random bits: 2n real numbers (probabilities)

But probabilities do not exhibit interference!

n quantum bits: 2n complex numbers |!! =2n!1!

x=0

"x|x!

From random walk to quantum walk

Graph G:

1 2

3 4

5

From random walk to quantum walk

Graph G:

1 2

3 4

5A =

!

""""#

0 1 1 0 01 0 0 1 11 0 0 1 00 1 1 0 10 1 0 1 0

$

%%%%&

adjacency matrix

From random walk to quantum walk

Graph G:

1 2

3 4

5A =

!

""""#

0 1 1 0 01 0 0 1 11 0 0 1 00 1 1 0 10 1 0 1 0

$

%%%%&

adjacency matrix

L =

!

""""#

!2 1 1 0 01 !3 0 1 11 0 !2 1 00 1 1 !3 10 1 0 1 !2

$

%%%%&

Laplacian

From random walk to quantum walk

Graph G:

1 2

3 4

5

Random walk on G

State: Probability pj(t) of being at vertex j at time t

Dynamics:ddt

!p = !"L!p

A =

!

""""#

0 1 1 0 01 0 0 1 11 0 0 1 00 1 1 0 10 1 0 1 0

$

%%%%&

adjacency matrix

L =

!

""""#

!2 1 1 0 01 !3 0 1 11 0 !2 1 00 1 1 !3 10 1 0 1 !2

$

%%%%&

Laplacian

From random walk to quantum walk

Graph G:

1 2

3 4

5

Random walk on G

State: Probability pj(t) of being at vertex j at time t

Dynamics:ddt

!p = !"L!p

A =

!

""""#

0 1 1 0 01 0 0 1 11 0 0 1 00 1 1 0 10 1 0 1 0

$

%%%%&

adjacency matrix

L =

!

""""#

!2 1 1 0 01 !3 0 1 11 0 !2 1 00 1 1 !3 10 1 0 1 !2

$

%%%%&

Laplacian

Quantum walk on G

State: Amplitude qj(t) to be at vertex j at time t

Dynamics: iddt

!q = !"L!q

2. Exponential speedupby quantum walk

•Childs, Farhi, and Gutmann, Quantum Inf. Proc. 1, 35-43 (2002).•Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman, in Proc. STOC (2003), pp. 59-68.

in out

Black box graph traversal problemNames of vertices are random bit strings (length 2 log |G|).Name of in vertex is known.

Name of out vertex is unknown; find it!Black box outputs the names of adjacent vertices.

Black box graph traversal problemNames of vertices are random bit strings (length 2 log |G|).Name of in vertex is known.

Name of out vertex is unknown; find it!Black box outputs the names of adjacent vertices.

Claim: There is a family of graphs Gn (with designated in and out vertices) for which a quantum walk starting at the in vertex finds the out vertex in time poly(n), but any classical algorithm using poly(n) queries finds the out vertex with exponentially small probability.

Black box graph traversal problemNames of vertices are random bit strings (length 2 log |G|).Name of in vertex is known.

Name of out vertex is unknown; find it!Black box outputs the names of adjacent vertices.

Claim: There is a family of graphs Gn (with designated in and out vertices) for which a quantum walk starting at the in vertex finds the out vertex in time poly(n), but any classical algorithm using poly(n) queries finds the out vertex with exponentially small probability.

in outTypical G4:

Reduction of the quantum walk

in out

Reduction of the quantum walk

in out

Column subspace

Nj =

!2j 0 ! j ! n

22n+1!j n + 1 ! j ! 2n + 1

|col j! =1!Nj

"

a!column j

|a!

where

Reduction of the quantum walk

in out

Column subspace

Nj =

!2j 0 ! j ! n

22n+1!j n + 1 ! j ! 2n + 1

|col j! =1!Nj

"

a!column j

|a!

where

Reduced adjacency matrix!col j|A|col j + 1"

=

!"#

"$

#2 0 $ j $ n% 1 ,

n + 1 $ j $ 2n

2 j = ncol 0 col 1 col 2 col 3 col 4 col 5 col 6 col 7 col 8 col 9

!2 2

!2

!2

!2

!2

!2

!2

!2

Implementing the walkProblem: Given a black box for G, implement the quantum walk on G, i.e., simulate the unitary time evolution e—iHt where H=L (or A).(Cf. implementing a random walk, which is easy.)

Main idea: Color the graph. Then the simulation breaks into small pieces that are easy to handle.

Implementing the walkProblem: Given a black box for G, implement the quantum walk on G, i.e., simulate the unitary time evolution e—iHt where H=L (or A).(Cf. implementing a random walk, which is easy.)

Main idea: Color the graph. Then the simulation breaks into small pieces that are easy to handle.

Implementing the walkProblem: Given a black box for G, implement the quantum walk on G, i.e., simulate the unitary time evolution e—iHt where H=L (or A).(Cf. implementing a random walk, which is easy.)

Main idea: Color the graph. Then the simulation breaks into small pieces that are easy to handle.

Implementing the walkProblem: Given a black box for G, implement the quantum walk on G, i.e., simulate the unitary time evolution e—iHt where H=L (or A).(Cf. implementing a random walk, which is easy.)

Main idea: Color the graph. Then the simulation breaks into small pieces that are easy to handle.

= + +

Implementing the walkProblem: Given a black box for G, implement the quantum walk on G, i.e., simulate the unitary time evolution e—iHt where H=L (or A).(Cf. implementing a random walk, which is easy.)

Main idea: Color the graph. Then the simulation breaks into small pieces that are easy to handle.

Any sufficiently sparse graph can be efficiently colored using only local information (Linial 1987).

= + +

Implementing the walkProblem: Given a black box for G, implement the quantum walk on G, i.e., simulate the unitary time evolution e—iHt where H=L (or A).(Cf. implementing a random walk, which is easy.)

Main idea: Color the graph. Then the simulation breaks into small pieces that are easy to handle.

Any sufficiently sparse graph can be efficiently colored using only local information (Linial 1987).

= + +

This idea is useful for simulating Hamiltonians whenever the graph of nonzero matrix elements is sufficiently sparse.

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

Proof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

Classical lower bound

Theorem: Any classical algorithm that makes at most 2n/6 queries to the black box finds the out vertex with probability at most 4⋅2—n/6.

inProof idea:

3. Spatial search by quantum walk

•Childs and Goldstone, Phys. Rev. A 70, 022314 (2004).•Childs and Goldstone, Phys. Rev. A 70, 042312 (2004).

Unstructured search

Search space: N items,

Goal: Find one marked item, wx ! {1, 2, . . . , N}

Query: “is w = x?”

i.e., black box function f(x) =

!0 x != w

1 x = w

Unstructured search

Search space: N items,

Goal: Find one marked item, wx ! {1, 2, . . . , N}

Query: “is w = x?”

i.e., black box function f(x) =

!0 x != w

1 x = w

Classical complexity: !(N)

Unstructured search

Search space: N items,

Goal: Find one marked item, wx ! {1, 2, . . . , N}

Query: “is w = x?”

i.e., black box function f(x) =

!0 x != w

1 x = w

Classical complexity: !(N)Quantum algorithm [Grover 1996]: O(

!N)

Unstructured search

Search space: N items,

Goal: Find one marked item, wx ! {1, 2, . . . , N}

Query: “is w = x?”

i.e., black box function f(x) =

!0 x != w

1 x = w

Classical complexity: !(N)Quantum algorithm [Grover 1996]: O(

!N)

Quantum lower bound [BBBV 1996]: !(!

N)

Unstructured search

Search space: N items,

Goal: Find one marked item, wx ! {1, 2, . . . , N}

Query: “is w = x?”

i.e., black box function f(x) =

!0 x != w

1 x = w

Classical complexity: !(N)Quantum algorithm [Grover 1996]: O(

!N)

Quantum lower bound [BBBV 1996]: !(!

N)

Grover searching can be applied to many computational problems.But can it be used to search a physical database, in which the N items are distributed in space?

Spatial search by quantum walkQuantum walk approach: H = !!L ! |w"#w| Lab =

!"#

"$

1 ab ! G

"deg(a) a = b

0 otherwise

L # $2

Spatial search by quantum walkQuantum walk approach: H = !!L ! |w"#w|

|s! =1"N

!

x!G

|x!Start in the state .

Lab =

!"#

"$

1 ab ! G

"deg(a) a = b

0 otherwise

L # $2

Spatial search by quantum walkQuantum walk approach: H = !!L ! |w"#w|

|s! =1"N

!

x!G

|x!Start in the state .

Choose ∞ so that for as small a T as possible, is large. |!w|e!iHT |s"|2

Lab =

!"#

"$

1 ab ! G

"deg(a) a = b

0 otherwise

L # $2

Spatial search by quantum walkQuantum walk approach: H = !!L ! |w"#w|

|s! =1"N

!

x!G

|x!Start in the state .

Choose ∞ so that for as small a T as possible, is large. |!w|e!iHT |s"|2

!

Lab =

!"#

"$

1 ab ! G

"deg(a) a = b

0 otherwise

L # $2

Spatial search by quantum walkQuantum walk approach: H = !!L ! |w"#w|

|s! =1"N

!

x!G

|x!Start in the state .

Choose ∞ so that for as small a T as possible, is large. |!w|e!iHT |s"|2

!

ground statefirst excited state

! |w"! |s"

H ! "|w#$w|

! ! 0

Lab =

!"#

"$

1 ab ! G

"deg(a) a = b

0 otherwise

L # $2

Spatial search by quantum walkQuantum walk approach: H = !!L ! |w"#w|

|s! =1"N

!

x!G

|x!Start in the state .

Choose ∞ so that for as small a T as possible, is large. |!w|e!iHT |s"|2

!

ground statefirst excited state

! |w"! |s"

H ! "|w#$w|

! ! 0

H ! "!L

ground state ! |s"

! !"

Lab =

!"#

"$

1 ab ! G

"deg(a) a = b

0 otherwise

L # $2

Spatial search by quantum walkQuantum walk approach: H = !!L ! |w"#w|

|s! =1"N

!

x!G

|x!Start in the state .

Choose ∞ so that for as small a T as possible, is large. |!w|e!iHT |s"|2

!

ground statefirst excited state

! |w"! |s"

H ! "|w#$w|

! ! 0

H ! "!L

ground state ! |s"

! !"

critical ∞! |s"+ |w"! |s" # |w"

ground statefirst excited state

time ! 1E1!E0

Lab =

!"#

"$

1 ab ! G

"deg(a) a = b

0 otherwise

L # $2

d = 4, N = 64 = 1296

Results

Graph Success amplitude Run time

Complete 1 – o(1) O(N 1/2)

Hypercube 1 – o(1) O(N 1/2)

Lattice, d > 4 O(1) O(N 1/2)

Lattice, d = 4 O(1/log1/2 N) O((N log N)1/2)

Lattice, d = 3 O(N -1/6) O(N 2/3)

Lattice, d = 2 O((log N/N)1/2) O(N /log N)

Behavior for d>4

where

Ij,d =1

(2!)d

! !

!!

dd"k

[E("k)]j

Critical ∞:

Run time:

Success probability:

!! = I1,d

T =!!

I2,dN

2I1,d

|!w|e!iHT |s"|2 =I21,d

I2,d

E(!k) = 2"!d!

"dj=1 cos kj

#

dispersion relation

Behavior for d>4

where

Ij,d =1

(2!)d

! !

!!

dd"k

[E("k)]j

Critical ∞:

Run time:

Success probability:

!! = I1,d

T =!!

I2,dN

2I1,d

|!w|e!iHT |s"|2 =I21,d

I2,d

E(!k) = 2"!d!

"dj=1 cos kj

#

dispersion relation

Behavior for d>4

where

Ij,d =1

(2!)d

! !

!!

dd"k

[E("k)]j

Critical ∞:

Run time:

Success probability:

!! = I1,d

T =!!

I2,dN

2I1,d

|!w|e!iHT |s"|2 =I21,d

I2,d

E(!k) = 2"!d!

"dj=1 cos kj

#

dispersion relation

Behavior for d>4

where

Ij,d =1

(2!)d

! !

!!

dd"k

[E("k)]j

Critical ∞:

Run time:

Success probability:

!! = I1,d

T =!!

I2,dN

2I1,d

|!w|e!iHT |s"|2 =I21,d

I2,d

!

0

dd!k

|!k|p!

!

0

kd!1dk

|!k|pNote:

converges for d>p.

E(!k) = 2"!d!

"dj=1 cos kj

#

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Lattice version: , Pj |!x! = i2 (|!x + ej! " |!x" ej!)

E(!k) = ±"!"d

j=1 sin2 kj

H0 = !!d

j=1 "jPj

dispersion relation

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Lattice version: , Pj |!x! = i2 (|!x + ej! " |!x" ej!)

E(!k) = ±"!"d

j=1 sin2 kj

H0 = !!d

j=1 "jPj

dispersion relation

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Lattice version: , Pj |!x! = i2 (|!x + ej! " |!x" ej!)

E(!k) = ±"!"d

j=1 sin2 kj

H0 = !!d

j=1 "jPj

dispersion relation

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Lattice version: , Pj |!x! = i2 (|!x + ej! " |!x" ej!)

E(!k) = ±"!"d

j=1 sin2 kj

H0 = !!d

j=1 "jPj

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Improved:

E(!k) = ±!

"2"d

j=1 sin2 kj + #2[2"d

j=1(1! cos kj)]2

H0 = !!d

j=1 "jPj + #$L

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Improved:

E(!k) = ±!

"2"d

j=1 sin2 kj + #2[2"d

j=1(1! cos kj)]2

H0 = !!d

j=1 "jPj + #$L

dispersion relation

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Improved:

E(!k) = ±!

"2"d

j=1 sin2 kj + #2[2"d

j=1(1! cos kj)]2

H0 = !!d

j=1 "jPj + #$L

dispersion relation

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Improved:

E(!k) = ±!

"2"d

j=1 sin2 kj + #2[2"d

j=1(1! cos kj)]2

H0 = !!d

j=1 "jPj + #$L

dispersion relation Algorithm:Let .Start in .Choose constants so that for as small a T as possible, is large.|!!, w|e!iHT |!, s"|2

H = H0 ! !|w"#w||!, s!

!, "

The Dirac equation: Faster search for d = 2, 3, 4Dirac Hamiltonian: ,HDirac =

!dj=1 !jpj + "m

{!j ,!k} = 2"jk , {!j ,#} = 0 , #2 = 1with

pj = i ddxj

Improved:

E(!k) = ±!

"2"d

j=1 sin2 kj + #2[2"d

j=1(1! cos kj)]2

H0 = !!d

j=1 "jPj + #$L

dispersion relation

Run time:O(!

N)O(!

N log N)in d > 2,

in d = 2.

Algorithm:Let .Start in .Choose constants so that for as small a T as possible, is large.|!!, w|e!iHT |!, s"|2

H = H0 ! !|w"#w||!, s!

!, "

d =

2d =

3

d = 4

d = 5

4. Evaluating Boolean formulas

•Childs, Cleve, Jordan, and Yeung, quant-ph/0702160.•Childs, Reichardt, Špalek, and Zhang, quant-ph/0703015.

Evaluating Boolean formulasFix a Boolean formula on N variables.

Given a black box for determining how the variables are assigned, how many variables must we query to determine the value of the formula?

Evaluating Boolean formulasFix a Boolean formula on N variables.

Given a black box for determining how the variables are assigned, how many variables must we query to determine the value of the formula?

Example: Unstructured search (OR)

or

Evaluating Boolean formulasFix a Boolean formula on N variables.

Given a black box for determining how the variables are assigned, how many variables must we query to determine the value of the formula?

Example: Unstructured search (OR)

or

Evaluating Boolean formulasFix a Boolean formula on N variables.

Given a black box for determining how the variables are assigned, how many variables must we query to determine the value of the formula?

Example: Unstructured search (OR)

0 0 0 0 0

or

Evaluating Boolean formulasFix a Boolean formula on N variables.

Given a black box for determining how the variables are assigned, how many variables must we query to determine the value of the formula?

Example: Unstructured search (OR)

0

0 0 0 0 0

or or

0 0 0 1 0

Evaluating Boolean formulasFix a Boolean formula on N variables.

Given a black box for determining how the variables are assigned, how many variables must we query to determine the value of the formula?

Example: Unstructured search (OR)

0

0 0 0 0 0

or or

0 0 0 1 0

Evaluating Boolean formulasFix a Boolean formula on N variables.

Given a black box for determining how the variables are assigned, how many variables must we query to determine the value of the formula?

Example: Unstructured search (OR)

0 1

0 0 0 0 0

or or

0 0 0 1 0

Evaluating Boolean formulasFix a Boolean formula on N variables.

Given a black box for determining how the variables are assigned, how many variables must we query to determine the value of the formula?

Example: Unstructured search (OR)

Classical complexity: !(N)Quantum algorithm [Grover 1996]: O(

!N)

Quantum lower bound [BBBV 1996]: !(!

N)

0 1

0 0 0 0 0

AND-OR trees (aka game trees)and

or or

and and and and

or or or or or or or or

AND-OR trees (aka game trees)and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

AND-OR trees (aka game trees)and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

1 0 1 1 1 1 0 0

AND-OR trees (aka game trees)and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

1 0 1 1 1 1 0 0

0 1 1 0

AND-OR trees (aka game trees)and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

1 0 1 1 1 1 0 0

0 1 1 0

1 1

AND-OR trees (aka game trees)and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

1 0 1 1 1 1 0 0

0 1 1 0

1 1

1

AND-OR trees (aka game trees)

Classical complexity: [Snir 85, Saks-Wigderson 86, Santha 95]!(N0.753)

and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

1 0 1 1 1 1 0 0

0 1 1 0

1 1

1

AND-OR trees (aka game trees)

Classical complexity: [Snir 85, Saks-Wigderson 86, Santha 95]!(N0.753)

and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

1 0 1 1 1 1 0 0

0 1 1 0

1 1

1

Quantum lower bound [Barnum-Saks 02]: !(!

N)

AND-OR trees (aka game trees)

Classical complexity: [Snir 85, Saks-Wigderson 86, Santha 95]!(N0.753)

and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

1 0 1 1 1 1 0 0

0 1 1 0

1 1

1

Quantum lower bound [Barnum-Saks 02]: !(!

N)

Quantum walk algorithm [FGG 07]: query time (in the Hamiltonian oracle model)

O(!

N)

AND-OR trees (aka game trees)

Classical complexity: [Snir 85, Saks-Wigderson 86, Santha 95]!(N0.753)

and

or or

and and and and

or or or or or or or or

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

1 0 1 1 1 1 0 0

0 1 1 0

1 1

1

Quantum lower bound [Barnum-Saks 02]: !(!

N)

Discretized version [Childs-Cleve-Jordan-Yeung 07]: O(!

N1+!)

Quantum walk algorithm [FGG 07]: query time (in the Hamiltonian oracle model)

O(!

N)

Evaluating AND-OR trees by scattering

Evaluating AND-OR trees by scatteringk

Evaluating AND-OR trees by scatteringk

Evaluating AND-OR trees by scattering

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

k

Evaluating AND-OR trees by scattering

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

k

Evaluating AND-OR trees by scattering

0 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0

k

Claim: For small k, the transmission coefficient is large if the formula (translated into NAND gates) evaluates to 0, and small if it evaluates to 1.

General formulasnand

nand

nandnand

nand nand nand

nandnand

nand nand

General formulasnand

nand

nandnand

nand nand nand

nandnand

nand nand

Quantum lower bound [Barnum-Saks 02]: !(!

N)

General formulasnand

nand

nandnand

nand nand nand

nandnand

nand nand

Quantum lower bound [Barnum-Saks 02]: !(!

N)

Quantum algorithm [Childs-Reichardt-Špalek-Zhang 07]: O(!

N1+!)

General formulas

Main idea: The quantum walk on the (expanded, appropriately weighted) tree has a zero eigenvalue if the formula evaluates to 1, and a smallest eigenvalue if it evaluates to 0. Use phase estimation.O( 1!

N)

nand

nand

nandnand

nand nand nand

nandnand

nand nand

Quantum lower bound [Barnum-Saks 02]: !(!

N)

Quantum algorithm [Childs-Reichardt-Špalek-Zhang 07]: O(!

N1+!)

Summary and outlook• Physics ↔ Information

• Quantum systems can encode and process information in a fundamentally non-classical way.

• While we have many examples of this phenomenon, we are far from having a general understanding of the information processing power of quantum mechanics.

• In particular, we would like to develop new ways of exploiting quantum interference to perform information processing tasks.