Multipartite Entanglement and its Role in Quantum Algorithms

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

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

3. 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. 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. 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. 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 examplebut only this partitioning gives a tensor product

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

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

9. Quantum Entanglement www.jpl.nasa.gov

10. 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. 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. Groverian Entanglement Phys Rev A 74, 022308 (2007)

13. 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. 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. Average Grover Iteration Rotate marked state Rotate all states around average Amplitude State Number

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

17. 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. 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

19. Shor’s Algorithm r r L1 L2

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

21. Shor’s Algorithm Post processing • Measuring gives a multiple of q/r • If r is even we definegiving • gcd(x+1,N) and gcd(x-1,N) give a divider

22. 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. Ent. In Shor’s Algorithm • Maybe DFT never changes entanglement Random states compared to Shor states Tensor product states compared to Shor states

24. 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. 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. 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)