We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real-time axis, that some number of its time derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is nondegenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time derivative of the Hamiltonian divided by the cube of the minimal gap.

1.
M.
Born
and
V.
Fock
,
Z. Phys.
51
,
165
(
1928
).
2.
T.
Kato
,
J. Phys. Soc. Jpn.
5
,
435
(
1950
).
3.
A.
Messiah
,
Quantum Mechanics
(
North-Holland
,
Amsterdam
,
1962
), Vol.
II
.
4.
E.
Farhi
,
J.
Goldstone
,
S.
Gutmann
, and
M.
Sipser
, eprint arXiv:quant-ph/0001106.
5.
E.
Farhi
,
J.
Goldstone
,
S.
Gutmann
,
J.
Lapan
,
A.
Lundgren
, and
D.
Preda
,
Science
292
,
472
(
2001
).
6.
J.
Roland
and
N. J.
Cerf
,
Phys. Rev. A
65
,
042308
(
2002
).
7.
J.
Du
,
L.
Hu
,
Y.
Wang
,
J.
Wu
,
M.
Zhao
, and
D.
Suter
,
Phys. Rev. Lett.
101
,
060403
(
2008
).
8.
S.
Teufel
,
Adiabatic Perturbation Theory in Quantum Dynamics
(
Springer-Verlag
,
Berlin
,
2003
).
9.
G.
Nenciu
,
Commun. Math. Phys.
152
,
479
(
1993
).
10.
J. E.
Avron
and
A.
Elgart
,
Commun. Math. Phys.
203
,
445
(
1999
).
11.
G. A.
Hagedorn
and
A.
Joye
,
J. Math. Anal. Appl.
267
,
235
(
2002
).
12.
S.
Jansen
,
M. -B.
Ruskai
, and
R.
Seiler
,
J. Math. Phys.
48
,
102111
(
2007
).
13.
M. J.
O’Hara
and
D. P.
O’Leary
,
Phys. Rev. A
77
,
042319
(
2008
).
14.
M. A.
Nielsen
and
I. L.
Chuang
,
Quantum Computation and Quantum Information
(
Cambridge University Press
,
Cambridge
,
2000
).
15.
D.
Aharonov
,
W.
van Dam
,
J.
Kempe
,
Z.
Landau
,
S.
Lloyd
, and
O.
Regev
,
SIAM J. Comput.
37
,
166
(
2007
).
16.
17.
J.
Kempe
,
A.
Kitaev
, and
O.
Regev
,
SIAM J. Comput.
35
,
1070
(
2006
).
18.
R.
Oliveira
and
B.
Terhal
,
Quantum Inf. Comput.
8
,
0900
(
2005
).
19.
A.
Mizel
,
D. A.
Lidar
, and
M.
Mitchell
,
Phys. Rev. Lett.
99
,
070502
(
2007
).
20.
S.
Sachdev
,
Quantum Phase Transitions
(
Cambridge University Press
,
Cambride
,
2001
).
21.
J. I.
Latorre
and
R.
Orus
,
Phys. Rev. A
69
,
062302
(
2004
).
22.
R.
Schützhold
and
G.
Schaller
,
Phys. Rev. A
74
,
060304
(R) (
2006
).
23.
A. M.
Zagoskin
,
S.
Savel’ev
, and
F.
Nori
,
Phys. Rev. Lett.
98
,
120503
(
2007
).
24.
G.
Schaller
,
S.
Mostame
, and
R.
Schützhold
,
Phys. Rev. A
73
,
062307
(
2006
).
25.
M.
Reed
and
B.
Simon
,
Methods of Modern Mathematical Physics IV: Analysis of Operators
(
Academic
,
San Diego
,
1978
).
26.
R.
Bhatia
,
Matrix Analysis
,
Graduate Texts in Mathematics
No.
169
(
Springer-Verlag
,
New York
,
1997
).
27.
R.
Tempo
,
G.
Calafiore
, and
F.
Dabbene
,
Randomized Algorithms for Analysis and Control of Uncertain Systems
(
Springer-Verlag
,
London
,
2005
).
28.
M.
Mohseni
,
A. T.
Rezakhani
, and
D. A.
Lidar
,
Phys. Rev. A
77
,
032322
(
2008
).
29.
M. S.
Birman
and
M. Z.
Solomjak
,
Spectral Theory of Self-Adjoint Operators in Hilbert Space
(
Reidel
,
Dordrecht
,
1986
).
30.
M. V.
Berry
,
Proc. R. Soc. London, Ser. A
429
,
61
(
1990
).
31.
33.
L. M.
Garrido
and
F. J.
Sancho
,
Physica (Amsterdam)
28
,
553
(
1962
).
34.
G.
Nenciu
,
Commun. Math. Phys.
82
,
121
(
1981
).
35.
G.
Nenciu
and
G.
Rasche
,
Helv. Phys. Acta.
62
,
372
(
1989
).
36.
H. L.
Haselgrove
,
M. A.
Nielsen
, and
T. J.
Osborne
,
Phys. Rev. Lett.
91
,
210401
(
2003
).
37.
P.
Deift
,
M. -B.
Ruskai
, and
W.
Spitzer
,
Quantum Inf. Process.
6
,
121
(
2007
).
38.
H. -P.
Breuer
and
F.
Petruccione
,
The Theory of Open Quantum Systems
(
Oxford University Press
,
Oxford
,
2002
).
39.
M. S.
Sarandy
and
D. A.
Lidar
,
Phys. Rev. A
71
,
012331
(
2005
).
40.
M. S.
Sarandy
and
D. A.
Lidar
,
Phys. Rev. Lett.
95
,
250503
(
2005
).
41.
P.
Thunström
,
J.
Åberg
, and
E.
Sjöqvist
,
Phys. Rev. A
72
,
022328
(
2005
).
42.
A.
Joye
,
Commun. Math. Phys.
275
,
139
(
2007
).
43.
A. M.
Childs
,
E.
Farhi
, and
J.
Preskill
,
Phys. Rev. A
65
,
012322
(
2001
).
44.
J.
Roland
and
N. J.
Cerf
,
Phys. Rev. A
71
,
032330
(
2005
).
45.
S.
Ashhab
,
J. R.
Johansson
, and
F.
Nori
,
Phys. Rev. A
74
,
052330
(
2006
).
46.
M.
Tiersch
and
R.
Schützhold
,
Phys. Rev. A
75
,
062313
(
2007
).
47.
M. H. S.
Amin
,
D. V.
Averin
, and
J. A.
Nesteroff
,
Phys. Rev. A
79
,
022107
(
2009
).
48.
49.
S. P.
Jordan
,
E.
Farhi
, and
P. W.
Shor
,
Phys. Rev. A
74
,
052322
(
2006
).
50.
K. -P.
Marzlin
and
B. C.
Sanders
,
Phys. Rev. Lett.
93
,
160408
(
2004
).
51.
Note that the g(2) factor present in Eq. (91) [evaluated at N=1 and absent in Eq. (88)] gives rise to a discrepancy between the two bounds, unless we set γ=1/14. This does not in fact impose a constraint on the family of Hamiltonians our proof applies to (recall Assumption 1), since in the application of Cauchy’s theorem we are free to choose an arbitrarily small integration contour around the real-time axis. In spite of having thus fixed its value, we continue to write γ rather than 1/14, as there is no fundamental importance to this value; it is merely an outcome of our rather loose bounds, e.g., as in Eq. (71).
52.
Another way to understand the need for the adjustment in the last line of Eq. (98) comes from this example: ψ2Gr[|f1|Φ̇+Pψ̇1]A5β3+A4β22A5β3. To get the last inequality we multiplied the term A4β2 by (Aβ)k with k=1. This is required in order to obtain a bound involving just a single power of A and of β. Failing to do this allows for the possibility that the two bounds (95) and (98) will not agree.
53.
A. T.
Rezakhani
,
W. J.
-
Kuo
,
A.
Hamma
,
D. A.
Lidar
, and
P.
Zanardi
,
Phys. Rev. Lett.
103
,
080502
(
2009
).
54.
Note that ξσ(τ) may in general depend on n. One can see this through a simple example. Imagine a cylinder of gaseous particles with short-ranged interactions. Any particle will interact with all particles inside a sphere of radius rint—the range of the interaction—around it. If we add new particles to the cylinder, at some point (i.e., at some n) all the space inside the shell will be occupied (close packed); hence, the new particles cannot interact with the particle in the center. For such particles, the coupling strength of the interaction with the particle in the center is effectively zero. The other condition we mention, namely time-independence of the graph or lattice, is designed to exclude folding of the system lattice, for example, in the case of large polymer or protein molecules, as this would also potentially allow a dependence of ξσ(τ) on n.
55.
D. A.
Lidar
,
P.
Zanardi
, and
K.
Khodjasteh
,
Phys. Rev. A
78
,
012308
(
2008
).
You do not currently have access to this content.