Implementation of a Controlled Phasegate
for Ultracold Calcium Atoms Using Optimal Control
1
2
Michael Goerz , Tommaso Calarco , Christiane P. Koch
1
1: Institut für Theoretische Physik, Freie Universität Berlin, 2: Institut für Quanteninformationsverarbeitung, Universität Ulm
Summary
1
t = T − ∆t
nt−2
2
T
20
10
nt−1
Ψbw(t0)
Ψbw(t)
Ψ.bw. .(t)
Ψbw(t)
µ̂
µ̂
. µ̂. .
µ̂
Ψfw(t)
.
Ψ.fw. (t)
Ψi
˜1
Ψt
Ψt
T = 1.23 ps
phase dynamics
(|00i, |01i, |0i)
= τ
Ψfw(T )
Ψfw(T )
Ψfw(t)
˜nt−2
˜2
-20 0
1P
1
1P
0 µ(t)
E1 0  +
+ 1x 2 ⊗
0 Ea
X (ij)
X (ij)
V̂trap (x1, x2)
+
V̂BO (|x2 − x1|) +
3
t0 + 23 ∆t
t0 + ∆t
2
...
T − 32 ∆t
T − ∆t
2
1 1
T = 15 ps
F = 0.773
200 iterations
x2
|aai
|a1i
|1ai
d
|11i
|0ai
|a0i
|01i
+
|10i
R
|00i
|00i target
|01i target
|10i target
|11i target





















d
µω 1/4 X
0
initialize to trap ground state Ψ00,i(R) ≈
4π~
e
φ00, φ01, φ10, (φ11)
|00i −→ ei(φ+φT ) |00i
|01i −→ eiφT |01i
gate phases
Look at different targets, and use one-photon or three-photon guess pulses; Use Gaussian guess
pulses that perform full Rabi cycles in the given time window.
0.2
0.4
0.6
t / ps
0.8
1
1
0.5
0
0.5
1.2
0.5
1
1
0.5
0
0.5
1
1
1
0
-5
-10
0.8
0.6
0.4
0.2
-15
2
4
6
8
t/ps
10
12
0 0
14
2
4
6
8
t / ps
10
12
14
20
1
15
10
5
0
-5
-10
0.8
0.6
0.4
0.2
-15
φ00
φ10 = φ01
φ11
= φ00
= φ0 + φ1
= 2φ1
true two-qubit phase χ = φ00 − φ01 − φ10 + φ11
χ = φ00 − 2φ0
True two-qubit phase χ from Cartan decomposition [4].
Û = Uˆ1Ô(χ)Uˆ2,
0
5
-20 0
χ00
φ00
φ0
where Û1 and Û2 are purely local operations.
Parameters:
T = 0.1 – 1.0 ps
0 0
1.2
10
|0i −→ eiφT /2 |0i
±
ˆ Optimize for different gate times: medium T = 1.23 – 50 ps, long T = 150 – 800 ps, and short
1
15
|00i −→ ei(φ+φT ) |00i
−mω0
2
(d±R)
2~
ˆ Spatial grid: Rmin = 5.0 a.u., Rmax = 300.0 a.u., nr = 512
ˆ Trap parameters: ω = 400 MHz, d = 5 nm.
0.8
20
T = 290 ps
F = 0.984
70 iterations
φ00, φ0, (φ1)
|10i −→ eiφT |10i

0.5
reduced optimization scheme
|1i
optimization targets
x1










=










full optimization scheme
|0i

0.6
t/ps
0
-20 0
ij
ij
0.4
0.5
˜nt−1


N
E
S(t̃) X 1 D
† †
Ψik Ô Û (T → t, (0))µ̂Û (0 → t, (1)) Ψik 
∆(t̃) =
=
α
2
system phases
1S
0
0.2
0.5
|ai
ωL = 23652 cm-1

0.4
1
One-Qubit and Two-Qubit Phases
⊗ 1x 1 +
-5
0.6
-15
/10−5 a.u.
E0 0
Ĥ2q =  0 E1
µ(t) 0
0
0.8
0.2
Qubit Encoding in Calcium

µ(t)
0  ⊗ 11q
Ea

E0
11q ⊗  0
µ(t)
5
-10
k=1

1
15
population |00i |01i
...
t = t0 + ∆t
T = 1.23 ps
F = 0.622
41 iterations
/10−5 a.u.
t0
J = −F +
population |00i |01i
The set of all one-qubit gates plus the two-qubit CNOT is universal. More generally, the CNOT
is equivalent to the controlled phasegate, combined with Hadamard and X-gates.

 iφ
e 0 0 0
 0 1 0 0
X
X


Ô(φ) = 
;
CN OT =

0 0 1 0
O(π)
H
X
X
H
0 0 0 1
Change pulse iteratively to minimize[2, 3]
α
1 h † i
∆(t) dt, F = < tr Ô Û
S(t)
N
population |00i |01i
Universal Quantum Computing
Z
Results
/10−5 a.u.
We consider a quantum computational system of ultracold Calcium atoms in an optical lattice
[1]. The qubits are encoded in the electronic states. A shaped laser pulse couples |0i to an
auxilliary state |ai. For the two-qubit system of two neighboring atoms, there is a dipole-type
(1/R3) interacting for the |0ai state, allowing the implementation of a controlled phasegate on
the |00i state. Via optimal control, for different pulse lengths T , we attempt to find laser pulses
that yield the target transformation. At very short trap distances (5 nm), we find that for short
pulses, a two-qubit phase can be reached; however, the atoms couple to the motional degree of
freedoms. Only for very long pulses that can resolve the trap frequency, good fidelities can be
found.
OCT: Finding an Optimized Pulse
T /ps
1.23
2.00
5.00
8.00
12.36
15.00
30.00
50.00
150.00
290.00
430.00
800.00
iters
41
15
15
15
50
200
4
10
20
70
40
30
F
0.622
0.639
0.719
0.787
0.779
0.773
0.630
0.653
0.898
0.984
0.998
0.999
50
100
χ/π |00i pur.
0.162
0.844
0.190
0.807
0.354
0.589
0.560
0.367
0.662
0.229
0.783
0.343
0.174
0.014
0.266
0.000
0.982
0.639
1.004
0.936
0.998
0.991
0.998
0.997
150
t/ps
200
0 0
250
T /ps
0.5
0.5
0.5
0.5
1.0
1.0
1.0
1.0
C3
1 × 106
4 × 108
8 × 108
1.6 × 109
1 × 106
4 × 108
8 × 108
1.6 × 109
iters
40
40
40
40
40
40
40
40
50
100
F
0.582
0.684
0.549
0.523
0.579
0.607
0.568
0.555
150
t / ps
200
250
χ/π |00i pur.
0.027
0.996
0.655
0.289
0.853
0.042
0.880
0.025
0.023
0.998
0.810
0.085
0.866
0.045
0.874
0.033
All optimizations for φ = π.
Dipole Interaction
Is it possible to get a two-qubit phasegate in
arbitrarily short time if we make the dipole interaction
sufficiently strong?
At d = 200 nm, try:
C3 = 1 × 106, 4 × 108, 8 × 108, 1.6 × 109.
References
|aai
|0ai
C3
= −R
3
|a0i
[1] T. Calarco et al., Phys. Rev. A 61, 022304 (2004)
[2] J. P. Palao, R. Kosloff, Phys. Rev. Lett. 89, 188301 (2002), Phys. Rev. A 68, 062308 (2003).
[3] C. P. Koch et. al. Phys. Rev. A 70, 013402 (2004).
|00i
[4] J. Zhang et. al. Phys. Rev. A. 67, 042313 (2003).