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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
University of British Columbia, 1 December 2007
Comm. Math. Phys. 270(2): 359 - 371 (March 2007) + New Work.
Simulating evolution: quantum state generation
ClassicalPreprocessor
ClassicalPreprocessor
Λ ≥sup H , &H , &&H3 ,K{ },dimH ,
ε ,T ,d = sparseness H( )
n (including ancillae) , %ti{ }
Background
t3/2
(d+1)2 n6
Lie-TrotterLie-Trotter
Graph ColouringGraph Colouring
t3/2
d2 n2
Lie-TrotterLie-Trotter
Graph ColouringGraph Colouring
t1+1/2k
d2 log*n
Lie-Trotter-Suzuki(kth order)
Lie-Trotter-Suzuki(kth order)
Deterministic Coin Tossing
Deterministic Coin Tossing
ATS 2003ATS 2003
Childs 2004Childs 2004
Our ResultsOur Results
Feynman 1982: Quantum Computer would efficientlysimulate dynamics of quantum systems.
Lloyd 1996: Formalized conjecture, assumed tensorproduct structure, showed efficient algorithm.
Optimal in t; nearly constant in n
t1+1/2k
d2 log*n
Lie-Trotter-Suzuki(kth order)
Lie-Trotter-Suzuki(kth order)
Deterministic Coin Tossing
Deterministic Coin Tossing
Our ResultsOur Results
log*n is the height of the smallesttower of powers of 2 that exceeds n:
%ψ
%ψ '
%U = %U jN
L %U j2%U j1
j1 j2 j3 j4
0L 0 0L 0
%U = %U jN
L %U j2%U j1
ψ = cl l∑
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)',max( ll=x
( )',min ll=y
%H x,y , colour(l , l ') = j
0 ,colour(l , l ') ≠ j
⎧⎨⎪
⎩⎪
x
y
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Hamiltonian H generates unitary: break up
H: sum of local Hamiltonians
Trotter (m=2): eiHt(eiH1t/2r eiH2t/r eiH1t/2r)r, HH1+H2.
Number of steps for quantum computer N t3/2.
Suzuki generalization of Trotter formula:
Suzuki proves for small :
H = Hii=1
m
∑
, pk = 4 −41/ 2k−1( )( )−1
5 terms5 terms
Lemma: Strict bound for Lie-Trotter-Suzuki
qk = 1−4pk'( )k'=2
k
∏
exp −it Hii=1
m
∑⎛⎝⎜
⎞⎠⎟− S2k −i
tr
⎛⎝⎜
⎞⎠⎟
⎡
⎣⎢
⎤
⎦⎥
r
≤22m5k−1qkτ( )
2k+1
2k+1( )!r2k
12m5k−1qkτ / r ≤1,
32
2m5k−1qkτ( )2k+1
2k+1( )!r2k ≤1.
τ =t × max H j
ε ≤1 ≤4m5k −1qkτ
2k +1( )!62k
Theorem: Simulation cost nearly linear in time
Theorem:
Optimal choice of k:
Then
N ≤m52k mqkτ( )
1+1/2k
2 2k+1( )!ε⎡⎣ ⎤⎦1/2k
k ≈12
log5
mτε
⎛⎝⎜
⎞⎠⎟
N ≤4m2τ exp 2 log5 mτ / ε( )⎡⎣
⎤⎦
2
t=0
4
34
Xj = 0 1 1 0 1 0 0 1
Simulation time cannot be sublinear in t
Lemma (decomposition of H unknown)
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∃ decomposition H = H j
j=1
m
∑ , with each H j 1- sparse,
such that m = 6d2, and each query to any H j can be
simulated by O log* n( ) queries to H.
Graph associated with Graph associated with HH
2
2
2
2
2
1
11
2
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
22
1
1
1
1 1
1
3
3
3
3
3
3
33
3
3
3
3
21
2
22
2
2
2
3
3
3
3
3
3
y1
yd
x :α1
αd
Connect x to yk (x) with an
edge of weight αk (x)
SymmetricallySymmetrically labeled graphs labeled graphs
2
1
2
1
3
1
31
1
2
1
2
1
3
2
1
1
2
1
1
3
2
1
3
13
1
3
3
3 2
3
2
1
2
3
3
1
32
3
3
2
2
22
2
31
2
3
2
3
1
1
1
2
2
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.
Math. Phys. 270(2): 359 - 371 (March 2007); quant-ph/0508139.