1. Multipartite Entanglement and its Role in Quantum Algorithms Special Seminar: Ph.D. Lecture by Yishai Shimoni

2. 2/26 Acknowledgement This work was carried out under the supervision of Prof. Ofer Biham & In collaboration with Dr. Daniel Shapira cam.qubit.org

3. 3/26 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

4. 4/26 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

5. 5/26 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

6. 6/26 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

7. 7/26 Quantum Entanglement Requirements of entanglement measures: 1.Vanishes only for tensor product states 2.Invariant to local (in party) unitary operations 3.Cannot increase using local operation and classical communication (LOCC)

8. 8/26 Quantum Entanglement Bipartite entanglement connected to entropy and information Resource for teleportation and communication protocols Not much known about multipartite entanglement

9. 9/26 Quantum Entanglement www.jpl.nasa.gov

10. 10/26 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

11. 11/26 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

12. 12/26 Groverian Entanglement Phys Rev A 74, 022308 (2007)

13. 13/26 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

14. 14/26 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

15. 15/26 Grover Iteration Amplitude State Number Average Rotate marked stateRotate all states around average

16. 16/26 Ent. In Grover’s Algorithm Phys Rev A 69, 062303 (2004)

17. 17/26 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

18. 18/26 Shor’s Algorithm Preprocessing: Choose an integer y so that gcd(y,N)=1 Find q=2 L >N Create the state Measure the second part, getting

19. 19/26 Shor’s Algorithm r r L1L1 L2L2

20. 20/26 Shor’s Algorithm Discrete Fourier Transform: Applies the transformation The resulting state is

21. 21/26 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

22. 22/26 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)

23. 23/26 Ent. In Shor’s Algorithm Maybe DFT never changes entanglement Random states compared to Shor states Tensor product states compared to Shor states

24. 24/26 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

25. 25/26 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

26. 26/26 More Information Can be found at:  Analysis of Grover’s quantum seardh as a dynamical system O. Biham, D. Shapira, and Y.shimoni Phys Rev A 68, 022326 (2003)  Charachterization of pure quantum states of multiple qubiots using the Groverian entanglement measure Y. Shimoni, D. Shapira, and O. Biham Phys Rev A 69, 062303 (2004)  Algebraic analysis of quantum search with pure and mixed states D. Shapira, Y. Shimoni, and O. Biham Phys Rev A 71, 042320 (2005)  Entanglement during Shor’s algorithm Y. Shimoni and O. Biham Phys Rev A 72, 062308 (2005)  Groverian measure of entanglement for mixed states D. Shapira, Y. Shimoni, and O. Biham Phys Rev A 73, 044301 (2006)  Groverian entanglement measure of pure states with arbitrary partitions Y. Shimoni and O. Biham Phys Rev A 74, 022308 (2007)

27. 27/26 Grover’s Algorithm To rotate the marked states:  we use an ancilla bit in the state  A call to the function applies the transformation  The result is

28. 28/26 Grover’s Algorithm To rotate around the average:  Rotate the |0> state, using the operator  Applying Hadamard gates to all qubits gives (with a minus sign)  The result on a state is