Barry C. Sanders

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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.
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Simulating evolution quantum state generation
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Classical Preprocessor
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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
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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
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j2
j1
j3
j4
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j2
j1
j3
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Cleanup
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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
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Lemma Strict bound for Lie-Trotter-Suzuki
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Theorem Simulation cost nearly linear in time

• Theorem
• Optimal choice of k
• Then

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Simulation time cannot be sublinear in t
p
3p/4
p/2
p/4
t0
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Lemma (decomposition of H unknown)
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Graph associated with H
Connect x to yk (x) with an edge of weight ak (x)
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Symmetrically labeled graphs
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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)
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Graph with monochromatic paths
To break up the paths, we increase the number of
colours
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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
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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
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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.