From a talk presented in JPS meeting 2011-09

1. 1 / 13
Adiabatic Theorem for Discrete Time Evolution
Atushi Tanaka ( )
Dept. Phys, Tokyo Metropolitan Univ. ( )
2011-09-23
2011 (2011-09-21/24), , 23pGt-8

2. Introduction 2 / 13
Discrete time evolution by Quantum Map
A unitary transformation (“quantum map”) describes the time evolution
from (n − 1)-th step to n-th step:
|ψn = U|ψn−1
ˆ
Examples
a periodically driven system
T
H(t) = K + V sin(2π t/T ),
ˆ ˆ ˆ ˆ
U = exp← [−i 0
ˆ
H(t)dt].
a kicked rotor
H(t) = K + V ∑n δ (t − nT ),
ˆ ˆ ˆ ˆ ˆ ˆ
U = exp(−i K T ) exp(−i V ).
a quantum circuit
U

3. Introduction 3 / 13
Adiabatic time evolution for quantum maps
AT and Miyamoto (2007) and AT and Nemoto (2010): Applications of
Cheon’s anholonomies to quantum state manipulations
|ξ (λ0 ) - U(λ )
ˆ 1 U(λ2 )
ˆ ··· U(λL−1 ) - e i γ |ξ (λL )
ˆ
Cabauy and Benioff (2003), Cyclic networks of quantum gates:
Figure: From Cabauy and Benioff, PRA 032315 (2003).

4. Introduction 4 / 13
(Why need to prove) adiabatic theorem for quantum maps?
The adiabatic theorem for periodically driven systems has been proved
when the modulation is smooth (Yound and Deal, 1970).
→ This cannot be applicable to quantum maps in general.
Evidences of the adiabatic behavior for quantum maps:
Takami’s numerical verification (private communication, 1995).
Hogg’s heuristic argument (PRA 2003)

5. Introduction 5 / 13
Outline
Make a discrete analog of Kato’s proof (JPSJ 1950; Messiah 1959).
Introduction
Setting and statement
Interaction picture
Main estimation
Summary and discussion

6. Setting and statement 6 / 13
Setting
Definition
Eigenangle θj (s) and eigenprojector Pj (s) satisty
ˆ
U(s)Pj (s) = e i θj (s) Pj (s).
ˆ ˆ ˆ
Definition
ˆ
Un describes the exact time evolution along the
discretized path {sn }, i.e.,
ˆ ˆ ˆ
Un = U(sn )Un−1 for n > 0,
ˆ
and U0 = 1.

7. Setting and statement 7 / 13
Theorem
Under the following assumptions,
N is a large parameter.
sn − sn−1 = O(N −1 ).
C , Pj (s) and θj (s) are smooth.
ˆ
There is no spectral crossing
we have
Pj (s )UN Pk (s ) = δjk + O(N −1 )
ˆ ˆ ˆ
for N → ∞.

8. Interaction picture 8 / 13
The adiabatic evolution
Kato’s “geometric” evolution operator
s ∂ Pj (s) ˆ
ˆ
UK (s, s ) ≡ exp −i
ˆ ˆ
HK (r )dr , where HK (s) ≡ i ∑
ˆ , Pj (s)
← s j ∂s
satisfies so-called intertwining property
ˆ ˆ ˆ ˆ
Pj (s)UK (s, s ) = UK (s, s )Pj (s ).
Time evolution involving only the dynamical phase:
UD,n ≡ ∑ Pj (s )e i ∑n =1 θj (sn ) .
n
ˆ ˆ
j

9. Main estimation 9 / 13
Main estimation
For the quantum map in the interaction picture:
ˆ ˆ ˆ
Wn = UW ,n Wn−1 ,
it is suffice to show
Pj (s )WN Pk (s ) = δjk + O(N −1 )
ˆ ˆ ˆ

10. Main estimation 10 / 13
ˆ
The quantum map for Wn is equivalent with “Volterra’s equation”:
n
ˆ
Wn = 1 + ∑ (UW ,n − 1)Wn −1 .
ˆ ˆ
n =1
We introduce
n
Vn ≡
ˆ ∑ (UW ,n − 1),
ˆ for n > 0.
n =1
Using a discrete analog of integration by parts, we obtain
n−1
Wn = 1 + Vn Wn−1 −
ˆ ˆ ˆ ∑ Vn (UW ,n − 1)Wn −1 .
ˆ ˆ ˆ
n =1
Hence, it is suffice to show UW ,N − 1 = O(N −1 ) and VN = O(N −1 ) to
ˆ ˆ
prove the main theorem.
From the smoothness of Pj (s), we have UW ,n − 1 = O(N −1 ) and
ˆ
Pj (s )VN Pj (s ) = O(N −1 ).
ˆ ˆ ˆ

11. Main estimation 11 / 13
Pj (s )VN Pk (s ) = O(N −1 ) for j = k
ˆ ˆ ˆ
Destructive inteference is the key for the evalutation of
N
ˆ ˆ ˆ
Pj (s )VN Pk (s ) = ∑ Zn −1,jk Rn ,jk ,
ˆ
n =1
where
n
Zn,jk ≡ exp −i ∑ [θj (sn ) − θk (sn )]
n =1
and
†
Rn,jk ≡ UK (sn , s )
ˆ ˆ ˆ ˆ ˆ
Pj (sn )Pk (sn−1 )UK (sn−1 , s ).
NOTE: Zn,jk is oscillatory, and Rn,jk = O(N −1 ).
ˆ

12. Main estimation 12 / 13
ˆ ˆ ˆ ˆ
An estimation of Pj (s )VN Pk (s ) = ∑N =1 Zn −1,jk Rn ,jk is shown. Due to
n
the absence of spectral crossing, we have
Zn,jk − Zn−1,jk
Zn−1,jk = −i θj (sn )+i θk (sn ) − 1
.
e
Using a discrete analog of integration by parts, we have,
ˆ
Zn,jk Rn,jk ˆ
R1,jk n−1
ˆ ˆ ˆ
Pj (s )Vn Pk (s ) = − + ∑ Zn ,jk Rn ,jk
ˆ (2)
zjk (sn ) − 1 zjk (s1 ) − 1 n =1
where
ˆ
Rn+1,jk ˆ
Rn,jk
ˆ (2)
Rn,jk ≡ − .
zjk (sn+1 ) − 1 zjk (sn ) − 1
Because Rn = O(N −2 ), we obtain
(2)
Pj (s )Vn Pk (s ) = O(N −1 )
ˆ ˆ ˆ .

13. Summary and discussion 13 / 13
Summary and discussion
Summary
It is straightforward to extend Kato’s proof of the adiabatic theorem to
quantum maps.
Discussion
Compare with Hogg
Compare with Ambainis and Regev (quant-ph/0411152, 2004)
Possible extensions
Ref. AT, arXiv:1107.2989.