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.
Simulating evolution:
quantum state generation

  sup H ,

&& ,K ,dimH ,
H& , 3 H
 ,T ,d  sparseness H 
Classical
Preprocessor
n (including ancillae) , t%
i
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
Childs 2004
a
Lie-Trotter
a t3/2
Graph Colouring
Our Results
(d+1)2 n6
a
Lie-Trotter-Suzuki
(kth order)
Deterministic
Coin Tossing
d2 n2
a t1+1/2k
a
d2 log*n
Optimal in t; nearly constant in n
log*n is the height of the smallest
tower of powers of 2 that exceeds n:
Our Results
Lie-Trotter-Suzuki
(kth order)
Deterministic
Coin Tossing
a t1+1/2k
a
d2 log*n
j1
j2
j3
0L 0
j4
0L 0

'
U  U jN
U j2 U j1
j2
j1
j3
0
0
0
0
0
0
0
0
0
0
0
0


0


0

0

0

~
U j1 

~ ~
U j2U j1 

0

0
  c
U  U jN
U j2 U j1
0
x  max( , ' )
x
0
0
y  min , '
y
0

0
 H x, y , colour(l , l ')  j

 0 , colour(l , l ')  j

0
U˜ j
U˜ j
0

0
U˜ j
U˜ j

0
Hamiltonian H generates unitary: break
up
m
 H: sum of local Hamiltonians H   H i
i 1
 Trotter (m=2): eiHt(eiH1t/2r eiH2t/r eiH1t/2r)r, HH1+H2.
 Number of steps for quantum computer N  t3/2.
 Suzuki generalization of Trotter formula:

, pk  4  4
1/ 2 k 1

1
 Suzuki proves for small :
5 terms
Lemma: Strict bound for Lie-Trotter-Suzuki


2k 1
2m5 qk

   t 
exp  it  H i    S2k  i    2
2k


 i 1  
r 
2k  1!r
r
m
k 1
  t  max H j
k
qk   1  4 pk ' 
k ' 2
4m5 k 1 qk
 1
2k  1!62k
12m5 k 1 qk / r  1,


2 k 1
3 2m5 qk
2k
2 2k  1!r
k 1
 1.
Theorem: Simulation cost nearly linear in time
Theorem:
N
m5
2k
mqk 
11/2 k
2 2k  1! 
1/2 k
1
 m 
log 5 
Optimal choice of k: k 


2

2
Then N  4m  exp  2 log 5 m /  
Simulation time cannot be sublinear
in t
Xj =
0
1
1
0
1
0
0
p
3p/4
p/2
p/4
t=0
1
Lemma (decomposition of H unknown)
m
 decomposition H   H j , with each H j 1- sparse,
j1
such that m  6d , and each query to any H j can be
2
simulated by Olog n  queries to H.
*
Graph associated with H
y1
α1
Connect x to yk (x) with
x
an edge of weight αk (x)
:
αd
yd
1
2 3
1
3 2
1
1
2
3
3
1
2
1
2
1
3
3
3
1
2
3
1
3
3
1
2
1
1
1
1
1
2
3
1
3
2
1
3
2
1
3
3
2
2
2
1
1
2
3
2
3
2
3
1
2
2
2
Symmetrically labeled graphs
1
1
3
2
2
1
3
3
3
1
2
3
2
1
3
2
2
3
1
3
1
1
2
2
3
1
1
2
2
3
3
1
2
3
1
2
3
2
2
2
2
3
1
1
2
1
3
2
3
3
1
1
1
3
1
2
1
2
Non-symmetric case
Modify labeling to be symmetric (with an overhead cost)
x
x
a
b
(a , b )
(a , b )
We now have d 2 labels
instead of d labels, but
a symmetric labeling
y
(1, 2)
Example:
(1, 3)
(1, 3)
1
3
x
with x < y
y
(1, 2)
1
z
with z < y
3
w
with y < w
2
y
1
(1, 3)
Problem!
(1, 3)
Graph with monochromatic paths
1
1
1
1
2
1
2
3
3
1
1
3
1
3
1
2
2
3
3
2
1
1
1
1
1
3
2
1
1
3
3
3
2
1
1
1
3
2
2
2
1
3
2
1
2
2
3
1
1
1
2
3
2
3
3
3
2
3
To break up the paths, we increase the number of colours
“Deterministic coin-tossing”
x<y<z<w
[Cole & Vishkin ’86]
x
x
(a , b ,
(a , b ,
z
z
z′
z
colours bits
Example: y = 01100101
z = 01001101
w′
w
n
010
Then y′ = (010,1)
w
w
d 2 2n
y′
y
(a , b ,
y′  (i, yi), where i = min{ j : yj  zj}
y
y
(a,b,
x′
x
log(n)+1
bits
Note: still a valid coloring!
x′  y′ & y′  z′ & z′  w′
Breaking up the paths II
x
x
(a , b ,
y
y
(a , b ,
...
z′′
z′′′
w
w
(a,b,
w
w′
d 2 2n
n
log(n)+1
log(log(n)+1)+1
bits
bits
colors bits
y′′′
z
z
w
w
...
y′′
z′
z
x′′′
y
y
z
z
x
...
x′′
y′
y
(a , b ,
x
x′
x
O(log*(n))
iterations
w′′
...
w′′′
6 elements
Just 5 iterations for n  1010
37
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.
Math. Phys. 270(2): 359 - 371 (March 2007); quant-ph/0508139.
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