Multipartite Entanglement and its Role in Quantum Algorithms

Special Seminar:Ph.D. Lecture

byYishai Shimoni

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Acknowledgement

This work was carried out under the supervision of Prof. Ofer Biham

&In collaboration with Dr. Daniel Shapira

cam.qubit.org

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Outline

• Quantum computation• Quantum entanglement• The Groverian measure of entanglement• Grover’s algorithm• Entanglement in Grover’s algorithm• Shor’s algorithm• Entanglement in Shor’s algorithm• Conclusion

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Quantum Computation

• Uses quantum bits and registers

• A function operator applied to the register can compute all possible values of the function

• Does this lead to exponential speed-up? Only one output can be read Using superposition this speed-up can be achieved

n

ii ia

2

0

||

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Quantum Computation

• Several quantum algorithms show speed-up over classical algorithms: Grover’s search algorithm – square root Shor’s factoring algorithm – exponential (?) Simulating quantum systems – exponential

• Any quantum algorithm can be efficiently simulated on a classical computer if it does not create entanglement

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Quantum Entanglement

• Correlations in the measurement outcome of different parts of the system

• A state is un-entangled if and only if it cannot be written as a tensor product

• Depends on partitioning, for example

but only this partitioning gives a tensor product

0|11|00|2

1110|000|2

1

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Quantum Entanglement

Requirements of entanglement measures:

1. Vanishes only for tensor product states2. Invariant to local (in party) unitary operations3. Cannot increase using local operation and

classical communication (LOCC)

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Quantum Entanglement

• Bipartite entanglement connected to entropy and information

• Resource for teleportation and communication protocols

• Not much known about multipartite entanglement

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Quantum Entanglement

www.jpl.nasa.gov

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Groverian Entanglement• A quantum algorithm with

well defined initial and final quantum states

• Using an arbitrary initial state, the probability of success of the algorithm

• Any algorithm can be described as starting from a tensor product state

22 ||| iAfP

fiA || || iAf

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Groverian Entanglement

• Allow local unitary operators to get the maximal probability of success

• Local unitary operators on a product state leave it as a product state

21max ||max

1 nUU UUiP

n

2|max |max tP Tt

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Groverian Entanglement

Phys Rev A 74, 022308 (2007)

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Groverian Entanglement• The Groverian entanglement measure

• Vanishes only for tensor product states• Invariant to local unitary operators• Cannot increase using LOCC• Relatively easy to compute• Multipartite• Suitable for algorithms

max1)( PG

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Grover’s Search Algorithm

• N elements, r of which are marked

• Classically this takes on average N/(r+1) calls to the function

• On a quantum computer the number of calls is only

unmarked 0

marked 1)(

ii

if

rN

4

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Grover Iteration

Amplitude

State Number

Average

Rotate marked stateRotate all states around average

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Ent. In Grover’s Algorithm

Phys Rev A 69, 062303 (2004)

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Shor’s Algorithm

• Given an integer N, find one divider of N• Best known classical algorithm is

exponential in the number of bits describing N

• The quantum algorithm is polynomial in the number of bits

• The algorithm is made of 3 part: preprocessing, fourier transform, and post processing

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Shor’s Algorithm

Preprocessing:• Choose an integer y so that gcd(y,N)=1 • Find q=2L>N• Create the state

• Measure the second part, getting

1

0

mod,|1 q

a

a Nyaq

A

j

jrlA 0

|1 Ny r mod1

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Shor’s Algorithm

0

0.1

0.2

0.3

0.4

0.5

0 8 16 240

0.1

0.2

0.3

0.4

0.5

0 8 16 24

r rL1 L2

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Shor’s Algorithm

Discrete Fourier Transform:• Applies the transformation

• The resulting state is

1

0

2' 1 q

jj

qijk

k aeq

a

c

rilc

rqce

r|1 2

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Shor’s Algorithm

Post processing• Measuring gives a multiple of q/r• If r is even we define

giving

• gcd(x+1,N) and gcd(x-1,N) give a divider

Nyx r mod2/

Nxxx mod0)1)(1(12

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Ent. In Shor’s Algorithm

• Preprocessing – constructing the quantum state• The post processing is classical• Is DFT where the speed-up happens?

Phys Rev A 72, 062308 (2005)

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Ent. In Shor’s Algorithm

• Maybe DFT never changes entanglement

Random states compared to Shor states

Tensor product states compared to Shor states

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Ent. In Shor’s Algorithm

• All the entanglement is created in the preprocessing stage

• Guesses (N,y) which create a small amount of ent. can be deduced classically

• The amount of ent. increases with the number of bits and approaches the theoretical bound

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Conclusion• The entanglement generated by Grover’s algorithm does

not depend on the size of the search space• Grover’s algorithm offers polynomial speed up• The amount of entanglement generated by Shor’s

algorithm approaches the theoretical limit• Shor’s algorithm provides exponential speed up over all

known classical algorithms• Hints at the fact that factoring really is exponential

classically (?)• All the entanglement in Shor’s algorithm is created in the

preprocessing stage• Entanglement is generated by Shor’s algorithm only in

those cases where the problem is classically difficult

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More Information• Can be found at:

Analysis of Grover’s quantum seardh as a dynamical systemO. Biham, D. Shapira, and Y.shimoniPhys Rev A 68, 022326 (2003)

Charachterization of pure quantum states of multiple qubiots using the Groverian entanglement measureY. Shimoni, D. Shapira, and O. BihamPhys Rev A 69, 062303 (2004)

Algebraic analysis of quantum search with pure and mixed statesD. Shapira, Y. Shimoni, and O. BihamPhys Rev A 71, 042320 (2005)

Entanglement during Shor’s algorithmY. Shimoni and O. BihamPhys Rev A 72, 062308 (2005)

Groverian measure of entanglement for mixed statesD. Shapira, Y. Shimoni, and O. BihamPhys Rev A 73, 044301 (2006)

Groverian entanglement measure of pure states with arbitrary partitionsY. Shimoni and O. BihamPhys Rev A 74, 022308 (2007)