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
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Barry C. Sanders Institute for Quantum Information Science, University of Calgary with G Ahokas (Calgary), D W Berry (Macquarie), R Cleve (Waterloo), P.
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Barry C. SandersInstitute for Quantum Information Science, University of
Calgarywith 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
Non-symmetric caseNon-symmetric caseModify 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 bx y with x < y
x y
(1, 3)with z < y
with y < w
(1, 2)
(1, 3)
x y
z
w
1 32
1
1
3
Example:
Problem!
(a, b)
(1, 3)
(1, 2)
(1, 3)
Graph with monochromatic pathsGraph with monochromatic paths1
2
1
3
1
1
33
1
2
1
3
1
3
3
3
1
1
3
1
3
2
3
3
33
2
1
1
3 2
3
2
2
1
3
3
2
21
1
1
1
2
21
1
21
2
1
1
3
2
2
1
1
2
To break up the paths, we increase the number of colours
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 2n
colourslog(n)+1 bits
y′ (i, yi), where i = min{ j : yj zj}
Then y′ = (010,1)
Example: y = 01100101
z = 01001101
010
x < y < z < w
Note: still a valid coloring!x′ y′ & y′ z′ & z′ w′
“Deterministic coin-tossing” [Cole & Vishkin ’86]
Breaking up the paths IIBreaking 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 2n
colorslog(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 101037
Sketch of Proof:
# of Hj’s is m = 6d2. Need to call the black-box O(log*n) times for each Hj.
Substituting into theorem for upper bound on Nexp gives result.
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.