1 / 26

Multipartite Entanglement and its Role in Quantum Algorithms

Multipartite Entanglement and its Role in Quantum Algorithms. Special Seminar: Ph.D. Lecture by Yishai Shimoni. Acknowledgement. This work was carried out under the supervision of Prof. Ofer Biham & In collaboration with Dr. Daniel Shapira. cam.qubit.org. Outline. Quantum computation

mohawk
Download Presentation

Multipartite Entanglement and its Role in Quantum Algorithms

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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)

More Related