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Barry C. Sanders

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Barry C. Sanders. Institute for Quantum Information Science, University of Calgary ... G. Ahokas, D. W. Berry, R. Cleve, and B. C. Sanders, Comm. Math. Phys. ... – PowerPoint PPT presentation

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Title: Barry C. Sanders


1
Efficiently algorithm for universal quantum
simulation
  • 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)

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
3
(No Transcript)
4
Classical Preprocessor
5
Background
Feynman 1982 Quantum Computer would
efficiently simulate dynamics of quantum
systems. Lloyd 1996 Formalized conjecture,
assumed tensor product structure, showed
efficient algorithm.
Lie-Trotter
a t3/2
ATS 2003
Graph Colouring
a (d1)2 n6
Lie-Trotter
a t3/2
Childs 2004
Graph Colouring
a d2 n2
Lie-Trotter-Suzuki (kth order)
a t11/2k
Our Results
Deterministic Coin Tossing
a d2 logn
6
Optimal in t nearly constant in n
logn is the height of the smallest tower of
powers of 2 that exceeds n
Lie-Trotter-Suzuki (kth order)
a t11/2k
Our Results
Deterministic Coin Tossing
a d2 logn
7
j2
j1
j3
j4
8
j2
j1
j3
9
Cleanup
10
(No Transcript)
11
Hamiltonian H generates unitary break up
  • H sum of local Hamiltonians
  • Trotter (m2) eiHt?(eiH1t/2r eiH2t/r eiH1t/2r)r,
    H?H1H2.
  • Number of steps for quantum computer N ? t3/2.
  • 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
Simulation time cannot be sublinear in t
p
3p/4
p/2
p/4
t0
15
Lemma (decomposition of H unknown)
16
Graph associated with H
Connect x to yk (x) with an edge of weight ak (x)
17
Symmetrically labeled graphs
18
Non-symmetric case
Modify labeling to be symmetric (with an overhead
cost)
We now have d 2 labels instead of d labels, but a
symmetric labeling
(a, b)
(a, b)
x
y
(1, 2)
with z lt y
(1, 2)
(1, 3)
(1, 3)
(1, 3)
with y lt w
Problem!
(1, 3)
19
Graph with monochromatic paths
To break up the paths, we increase the number of
colours
20
x lt y lt z lt w
Deterministic coin-tossing Cole Vishkin 86
x
(a,b, x
y' ? (i, yi), where i min j yj ? zj
y
(a,b, y
z
(a,b, z
w
(a,b, w
Note still a valid coloring! x' ? y' y' ? z'
z' ? w'
n bits
log(n)1 bits
d 2 2n colours
21
Breaking up the paths II
O(log(n)) iterations
x
x
(a,b, x
x'
y
y
(a,b, y
y'
z
z
z'
(a,b, z
w
w
w'
(a,b, w
n bits
log(n)1 bits
log(log(n)1)1 bits
d 2 2n colors
6 elements
Just 5 iterations for n ? 101037
22
(No Transcript)
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.
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