generalized deutsch algorithms ipqi 5 jan 2010. background basic aim : efficient determination of...
TRANSCRIPT
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Generalized Deutsch Algorithms IPQI 5 Jan 2010
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Background
Basic aim : Efficient determination of properties of functions.
Efficiency: No complete evaluation of the function itself.
Tools: Quantum Circuits.
Principles: Superposition, Entanglement
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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
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Deutsch contd…..
Is There a Quantum Method?Answer: determine f(0) f(1)
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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.
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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
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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)
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Experimental Implementation: IISc group
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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 ?
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The Circuit
Single Measurement/query does the job
H
| 0 n
|1
| |
| | ( )
x y
x y f x
nH nH
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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
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AIM :Generalization to more general functions
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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.
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Approach contd…
Constructive approach : Circuit is designed.
Allows the study of the relationship between quantum circuits and properties of functions,
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Larger Aim
Theory of Quantum Circiits
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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
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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
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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
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| |
| | ( )
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
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(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:…
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| |
| | ( )
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…
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Important Point
Invariants within each category are in the parentheses.
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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>
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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
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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
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Questions
Is a further generalization Possible? Note: We still have 240 functions untouched Is it possible to understand why the above
circuit works?
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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
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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
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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
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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.
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Claim
within each coset, we can again distinguish four categories consisting of 4 functions each.
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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
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Modification of the circuit: Introduce
| |
| | ( )
x y
x y f x
2| 0
2| 0 2H
2H 2H
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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
=
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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
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Cost of the new gate
Depends on the “entanglement” in the state!
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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
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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
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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).
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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
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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)
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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
= =
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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
= =
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Thank You
Collaborators :
Vipul Ambasht
Pronoy Sircar
Sunil Yeshwanth