Although superconducting systems provide a promising platform for quantum computing, their networking poses a challenge as they cannot be interfaced to light---the medium used to send quantum signals through channels at room temperature. We show that mechanical oscillators can mediated such coupling and light can be used to measure the joint state of two distant qubits. The measurement provides information on the total spin of the two qubits such that entangled qubit states can be postselected. Entanglement generation is possible without ground-state cooling of the mechanical oscillators for systems with optomechanical cooperativity moderately larger than unity; in addition, our setup tolerates a substantial transmission loss. The approach is scalable to generation of multipartite entanglement and represents a crucial step towards quantum networks with superconducting circuits.

1. Measurement-induced long-distance
entanglement of superconducting qubits
using optomechanical transducers
Ondřej Černotík and Klemens Hammerer
Leibniz Universität Hannover
DPG Spring meeting, 3 March 2016
arXiv:1512.00768

2. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
Superconducting systems are among the
best candidates for quantum computers.
2
• Controlling microwave fields with qubits
Hofheinz et al., Nature 454, 310 (2008); Nature 459, 546 (2009)
• Feedback control of qubits
Ristè et al., PRL 109, 240502 (2012); Vijay et al., Nature 490, 77 (2012);
de Lange et al., PRL 112, 080501 (2014)
• Quantum error correction
Córcoles et al., Nature Commun. 6, 6979 (2015), Kelly et al., Nature 519, 66
(2015), Ristè et al., Nature Commun. 6, 6983 (2015)
• Entanglement generation
Ristè et al., Nature 502, 350 (2013);
Roch et al., PRL 112, 170501 (2014);
Saira et al., PRL 112, 070502 (2014)
Schoelkopf

3. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
Entanglement between two qubits can be
generated by measurement and
postselection.
3
C. Hutchison et al., Canadian J. Phys. 87, 225 (2009)
N. Roch et al., PRL 112, 170501 (2014)
Hint = za†
aDispersive coupling
|11i
|00i
|01i + |10i

4. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
Mechanical oscillators can mediate
coupling between microwaves and light.
4
T. Bagci et al., Nature 507, 81 (2014)R. Andrews et al., Nature Phys. 10, 321 (2014)
Z. Yin et al., PRA 91, 012333 (2015)

5. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
Optomechanical transducer acts as a
force sensor.
5
F = ~ /(
p
2xzpf )
S2
F (!) = x2
zpf /[8g2 2
m(!)]Sensitivity:
! ⌧ !m
⌧meas =
S2
F (!)
F2
=
!2
m
16 2g2
⌧ T1,2Measurement time:
H = z(b + b†
) + !mb†
b + g(a + a†
)(b + b†
)

6. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
The thermal mechanical bath affects the
qubit.
6
mech = S2
f (!) =
2 2
!2
m
¯nDephasing rate:
⌧meas <
1
mech
! C =
4g2
 ¯n
>
1
2

7. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
The system can be modelled using a
conditional master equation.
7
D[O]⇢ = O⇢O† 1
2 (O†
O⇢ + ⇢O†
O)
H[O]⇢ = (O hOi)⇢ + ⇢(O†
hO†
i)
H. Wiseman & G. Milburn, Quantum
measurement and control (Cambridge)
d⇢ = i[H, ⇢]dt + Lq⇢dt +
2X
j=1
{(¯n + 1)D[bj] + ¯nD[b†
j]}⇢dt
+ D[a1 a2]⇢dt +
p
H[i(a1 a2)]⇢dW
H =
2X
j=1
j
z(bj + b†
j) + !mb†
jbj
+ g(aj + a†
j)(bj + b†
j) + i

2
(a1a†
2 a2a†
1)

8. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
We can obtain an effective equation for
the qubits.
8
meas = 16
2
g2
!2
m
, mech =
2
!2
m
(2¯n + 1)
OC et al., PRA 92, 012124 (2015)ˇ
d⇢q = Lq⇢qdt +
2X
j=1
mechD[ j
z]⇢qdt
+ measD[ 1
z + 2
z]⇢qdt +
p
measH[ 1
z + 2
z]⇢qdW

9. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
Optical losses introduce additional
dephasing.
9
(1 ⌧) measD[ 1
z]⇢q
p
⌘ measH[ 1
z + 2
z]⇢q

10. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
A transmon qubit can capacitively couple
to a nanobeam oscillator.
10
G. Anetsberger et al., Nature Phys. 5, 909 (2009)
J. Pirkkalainen et al., Nat. Commun. 6, 6981 (2015)
= 2⇡ ⇥ 5.8 MHz
g = 2⇡ ⇥ 900 kHz
 = 2⇡ ⇥ 39MHz
!m = 2⇡ ⇥ 8.7 MHz
Qm = 5 ⇥ 104
T = 20 mK
¯n = 48
T1,2 = 20 µs
C = 10
⌘
Psucc
Psucc

11. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
The mechanical oscillator can also be
formed by a membrane.
11
R. Andrews et al., Nature Phys. 10, 312 (2014)
T. Bagci et al., Nature 507, 81 (2014)
J. Pirkkalainen et al., Nature 494, 211 (2013)

12. Cernotík (LUH): Entanglement of superconducting qubits, arXiv:1512.00768ˇ
Mechanical oscillators can mediate
interaction between light and SC qubits.
12
• Strong optomechanical cooperativity,
• Sufficient qubit lifetime
OC & K. Hammerer, arXiv:1512.00768ˇ
-
C =
4g2
 ¯n
>
1
2