Title goes here
Building a superconducting
quantum computer with the
surface code
Matteo Mariantoni
Fall INTRIQ meeting, November 5th & 6th 2013
the DQM lab team
•
Matteo Mariantoni
Principal Investigator
Thomas G. McConkey
Doctoral Student
Corey Rae H. McRae
Doctoral Student
Collaborators:
1) Prof. Michael J. Hartmann
Heriot Watt University
2) Prof. Frederick W. Strauch
Williams College
Daryoush Shiri
3) Prof. Adrian Lupaşcu
Postdoctoral Fellow
IQC
4) Prof. Christopher M. Wilson
IQC
5) Prof. Zbig R. Wasilewski
WIN
6) Dr. Austin G. Fowler
John R. Rinehart
Carolyn “Cary” T. Earnest
Doctoral Student
UC Santa Barbara
Doctoral Student
7) Prof. David G. Cory
IQC
8) Prof. Guo-Xing Miao
IQC
9) Prof. Roger G. Melko
UW
Jérémy Béjanin
10) Sadegh Raeisi
Master’s Student
Yousef Rohanizadegan
IQC
Research Assistant
11) Yuval R. Sanders
IQC
lab virtual walkthrough
DR
the lab is being setup in these very days;
it will be up and running by February 2014
photo credit
BlueFors
Cryogenics Oy
lab real walkthrough
the lab is being setup in these very days;
it will be up and running by February 2014
on the edge
nano/micro meter
photo credit – M. Mariantoni and E. Lucero
University of California Santa Barbara
space
on the edge
milli kelvin
photo credit – BlueFors Cryogenics Oy
temperature
on the edge
giga hertz
photo credit – M. Mariantoni and E. Lucero
frequency
time
superconducting quantum circuits
• LC resonator
C
L
Hˆ LC
2
2
ˆ
ˆ

Q


2L 2C
superconducting quantum circuits
• LC resonator
~ 7 GHz
4
3
2
1
0
superconducting quantum circuits
• transmission-line resonator
dielectric material
lres
superconducting quantum circuits
• coplanar waveguide resonator
Ca
Cb
T1 ~ 5 ms
T2 ~ 2T1
M. Mariantoni et al., Nature Phys. 7, 287 (2011)
superconducting quantum circuits
• qubit
L
C
Josephson junction
ˆ 2→ nonlinearity
ˆ 



ˆ2
Q

Hˆ Q 
 EJ cos 2
 2C
2L

0 

superconducting quantum circuits
• qubit
dan ~ 200 MHz
~ 6.8 GHz
~ 7 GHz
f
e
g
superconducting quantum circuits
• qubit
junction
capacitor C
inductor L
T1 ~ 500 ns
T2 ~ 150 ns
M. Mariantoni et al., Nature Phys. 7, 287 (2011)
superconducting quantum circuits
• resonator + qubit + control
X,Y (, /2); Z
junction
capacitor C
inductor L
resonator
+ qubit
g(C ) ~ 100 MHz  10 ns
A. Blais, R.-S. Huang, A. Wallraff, bS.M. Girvin, and R.J. Schoelkopf, Phys. Rev. A 69,
M. Mariantoni
et al.,
Nature 431,
Phys.162
7, 287
(2011)
062320 (2004); A. Wallraff
et al., Nature
(London)
(2004)
one-qubit pulses and one-qubit quantum errors
• pulses
1. 𝑋 𝜋 (𝑅𝑥𝜋 )  energizes the qubit from g to e
𝜋/2
2. 𝑋 𝜋/2 (𝑅𝑥 )  prepares the qubit in state g − e
3. 𝑍 (𝑅𝑧 )  shifts the qubit by a certain phase,
𝑒 𝑖𝜙 e
• errors
1. bit-flip (𝑋)  brings the qubit from e to g
𝐼 + 𝜀 𝑋, 𝜀 ≪ 1
2. phase-flip (𝑍)  shifts the qubit from g + e to
g − e
𝐼 + 𝜀𝑍
• any error
𝑈𝑒 = 𝛼 𝐼 + 𝛽𝑋 + 𝛾𝑌 + 𝛿 𝑍
P.W. Shor, Phys. Rev. A 52, 2493 (1995)
create, write, re-create, zero, read
entanglement
M1
B
Q1
Z1
Z2
Q2
M2
M. Mariantoni et al.,
Science 334, 61 (2011)
i
the CZ- gate
• qutrit
qubit
the CZ- gate
• qutrit
phase qubit
qutrit
the CZ- gate
• qutrit-resonator interaction
the CZ- gate
• qutrit-resonator interaction
the CZ- gate
• qutrit-resonator interaction
g1  e0 : gQ1B
e1  f0 : ~
gQ1B

the CZ- gate
• qutrit-resonator interaction
g1  e0 : gQ1B
dQ B  0
~
e1  f0 : gQ1B
1
semi-resonant

dQ B  0
1
resonant
the CZ- gate
• two-qubit CZ-f gate
dQ B  0
1
semi-resonant
dQ B  0
1
resonant
the CZ- gate
• two-qubit CZ-f gate
dQ B  0
1
semi-resonant
control
1

0
Uˆ  
0

0





0 0 e i  1
0 0
1 0
0 1
0
0
0
target
dQ B  0
1
resonant
THEORY:
F. W. Strauch et al., Phys. Rev. Lett. 91, 167005
(2003)
G. Haack,…, M.M.,... et al., Phys. Rev. B 82, 024514
(2010)
EXPERIMENT:
L. DiCarlo et al., Nature (London) 460, 240-244 (2009)
T. Yamamoto ,…, M.M.,... et al., Phys. Rev. B 82,
184515 (2010)
the CZ- gate
• CZ- gate truth table
gg
0
ge
eg
0
0
ee
π
control
1

0
Uˆ  
0

0





0 0 e i  1
0 0
1 0
0 1
0
0
0
target
dQ B  0
1
semi-resonant
dQ B  0
1
resonant
the CZ- gate
• two-qubit CZ-f gate
dQ B  0
1
semi-resonant
dQ B  0
1
1

0
Uˆ  
0

0





0 0 e i  1
0 0
1 0
0 1
0
0
0
resonant
the CZ- gate
• two-qubit CZ-f gate
1

0
Uˆ  
0

0





0 0 e if 
0 0
1 0
0 1
0
0
0
dQ B  0
1
semi-resonant
dQ B  0
1
1

0
Uˆ  
0

0





0 0 e i  1
0 0
1 0
0 1
0
0
0
resonant
M. Mariantoni et al.,
Science 334, 61 (2011)
the CZ- gate
• two-qubit CZ-f gate
dQ B  0
1
semi-resonant
dQ B  0
1
1

0
Uˆ  
0

0





0 0 e i  1
0 0
1 0
0 1
0
0
0
resonant
M. Mariantoni et al.,
Science 334, 61 (2011)
the CZ- gate
• f-meter: Generalized Ramsey (a)
the CZ- gate
• f-meter: Generalized Ramsey (a)
i. compensate dynamic phase
ii. varying zcmp  Ramsey fringe
the CZ- gate
• f-meter: Generalized Ramsey (b)
the CZ- gate
• f-meter: Generalized Ramsey (a-b)
the CZ- gate
• f-meter: Generalized Ramsey (a-b)
the CZ- gate
• f-meter: Generalized Ramsey (a-b)
f = 0.01
f = /2
f=
the CZ- gate
• process tomography
fidelity ~70%
fidelity ~60%
qubit T1~500 ns, T2~150 ns
superconducting surface code
A.G. Fowler, M. Mariantoni, J.M.
Martinis, and A.N. Cleland, Phys.
Rev. A 86, 032324 (2012)
~ 50 pages of details
2D lattice with nearest neighbor
interactions
A.G. Fowler, M. Mariantoni, J.M.
Martinis, and A.N. Cleland, Phys.
Rev. A 86, 032324 (2012)
~ 50 pages of details
surface code
• data and syndrome qubit
data
syndrome → measured
surface code
• face and vertex
A.Yu. Kitaev, Annals of Physics
303, 2 (2003)
stabilizers
• Z-stabilizer
1
2
3
4
stabilizers
• Z-stabilizer
1
 zeroing gate
1
3
2
g
3
2
4
4
stabilizers
• Z-stabilizer
1
 projects
1
3
2
MˆI z
3
2
4
4
 1
Zˆ1Zˆ2Zˆ3Zˆ 4   Z1234   Z1234  
 1
stabilizers
• X-stabilizer
1
 projects
1
3
2
M
Hˆ z
3
2
4
4
 1
Xˆ 1Xˆ 2 Xˆ 3 Xˆ 4   X1234   X1234  
 1
stabilizers
one qubit
𝐼 + 𝜀𝑋
𝐼 + 𝜀𝑍
detected by repeatedly measuring the qubit
with combined 𝑋 and 𝑍 measurements
𝑋, 𝑍 ≠ 0
 qubit state destroyed
stabilizers
pair of qubits, 𝑎 and 𝑏
𝐼𝑎 + 𝜀𝑋𝑎 , 𝐼𝑏 + 𝜀𝑋𝑏
𝐼𝑎 + 𝜀 𝑍𝑎 , 𝐼𝑏 + 𝜀𝑍𝑏
detected by repeatedly measuring the pair of qubits
with 𝑋𝑎 𝑋𝑏 and 𝑍𝑎 𝑍𝑏 measurements
stabilizers
pair of qubits, 𝑎 and 𝑏
𝐼𝑎 + 𝜀𝑋𝑎 , 𝐼𝑏 + 𝜀𝑋𝑏
𝐼𝑎 + 𝜀 𝑍𝑎 , 𝐼𝑏 + 𝜀𝑍𝑏
detected by repeatedly measuring the pair of qubits
with 𝑋𝑎 𝑋𝑏 and 𝑍𝑎 𝑍𝑏 measurements
stabilizers
𝐼𝑏 + 𝜀𝑋𝑏
error detection event
BUT
cannot be distinguished from an
𝐼𝑎 + 𝜀 𝑋𝑎 error!
stabilizers
• quiescent state
+1
-1
stabilizers
• quiescent state
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
+1
-1
-1
+1
+1
+1
-1
quantum error detection
time
• quiescent state
-1
+1
-1
+1
-1
+1
-1
+1
quantum error detection
• quantum error detection
time
 bit-flip error Xˆ 2
+1
-1
+1
Xˆ 2 -1
-1
+1
-1
+1



Zˆ1Zˆ 2Zˆ3Zˆ 4 Xˆ 2    Z1234 Xˆ 2 

quantum error detection
• protected memory
 bit-flip error Xˆ 2
+1
-1
+1
+1
-1
-1
Xˆ 2 -1 +1
+1
+1
+1
-1
Zˆ1



Zˆ1Zˆ2Zˆ3Zˆ 4 Xˆ 2    Z1234 Xˆ 2 
+1
-1
-1
 phase-flip error Zˆ1



Xˆ 1Xˆ 2 Xˆ 3 Xˆ 4 Zˆ1    X1234 Zˆ1 
-1
time

1)
2)
3)

any error Uˆ e
boundaries
measurement errors
“minimum weight matching”
→ polynomial

error chains and logical qubits
• Xˆ error on first qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
+1
-1
-1
+1
+1
+1
-1
error chains and logical qubits
• Xˆ error on first qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
+1
+1
+1
-1
error chains and logical qubits
• Xˆ error on second qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
+1
+1
+1
-1
error chains and logical qubits
• Xˆ error on second qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
-1
-1
-1
+1
+1
+1
-1
error chains and logical qubits
• Xˆ error on third qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
-1
-1
-1
+1
+1
+1
-1
error chains and logical qubits
• Xˆ error on third qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
+1
-1
+1
+1
-1
+1
-1
+1
-1
-1
+1
+1
+1
-1
error chains and logical qubits
• Xˆ error on fourth qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
+1
-1
+1
+1
-1
+1
-1
+1
-1
-1
+1
+1
+1
-1
error chains and logical qubits
• quiescent state
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
+1
-1
-1
+1
+1
+1
-1
error chains and logical qubits
• Xˆ error on last qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
+1
-1
-1
+1
+1
+1
-1
error chains and logical qubits
• Xˆ error on third qubit
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
+1
-1
-1
+1
+1
+1
-1
error chains and logical qubits
• back to original quiescent state
-1
-1
+1
+1
-1
+1
-1
+1
+1
-1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
-1
+1
-1
-1
-1
-1
-1
+1
+1
-1
+1
-1
+1
-1
-1
+1
+1
+1
-1
fault tolerance – surface codes
A.G. Fowler et al., Phys. Rev. A 86, 032324 (2012)
• physical → logical qubit = error rate 10-14
fault tolerance – surface codes
• physical qubits → logical qubit = error rate 10-14
1) nearest neighbor interactions

2) CNOT physical gates
• fast ~ 100 ns
• with F ≥ 99 %
3) readout
• fast ~ 100 ns
• with F ≥ 90 %
4) interface with classical electronics
5) overhead
• proof-of-concept → 3/5 physical qubits
• quantum memory → 1010 = 102 physical qubits
• Shor → 103 to 104 physical qubits
fault tolerance – surface codes
1)
CNOT physical gates
•
Xmon lifetime T1 ~ 50 ms
•
Tgate ~ 50 ns = 0.05 ms
→ F  exp(- Tgate / T1)
= exp(- 0.05 ms / 50 ms) ~ 99.9 %
Xmon
R. Barends et al., Phys. Rev. Lett.
111, 080502 (2013)
R.
A. Barends
Megrant et al., App. Phys. Lett.
99,
100,113507
113510(2011)
(2012)
fault tolerance – surface codes
• physical qubits → logical qubit = error rate 10-14
1) nearest neighbor interactions
2) CNOT physical gates
• fast ~ 100 ns
• with F ≥ 99 %


3) readout
• fast ~ 100 ns
• with F ≥ 90 %
4) interface with classical electronics
5) overhead
• proof-of-concept → 3/5 physical qubits
• quantum memory → 1010 = 102 physical qubits
• Shor → 103 to 104 physical qubits
fault tolerance – surface codes
2)
readout
•
readout time r ~ 100 ns
•
F > 90 %
R. Vijay et al., Nature (London) 490, 77 (2012)
fault tolerance – surface codes
• physical qubits → logical qubit = error rate 10-14
1) nearest neighbor interactions
2) CNOT physical gates
• fast ~ 100 ns
• with F ≥ 99 %
3) readout
• fast ~ 100 ns
• with F ≥ 90 %



4) interface with classical electronics

5) overhead
• proof-of-concept → 3/5 physical qubits
• quantum memory → 1010 = 102 physical qubits
• Shor → 103 to 104 physical qubits
perspective
•
surface code proof-of-concept
→ 3/5 physical qubits
→ 3-4 years
•
quantum memory
→ 102 physical qubits
→ 8-9 years
•
Shor to factor a 2000 bit number in 24 h with 1 nuclear
power plant
→ 300106 physical qubits
 on the best classical super-cluster: many times the age
of the universe and virtually infinite power
resonator-qubit 2D lattice
Z
R
Q,|
B
…
A,|g
R
B
A,|g
R
unit cell
…
…
…
resonator-qubit 2D lattice
 higher isolation → OFF coupling
Q,|
B
A,|g
B
Q,|
…
…
…
…
resonator-qubit 2D lattice
R
Q,|
B
 higher isolation → OFF coupling
A,|g
 encoding → multiple
measurement
R
B
R
A,|g
…
…
…
resonator-qubit 2D lattice
Z
 higher isolation → OFF coupling
Q,|
 encoding → multiple
measurement
 zero
leakage to third state
…
…
…
resonator-qubit 2D lattice
Z
R
Q,|
B
 higher isolation → OFF coupling
A,|g
 encoding → multiple
measurement
R
B
R
 zero and/or store
A,|g
…
…
…
see also D.P. DiVincenzo, Phys.
Scr., T 137, 014020 (2009)