[PDF] - Hidden Subgroup Hidden Subgroup Def. A Map is PDF Document

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Samuel J. Lomonaco, Jr.

Dept. of Comp. Sci. & Electrical Engineering

University of Maryland Baltimore County Baltimore, MD 21250 Email: Lomonaco@UMBC.EDU WebPage: http://www.csee.umbc.edu/~lomonaco

Continuous Quantum Continuous Quantum Hidden Subgroup Hidden Subgroup Algorithms Algorithms

Defense Advanced Research Projects Agency (DARPA) & Defense Advanced Research Projects Agency (DARPA) & Air Force Research Laboratory, Air Force Materiel Command, USAF Air Force Research Laboratory, Air Force Materiel Command, USAF Agreement Number F30602 Agreement Number F30602-

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01-

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2-

0522

0522

This work is in collaboration with This work is in collaboration with Louis H. Kauffman Louis H. Kauffman

The Defense Advance Research Projects

Agency (DARPA) & Air Force Research Laboratory (AFRL), Air Force Materiel Command, USAF Agreement Number F30602-01-2-0522.

The National Institute for Standards

and Technology (NIST)

The Mathematical Sciences Research

Institute (MSRI).

The L-O-O-P Fund.

L L-

O

O-

O

O-

P

P

This work is supported by: This work is supported by:

Lomonaco & Kauffman,

Lomonaco & Kauffman, Continuous Continuous Quantum Hidden Subgroup Algorithms, Quantum Hidden Subgroup Algorithms, http://xxx.lanl.gov/abs/quant http://xxx.lanl.gov/abs/quant-

ph/0304084

ph/0304084

Kauffman &

Kauffman & Lomonaco, Lomonaco, Entanglement Entanglement Criteria Criteria – – Quantum and Topological Quantum and Topological, , http://xxx.lanl.gov/abs/quant http://xxx.lanl.gov/abs/quant-

ph/0304091

ph/0304091

Hidden Subgroup Hidden Subgroup Algorithms Algorithms

A subgroup of , and

A subgroup of , and

An injection

An injection

s. t. the diagram
s. t. the diagram

is commutative. is commutative.

/ A S A K

ϕ ϕ

ν ι → ↑

Hidden

Hidden Subgroup Subgroup Structure Structure

Def.

Def. A Map is said to have

A Map is said to have hidden hidden subgroup subgroup structure structure if there exist if there exist

: A S ϕ →

Kϕ

A

: / A K S

ϕ ϕ ϕ ϕ

ι → ϕ

Ambient Ambient Group Group Target Target Set Set Hidden Hidden Subgroup Subgroup Set of Right Set of Right Cosets Cosets Hidden Natural Hidden Natural Surjection Surjection

SLIDE 2

2

/ A S A K

ϕ ϕ

ν ι → ↑

Hidden Quotient

Hidden Quotient Group Group

Hidden Hidden Subgroup Subgroup Structure Structure (Cont.) (Cont.)

ϕ

If is an If is an invariant invariant subgroup subgroup of , then

f , then

is a group, and is an is a group, and is an epimorphism epimorphism

Kϕ A / H A K

ϕ ϕ ϕ ϕ

= : / A A Kϕ ν →

Hidden Hidden Epimorphism Epimorphism

Kitaev Kitaev observed that finding the period

bserved that finding the period

is equivalent to finding the subgroup , is equivalent to finding the subgroup , i.e., the kernel of . i.e., the kernel of .

P ⊂ Z Z mod mod

n

N n a N

ϕ

  →

Z

Z P

ϕ

Shor Shor’ ’s s Quantum factoring algorithm Quantum factoring algorithm reduces the task of factoring an integer reduces the task of factoring an integer to the task of finding the period to the task of finding the period

f a function
f a function

P N

Origin of QHS Algorithms Origin of QHS Algorithms

Three Methods for Three Methods for Creating New Quantum Creating New Quantum Algorithms Algorithms

Two Ways to Create New Quantum Algorithms Two Ways to Create New Quantum Algorithms Given Given

: A S ϕ →

Push Push Lift Lift

ι

ϕ

η L

Lifted Lifted Gp Gp

ν H

ϕ

ϕ ϕ ϕ ι =

Approx

Approx Gp Gp

S

Amb Amb. . Gp Gp

ϕ

Target Set Target Set

A

Lifting and Pushing Lifting and Pushing A 3rd Way to Create New Quantum Algorithms A 3rd Way to Create New Quantum Algorithms Duality Duality

A S →

ϕ

Amb Amb. . Gp Gp

A

S ′ →

Φ

Dual Dual Gp Gp

Dual Dual QHS QHS Alg Alg QHS QHS Alg Alg Summary Summary 3 Ways to create New Quantum Algorithms 3 Ways to create New Quantum Algorithms

Lifting

Lifting

Pushing

Pushing

Duality

Duality

SLIDE 3

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Some Past Algorithms Some Past Algorithms

Hidden Subgroup Algorithms Hidden Subgroup Algorithms

Lomonaco & Kauffman,

Lomonaco & Kauffman, A Continuous A Continuous Variable Variable Shor Shor Algorithm Algorithm, , http://xxx.lanl.gov/abs/quant http://xxx.lanl.gov/abs/quant-

ph/0210141

ph/0210141

Lomonaco & Kauffman,

Lomonaco & Kauffman, Quantum Hidden Quantum Hidden Subgroup Algorithms: Subgroup Algorithms: A Mathematical A Mathematical Perspective, Perspective, AMS, CONM/305, (2002). AMS, CONM/305, (2002). http://xxx.lanl.gov/abs/quant http://xxx.lanl.gov/abs/quant-

ph/0201095

ph/0201095

Wandering

Wandering Shor Shor

Continuous

Continuous Shor Shor

Wandering Wandering Shor Shor

Q

S ι ν → ↑↓ ↑↓

ϕ

ϕ ϕ ϕ ι =

ϕ

Free Abel Free Abel Finite Finite Rk Rk Amb Amb GP GP Approx Approx Gp Gp Shor Shor Transv Transv Approx Approx Map Map Target Target Set Set

Push Push

A ⊕ ⊕ ⊕ ⊕

⊕

⊕ ⊕ ⊕

⊕

⊕ ⊕ ⊕

⊕

⊕ ⊕ ⊕

⊕

⊕ ⊕ ⊕

⊕

⊕ ⊕ ⊕

Continuous

Continuous Shor Shor

A S →

ϕ

Ambient Group Ambient Group Key Idea: Key Idea:

f discrete algorithms to
f discrete algorithms to

a continuous groups a continuous groups

S →

Lifting

Lifting Add.

Add. Gp

Gp of

f Reals

Reals ? Quantum algorithm for ? Quantum algorithm for the Jones polynomial the Jones polynomial

A highly speculative quantum algorithm for

A highly speculative quantum algorithm for Three Recent QHS Algorithms Three Recent QHS Algorithms

A quantum algorithm on the

A quantum algorithm on the

⇒

A quantum algorithm to

A quantum algorithm to Shor Shor’ ’s s algorithm algorithm Circle Circle dual dual functional integrals functional integrals

Road Map Road Map

Shor Shor’ ’s s Alg Alg QHS QHS Algs Algs for for Functional Functional Integrals Integrals Pushing Pushing Dual of Dual of Shor Shor’ ’s s Alg Alg QHS QHS Alg Alg on

n

/

Duality Duality QHS QHS Alg Alg on

n
Lifting

Lifting

S

ϕ

Q

/

Q

ϕ

S

ϕ

~

ϕ

~

Lift of Lift of Shor Shor Algorithm Algorithm Shor Shor Algorithm Algorithm Dual Lifted Dual Lifted Algorithm Algorithm Dual Dual Shor Shor Algorithm Algorithm Dual Dual

Lifting Lifting & & Duality Duality

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4

A Lifting of A Lifting of Shor Shor’ ’s s Quantum Factoring Quantum Factoring Algorithm to Algorithm to Integers Integers

Fourier Analysis Fourier Analysis

n the
n the

Circle Circle

A Momentary Digression A Momentary Digression

The Circle as a Group The Circle as a Group

The The circle circle group group can be viewed as can be viewed as

A

A multiplicative multiplicative group group, i.e., as the unit , i.e., as the unit circle in the complex plane circle in the complex plane

{ }

2

:

ix

e x

π

∈

( ) ( )

2 2 2 i x y ix iy

e e e

π π π π π +

= i

where denotes the additive group of where denotes the additive group of reals reals. .

The Circle as a Group

The Circle as a Group

The The circle group circle group can can also

also be viewed as

be viewed as

An

An additive additive group group, i.e., as , i.e., as where denotes the additive group of where denotes the additive group of integers. integers.

/

mod1 reals = mod 1 x y + The Character Group The Character Group

The The character group character group

f an
f an abelian

abelian group group is defined as is defined as

(

) ( )

, A Hom A Circle =

{ }

: : A Circle a morphism χ χ χ χ = → = →

with group operation (in multiplicative notation), with group operation (in multiplicative notation),

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2

a a a χ χ χ χ χ χ = i i

r (in additive notation) as
r (in additive notation) as

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

1 2 1 2

a a a χ χ χ χ χ χ + = + = +

A

A

The Character Groups of The Character Groups of and and

The

The character group character group of is

f is
The

The character group character group of is

f is

/

{

}

2

: : /

inx x n

e x

π

χ = ∈ = ∈ =

/
{

}

{ } { }

2

/ : : : mod 1:

inx n n

x e n x nx n

π

χ χ ≅ ∈ ≅ ∈ ≅ ∈ ≅ ∈ =

/

⇔

SLIDE 5

5

Fourier Analysis on the Circle Fourier Analysis on the Circle

/

The The Fourier transform Fourier transform of

f

is defined as the map is defined as the map given by given by The The inverse Fourier transform inverse Fourier transform is defined as is defined as

: / f →

:

f →

2

( ) ( )

inx

f n dxe f x

π −

= ∫

2

( ) ( )

inx n

f x e f n

π ∈

= ∑

( )

( )

1

1 P

P n

n x x P P δ δ δ δ

− =

  = − = −    

∑

Dirac

Dirac Delta function on Delta function on

( ) ( )

x δ

/

For a non

For a non

z

ero integer, we will z ero integer, we will also need on the generalized also need on the generalized function function

P

/

Needed Mathematical Machinery Needed Mathematical Machinery

The elements of are formal integrals

The elements of are formal integrals

f the form
f the form
denotes the rigged Hilbert space

denotes the rigged Hilbert space

n with
n with orthonormal
rthonormal basis

basis , i.e., , i.e.,

/

H

( ) ( )

dx f x x

∫

/

H

{ }

: / x x∈

( ) ( )

x y x y δ = − = − /

Rigged Hilbert Space Rigged Hilbert Space Finally, let denote the space of formal Finally, let denote the space of formal sums sums with with orthonormal

rthonormal basis

basis

H

:

n n n

a n a n

∞ =− =−∞

  ∈ ∀ ∈ ∀ ∈    

∑

{

}

: n n∈

A Lifting of A Lifting of Shor Shor’ ’s s Quantum Factoring Quantum Factoring Algorithm to Algorithm to Integers Integers

S

ϕ

Q

ϕ

~

Lift of Lift of Shor Shor Algorithm Algorithm Shor Shor Algorithm Algorithm

Lifting Lifting & & Duality Duality

SLIDE 6

6

Let be periodic function with hidden minimum period .

Ob Objective: Find

: ϕ →

P

P

Periodic Functions on Periodic Functions on

Step 0.

Step 0. Initialize

Step 1.

Step 1. Apply

Step 2.

Step 2. Apply

/

0 0 ψ = ∈ = ∈ ⊗ H

H

2 1 in n n

e n n

π

ψ

∈ ∈ ∈ ∈

= = = = ∈ ⊗

∑ ∑

H H H H

i

1
1 ⊗

F : ( ) U n u n u n

ϕ

ϕ +

2

( )

n

n n ψ ϕ ψ ϕ

∈

= ∑

Step 3.

Step 3. Apply

1 ⊗ F

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1

2 3 / 1 2 1 1 2 2 1 2 1 1 2 1

1

inx n P i n P n x n n P in x in Px n n P in x P n P P in x n n P n

dx x e n dx x e n P n dx x e e n dx x x e n n e n P P n n P P

π π π π π π

ψ ϕ ψ ϕ ϕ ϕ δ ϕ δ ϕ ϕ

− ∈ − − + − + ∈ = ∈ = − − − ∈ = ∈ = − − = − − − − − = = = = − =

= ∈ = ∈ ⊗ = + = +   =     =   =       = Ω = Ω   

∑ ∫ ∑ ∑ ∫ ∑ ∑ ∫ ∑ ∫ ∑ ∑ ∑

H H H H

Step 4.

Step 4. Measure with respect to the observable to produce a random eigenvalue and then proceed to find the corresponding using the continued fraction recursion. (We assume )

1 3 P n

n n P P ψ

− =

  = Ω = Ω   

∑

Qy dy y y Q     = ∫

O

/ m Q / n P

2

2 Q P ≥

The The Actual Actual Shor Shor Algorithm Algorithm Un Un-

Lifted

Lifted

The Actual (Un The Actual (Un-

Lifted)

Lifted) Shor Shor Algorithm Algorithm Make the following approximations by selecting Make the following approximations by selecting a sufficiently large integer : a sufficiently large integer :

Q

is only approximately periodic ! is only approximately periodic !

ϕ

{ }

:0

Q

k k P ≈ = ≈ = ∈ ≤ <

/

mod 1: 0,1, , 1

Q

r r Q Q   ≈ = ≈ = = −    

…
:

:

Q

ϕ ϕ ϕ ϕ → ≈ → ≈ →

SLIDE 7

7

Run the algorithm in Run the algorithm in and measure the observable and measure the observable

Q

S

⊗

H

H

1 Q r

r r r Q Q Q

− =

= ∑ O

A Quantum Hidden A Quantum Hidden Subgroup Algorithm Subgroup Algorithm

n the
n the

Circle Circle

The Dual Algorithm The Dual Algorithm

n the
n the

Circle Circle

S

ϕ

Q

/

ϕ

S

ϕ

~

Lift of Lift of Shor Shor Algorithm Algorithm Shor Shor Algorithm Algorithm Dual Lifted Dual Lifted Algorithm Algorithm Dual Dual

Lifting Lifting & & Duality Duality

The elements of are formal

The elements of are formal integrals of the form integrals of the form

denotes the rigged Hilbert space

denotes the rigged Hilbert space

n with
n with orthonormal
rthonormal basis

basis , i.e., , i.e.,

/

H

/

H

{ }

: / x x∈

( ) ( )

x y x y δ = − = −

( ) ( )

dx f x x

∫

/

Rigged Hilbert Space Rigged Hilbert Space Finally, let denote the space of formal Finally, let denote the space of formal sums sums with with orthonormal

rthonormal basis

basis

H

:

n n n

a n a n

∞ =− =−∞

  ∈ ∀ ∈ ∀ ∈    

∑

{

}

: n n∈

SLIDE 8

8

Let be an admissible periodic function of minimum rational period Proposition: Let (with ) be a period of . Then is also a period of .

Remark: Hence, the minimum rational period is always the reciprocal of an integer modulo 1 .

: / f →

/

α ∈

2

1/a f f

Periodic Admissible Functions on Periodic Admissible Functions on

/

1 2

/ a a α =

( ) ( )

1 2

gcd , 1 a a =

Step 0.

Step 0. Initialize

Step 1.

Step 1. Apply

Step 2.

Step 2. Apply

0 0 ψ = ∈ = ∈ ⊗

H

H H H 1

1 ⊗

F

2 1 / ix

dxe x dx x

π

ψ = = = = ∈ ⊗

∫ ∫

i

H

H

: ( ) U x u x u x

ϕ

ϕ +

2

( ) dx x x ψ ϕ ψ ϕ = ∫

Step 3.

Step 3. Apply

1 ⊗ F

( ) ( ) ( ) ( )

2 3 2 inx n inx n

dxe n x n dxe x

π π

ψ ϕ ψ ϕ ϕ

− ∈ − ∈

= = ∈ = ∈ ⊗

∑∫ ∑ ∫

H

H

Letting , we have

m

m x x a = − = −

( ) ( ) ( ) ( ) ( ) ( )

1 1 2 2 1 1 2 0 0 1 2 1 2

m

m a a inx inx m m a m a a in x a m m m inm a a inx a m

dx e x dx e x m dx e x a e dx e x

π π π π π π π

ϕ ϕ ϕ ϕ ϕ ϕ

+ − − − − − =   − − + − +     = − − − =

=   = + = +       =    

∑ ∫ ∫ ∑∫ ∑ ∫

But

But Thus, Thus,

2 1 0mod

0mo f d i

inm a a n a m

a

therw s

e a i n e a

π

δ

− − = =

=  = = = =  

∑

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 3 1/ 2 0mod 1/ 2 inx n a inx n a n a i ax

n dx e x n dx e x a dx e x a a

π π π

ψ ϕ ψ ϕ δ ϕ δ ϕ ϕ

− ∈ − = ∈ − ∈ ∈

= =   =     = Ω = Ω

∑ ∫ ∑ ∫ ∑ ∫ ∑

Step 4.

Step 4. Measure with respect to the observable to produce a random eigenvalue

( ) ( )

3

a a ψ

∈

= Ω = Ω

∑

n

n n n

∈

= ∑

O

a

SLIDE 9

9

The The corresponding corresponding algorithm algorithm discrete discrete The Algorithmic Dual The Algorithmic Dual

f
f

Shor Shor’ ’s s Quantum Quantum Factoring Algorithm Factoring Algorithm

S

ϕ

Q

/

Q

ϕ

S

ϕ

~

ϕ

~

Lift of Lift of Shor Shor Algorithm Algorithm Shor Shor Algorithm Algorithm Dual Lifted Dual Lifted Algorithm Algorithm Dual Dual Shor Shor Algorithm Algorithm Dual Dual

Lifting Lifting & & Duality Duality

is only approximately periodic ! is only approximately periodic !

We now create a corresponding We now create a corresponding discrete algorithm discrete algorithm The approximations are: The approximations are:

:

:

Q

ϕ ϕ ϕ ϕ → ≈ → ≈ →

/

mod 1: 0,1, , 1

Q

r r Q Q   ≈ = ≈ = = −    

…

{ }

:0

Q

k k P ≈ = ≈ = ∈ ≤ <

ϕ

Run the algorithm in Run the algorithm in and measure the observable and measure the observable

Q

S

⊗

H

H

1 Q k

k k k

− =

= ∑ O Quantum Algorithms based on Quantum Algorithms based on Feynman Functional integrals Feynman Functional integrals

The following algorithm is The following algorithm is highly speculative highly speculative. . In the spirit of Feynman, the following In the spirit of Feynman, the following quantum algorithm is quantum algorithm is based on functional based on functional integrals whose existence is difficult to integrals whose existence is difficult to determine determine, let alone approximate. , let alone approximate.

Caveat Caveat Emptor

Emptor

SLIDE 10

10

The Space The Space Paths Paths Paths Paths = all continuous paths = all continuous paths which are with respect to the inner which are with respect to the inner product product Paths Paths is a vector space over with is a vector space over with respect to respect to

[ ] [ ]

: 0,1

n

x →

2

L

1

( ) ( ) x y ds x s y s = ∫ i

(

) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) x s x s x y s x s y s λ λ λ λ =     + = + = +  The Problem to be Solved The Problem to be Solved

Let be a functional with a Let be a functional with a hidden subspace of such that hidden subspace of such that

: Paths ϕ → V Paths

( ) ( ) ( )

x v x v V ϕ ϕ ϕ ϕ + = + = ∀ ∈

Objective. Create a quantum algorithm

Create a quantum algorithm that finds the hidden subspace . that finds the hidden subspace .

V

The Ambient Rigged Hilbert Space The Ambient Rigged Hilbert Space

Let be the rigged Hilbert space with Let be the rigged Hilbert space with

rthonormal
rthonormal basis ,

basis , and with bracket product and with bracket product

Paths

H { }

: x x Paths ∈

( ) ( )

| x y x y δ = − = − Parenthetical Remark

Please note that can be written as the Please note that can be written as the following disjoint union: following disjoint union:

( ) ( )

v V

Paths v V ⊥

∈

= + = +

∪

Paths

Step 0.

Step 0. Initialize Initialize

Step 1.

Step 1. Apply Apply

Step 2.

Step 2. Apply Apply

0 0

Paths

ψ = ∈ = ∈ ⊗ H H H H 1

1 ⊗

F

2 1 ix Paths Paths

x e x x x

π

ψ = = = =

∫ ∫

i

D D

: ( ) U x u x u x

ϕ

ϕ +

2

( )

Paths

x x x ψ ϕ ψ ϕ = ∫ D

Step 3. Apply

1 ⊗ F

( ) ( ) ( ) ( )

2 3 2 ix y Paths Paths ix y Paths Paths

y x e y x y y x e x

π π

ψ ϕ ψ ϕ ϕ

− −

= =

∫ ∫ ∫ ∫

i i

D D D D

SLIDE 11

11

But But

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

2 2 2 2 2 ix y ix y Paths V v V i v x y V V iv y ix y V V

xe x v xe x v xe v x ve xe x

π π π π π π π π π

ϕ ϕ ϕ ϕ ϕ ϕ

⊥ ⊥ ⊥

− − − − + − + − + − − − −

= = + = + =

∫ ∫ ∫ ∫ ∫ ∫ ∫

i i i i i

D D D D D D D

However, However, So, So,

( ) ( )

2 iv y V V

ve u y u

π

δ

⊥

−

= − = −

∫ ∫

i

D D

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

2 2 3 2 2

n n

iv y ix y Paths V V ix y Paths V V ix u V V V

y y v e x e x y y u y u x e x u u x e x u u u

π π π π π π

ψ ϕ ψ ϕ δ ϕ δ ϕ ϕ

⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥

− − − − − −

= = − = − = = Ω = Ω

∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

i i i i

D D D D D D D D D

Step 4.

Step 4. Measure Measure with respect to the observable with respect to the observable to produce a random element of to produce a random element of

( ) ( )

3 V

u u u ψ

⊥

= Ω = Ω

∫ D

Paths

A w w w w = ∫ D

V ⊥

Can the above path integral quantum algorithm Can the above path integral quantum algorithm be modified in such a way as to create a be modified in such a way as to create a quantum algorithm for the Jones polynomial ? quantum algorithm for the Jones polynomial ? I.e., can it be modified by replacing I.e., can it be modified by replacing by the by the space of gauge connections space of gauge connections, and by , and by making suitable modifications? making suitable modifications?

Question Question

Paths (

) ( ) ( ) ( )

K

K A A A ψ ψ ψ ψ = ∫ D W

where is the where is the Wilson loop Wilson loop

( ) ( )

( ) ( ) ( ) ( )

exp

K K

A tr P A =

∫

W

( ) ( )

K A

W

Quantum Computation: Quantum Computation: A Grand Mathematical Challenge A Grand Mathematical Challenge for the Twenty for the Twenty-

First Century and the Millennium,

First Century and the Millennium, Samuel J. Lomonaco, Jr. Samuel J. Lomonaco, Jr. (editor), (editor), AMS PSAPM/58, AMS PSAPM/58, (2002). (2002).

SLIDE 12

12

Quantum Computation and Information Quantum Computation and Information, , Samuel J. Samuel J. Lomonaco, Jr. and Howard E. Brandt Lomonaco, Jr. and Howard E. Brandt (editors), (editors), AMS AMS CONM/305, (2002). CONM/305, (2002).