Generalized Deutsch Algorithms IPQI 5 Jan 2010

Background

Basic aim : Efficient determination of properties of functions.

Efficiency: No complete evaluation of the function itself.

Tools: Quantum Circuits.

Principles: Superposition, Entanglement

Example: Deutsch AlgorithmExample: Deutsch Algorithm

Let f : {0,1} → {0,1} f

There are four possibilities:

x f1(x)

0

1

0

0

x f2(x)

0

1

1

1

x f3(x)

0

1

0

1

x f4(x)

0

1

1

0

Goal: Distinguish Constant from Non-constant Any classical method requires two queries

Deutsch contd…..

Is There a Quantum Method?Answer: determine f(0) f(1)

Quantum Oracle

Function black box or Oracle:- Unitary operation implementing unknown function

After applying a series of Gates and the Oracle, we “measure” the final state of the qubit in a suitable basis.

Membership to sets of functions with orthogonal final states determinable by measurement.

Deutsch Algorithm

The final state is |0> for constant and |1> for non-constant Operate |0><0|-|1><1| : one measurement distinguishes

between constant and non constant functions.

|0>

|1>

| | | | ( )x y x y f x

H

H|0>-|1>

|0>+|1>H

Note: Vectors vis-à-vis RaysNote: Vectors vis-à-vis Rays

x f1(x)

0

1

0

0

x f2(x)

0

1

1

1

x f3(x)

0

1

0

1

x f4(x)

0

1

1

0

(0 + 1)

(0 – 1)

Experimental Implementation: IISc group

Generalization: Deutsch Jozsa algorithm

F:{0,..,2n-1} → {0,1} Strong Restriction Further Restrictions: Either balanced or constant Recall: balanced functions send half the domain

points to 0 and the other half to 1 Question Posed : constant or balanced ?

The Circuit

Single Measurement/query does the job

H

| 0 n

|1

| |

| | ( )

x y

x y f x

nH nH

Deutsch Jozsa algorithm contd

Constant : outcome is |0> with probability 1 Balanced : outcome is non |0> with probability 1 Classical algorithm requires minimum of 2 and

maximum of 2n-1+1 measurements

H

| 0 n

|1

| |

| | ( )

x y

x y f x

nH nH

AIM :Generalization to more general functions

Approach to generalization

Question posed must be ‘non-trivial’

Functions must have symmetries that can be exploited.

Generalize Deutsch-Josza : Include a larger range and hence a larger class of functions.

Approach contd…

Constructive approach : Circuit is designed.

Allows the study of the relationship between quantum circuits and properties of functions,

Larger Aim

Theory of Quantum Circiits

Illustration: The 2-qubit case 256 such functions

4 functions in each category obtained by uniform translation

:{0,1, 2,3} {0,1, 2,3}f

0

1

2

3

0 1 2 3 0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP

Circuit: Ansatz

Category |0>,|1>,|2> or |3>. Deutsch Jozsa is a subset = identity, for the particular classification being

attempted

| |

| | ( )

x y

x y f x

2| 0

2| 0 2H

2H 2H

| |

| | ( )

x y

x y f x

2| 0

2| 0 2H

2H 2H

Circuit Continued…

20

4

21

4

22

4

22

4

0 0 01 0 0 0

0 0 0 0 0 0

0 0 1 00 0 0

0 0 0

0 0 0

i

i

i

i

e

e i

ei

e

3

Can be written as a product of single qubit gates

(0) (1) (2) (3)

(| 0 |1 | 2 | 3 )(| 0 |1 | 2 | 3 )

(( ) | 0 ( ) |1 ( ) | 2 ( ) | 3 ) |f f f f

i i

i i i i

Applying the oracle gives :

| |

| | ( )

x y

x y f x

2| 0

2| 0 2H

2H 2H

Circuit Explanation:…

| |

| | ( )

x y

x y f x

2| 0

2| 0 2H

2H 2H

The final state is :(0) (1) (2) (3)

(0) (1) (2) (3)

(0) (1) (2) (3)

(0) (1) (2) (3)

| 00 {( ) ( ) ( ) ( ) }

| 01 {( ) ( ) ( ) ( ) }

|10 {( ) ( ) ( ) ( ) }

|11 {( ) ( ) ( ) ( ) }

f f f f

f f f f

f f f f

f f f f

i i i i

i i i i

i i i i

i i i i

Circuit Continued…

Important Point

Invariants within each category are in the parentheses.

Basic requirement : functions from different categories produce orthogonal final states

i.e. : if f belongs to the i-th category, |i> is obtained with probability 1 on measurement.

Eg : constant functions is the 0-th category

CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

|00> |11> |01> |10>

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP

Characteristics of categories: Functions in a given category give the same ray

as the output. Such functions cannot be distinguished by the

circuit. These indistinguishable functions form a category

(0) (1) (2) (3)| {( ) | 0 ( ) |1 ( ) | 2 ( ) | 3 }

( ) ( ) ,

| ( ) |

f f f ff

kg f

i i i i

g x f x k x

i

Questions

Is a further generalization Possible? Note: We still have 240 functions untouched Is it possible to understand why the above

circuit works?

A unique Unitary transformation U(f) : action of the oracle on |x>|y>, corresponding to a function f.

{U(fi)} corresponding to the 256 functions form an abelian group, with composition as the group operation.

.

( ) | | | | ( )

( ) ( ) | | | | ( ) ( )i i

j i i j

U f x y x y f x

U f U f x y x y f x f x

The Underlying Group Structure

For a given f which has f(0)=F0, f(1)=F1, f(2)=F2, f(3)=F3, U(f) is given by a 16 x 16 matrix

0

1

2

3

0 0 0

0 0 0

0 0 0

0 0 0

F

F

F

F

A

A

A

A

0 1 2 3

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

0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0; ; ;

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

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

A A A A

0 1 2 3( ) ( , , , )U f U F F F F

Explicit form of U

Note: A A = A

a b a b

Basic idea behind the generalization Cosets of the subgroup consisting of U(fi

’) Left and right cosets equivalent since the

group is abelian 16 cosets of order 16 each

Cosets Each coset is “labelled” by the set {ki}

The 16 cosets {ki} required to exhaust all 256 group elements are as shown

{0,0,0,0} {1,0,0,0} {2,0,0,0} {3,0,0,0}{0,1,1,0} {0,1,0,0} {1,0,3,0} {0,3,0,0}{0,0,1,1} {0,0,1,0} {1,3,0,0} {0,0,3,0}{0,1,0,1} {0,0,0,1} {1,0,0,3} {0,0,0,3}

Note that within each coset, we can again distinguish between categories consisting of 4 functions each, as we shall now see.

Claim

within each coset, we can again distinguish four categories consisting of 4 functions each.

An Example: Coset generation and its labeling: {0,1,1,0}

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

CONSTANT PLATEAU/BASIN SAWTOOTH EVEN STEP

{0,1,1,0}

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

0

1

2

3

0 1 2 3

Modification of the circuit: Introduce

| |

| | ( )

x y

x y f x

2| 0

2| 0 2H

2H 2H

The structure of the Gate :

0

1

2

3

4

4

4

4

0 0 0

0 0 0

0 0 0

0 0 0

k

k

k

k

i

i

i

i

=

Example

|00> |11> |01> |10>

0000→0110 0220 → 0330 0202 → 0312 0022 → 0132

1111 → 1221 1331 → 1001 1313 → 1023 1133 → 1203

2222 → 2332 2002 → 2112 2020 → 2130 2200 → 2310

3333 → 3003 3113 → 3223 3131 → 3201 3311 → 3021

Cost of the new gate

Depends on the “entanglement” in the state!

Entanglement ?

Preparation of an entangled state is not required at any step in the Deutsch/Deutsch-Jozsa.

It is also not required if we were to consider only those 16 functions corresponding to the subgroup

However, the state shown, which is necessary for categorization of a coset, may be entangled for certain cosets.

0 31 24 44 4{| ( ) | 0 ( ) |1 ( ) | 2 ( ) | 3 }k kk ki i i i |

| | {| 0 |1 | 2 | 3 }{| 0 |1 | 2 | 3 }

{| 0 |1 }{| 0 |1 }{| 0 |1 }{| 0 |1 }I i i

i

is a measure of entanglement for :

Entanglement of the initial state

{0,0,0,0} {1,0,0,0} {2,0,0,0} {3,0,0,0}

{0,1,1,0}{0,1,0,0} {1,0,30} {0,3,0,0}

{0,0,1,1} {0,0,1,0} {1,3,0,0 }{0,0,3,0}

{0,1,0,1} {0,0,0,1} {1,0,0,3} {0,0,0,3}

(k0 k1 k2 k3 for each coset)

Partially entangled (E = 0.707)

Entangled (E = 1)

Not entangled (E = 0)

| 0 |1 | 2 | 3a b c d | |

2

ad bc

0 31 24 44 4{| ( ) | 0 ( ) |1 ( ) | 2 ( ) | 3 }k kk ki i i i

Entanglement of initial state Identify a global property of the functions that is

invariant within each coset

Measure of entanglement explicitly for the initial state.

sin(π(k1 + k2 – k3 – k0)/4).

Entanglement of initial state In the subgroup f(1)+f(2)-f(0)-f(3) = 0, f(1)+f(2)-f(0)-f(3) = 0, we do not need an

entangled state to categorize this coset.

f(1)+f(2)-f(0)-f(3) = 1 or 3, we need a partially entangled state

If f(1)+f(2)-f(0)-f(3) = 2, then we need a maximally entangled state

Points to note

The measure of entanglement which we have used has no non-trivial generalization to multipartite systems

In such cases, entropy of reduced density matrices is an indicator of entanglement

(In this particular case (bipartite), Entropy is a monotonic function of the measure used)

Generalization to n Qubits

| |

| | ( )

x y

x y f x

| 0 n

| 0 n

nH

nH

nH

2

2{ }

0...2 1

n

ij

n

diag e

j

2( )

2{ }

0...2 1

jn

ik

n

diag e

j

= =

Generalization to n Qubits There are NN functions. ( N = 2n ) Subgroup – N2 elements NN-2 cosets

2

2{ }

0...2 1

n

ij

n

diag e

j

2( )

2{ }

0...2 1

jn

ik

n

diag e

j

= =

Thank You

Collaborators :

Vipul Ambasht

Pronoy Sircar

Sunil Yeshwanth