1. Barry C. Sanders Institute for Quantum Information Science, University of Calgary with G Ahokas (Calgary), D W Berry (Macquarie), R Cleve (Waterloo), P Hoyer (Calgary), N Wiebe (Calgary) Efficiently algorithm for universal quantum simulation Quantum Information and Many Body Physics Workshop University of British Columbia, 1 December 2007 Comm. Math. Phys. 270(2): 359 - 371 (March 2007) + New Work.

2. Simulating evolution: quantum state generation

4. Classical Preprocessor Classical Preprocessor

5. Background  t 3/2  (d+1) 2 n 6 Lie-Trotter Graph Colouring  t 3/2  d 2 n 2 Lie-Trotter Graph Colouring  t 1+1/2k  d 2 log * n Lie-Trotter-Suzuki (k th order) Lie-Trotter-Suzuki (k th order) Deterministic Coin Tossing Deterministic Coin Tossing ATS 2003 Childs 2004 Our Results Feynman 1982: Quantum Computer would efficiently simulate dynamics of quantum systems. Lloyd 1996: Formalized conjecture, assumed tensor product structure, showed efficient algorithm.

6. Optimal in t; nearly constant in n  t 1+1/2k  d 2 log * n Lie-Trotter-Suzuki (k th order) Lie-Trotter-Suzuki (k th order) Deterministic Coin Tossing Deterministic Coin Tossing Our Results log*n is the height of the smallest tower of powers of 2 that exceeds n:

7. j1j1 j2j2 j3j3 j4j4

8. j1j1 j2j2 j3j3

11. Hamiltonian H generates unitary: break up H: sum of local Hamiltonians Trotter (m=2) : e iHt  (e iH 1 t/2r e iH 2 t/r e iH 1 t/2r ) r, H  H 1 +H 2. Number of steps for quantum computer N  t 3/2. Suzuki generalization of Trotter formula: Suzuki proves for small : 5 terms

12. Lemma: Strict bound for Lie-Trotter-Suzuki

13. Theorem: Simulation cost nearly linear in time Theorem: Optimal choice of k : Then

14.  t=0    X j = 0 1 1 0 1 0 0 1 Simulation time cannot be sublinear in t

15. Lemma (decomposition of H unknown)

16. Graph associated with H 2 2 2 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 11 1 3 3 3 3 3 3 3 3 3 3 3 3 2 1 2 2 2 2 2 2 3 3 3 3 3 3 y1y1 ydyd x : α1α1 αdαd Connect x to y k (x) with an edge of weight α k (x)

17. Symmetrically labeled graphs 2 1 2 1 3 1 3 1 1 2 1 2 1 3 2 1 1 2 1 1 3 2 1 3 1 3 1 3 3 32 3 2 1 2 3 3 1 3 2 3 3 2 2 2 2 2 3 1 2 3 2 3 1 1 1 2 2

18. Non-symmetric case Modify labeling to be symmetric (with an overhead cost) ( a, b ) We now have d 2 labels instead of d labels, but a symmetric labeling a b xy with x < y xy ( 1, 3 ) with z < y with y < w ( 1, 2 ) ( 1, 3 ) xy z w 13 2 1 1 3 Example: Problem! ( a, b ) ( 1, 3 ) ( 1, 2 ) ( 1, 3 )

19. Graph with monochromatic paths 1 2 1 3 1 1 3 3 1 2 1 3 1 3 3 3 1 1 3 1 3 2 3 3 3 3 2 1 1 32 3 2 2 1 3 3 2 2 1 1 1 1 2 2 1 1 2 1 2 1 1 3 2 2 1 1 2 To break up the paths, we increase the number of colours

20. x y z w ( a, b, x ( a, b, y ( a, b, z ( a, b, w n bits x y z w x′ y′ z′ w′ d 2 2 n colours log( n )+1 bits y′  ( i, y i ), where i = min{ j : y j  z j } Then y′ = (010,1) Example: y = 01100101 z = 01001101 010 x < y < z < wx < y < z < w Note: still a valid coloring! x′  y′ & y′  z′ & z′  w′ “Deterministic coin-tossing” [Cole & Vishkin ’86]

21. Breaking up the paths II x y z w ( a, b, x ( a, b, y ( a, b, z ( a, b, w n bits x y z w x′ y′ z′ w′ x y z w x′′ y′′ z′′ w′′ d 2 2 n colors log( n )+1 bits log(log( n )+1)+1 bits x y z w x′′′ y′′′ z′′′ w′′′ 6 elements... O ( log*( n ) ) iterations Just 5 iterations for n  10 10 37

22. Sketch of Proof: # of H j ’s is m = 6d 2. Need to call the black-box O(log * n) times for each H j. Substituting into theorem for upper bound on N exp gives result.

23. Further Reading S. Lloyd, Science 273, 1073 (1996). R. P. Feynman, Int. J. Th.. Phys. 21, 467 (1982). D. Aharonov and A. Ta-Shma, Proc. ACM STOC, 20 (2003). M. Suzuki, Phys. Lett. A 146, 319 (1990); JMP 32, 400 (1991). A. Childs, Ph.D. Thesis, MIT (2004). R. Cole and U. Vishkin, Inform. and Control 70, 32 (1986). N. Linial, SIAM J. Comp. 21, 193 (1992). A. Childs, R. Cleve, E. Deotto, E. Farhi, S. Guttman, and D. Spielman, Proc. ACM STOC, 59 (2003). R. Beals, H. Buhrman, R. Cleve, M. Mosca, and R. de Wolf, J. ACM 48, 778 (2001). G. Ahokas, D. W. Berry, R. Cleve, and B. C. Sanders, Comm. Math. Phys. 270(2): 359 - 371 (March 2007); quant-ph/0508139.