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High-Fidelity Josephson qubit gates – winning a battle against decoherence
• “Quantum Integrated Circuit” – scalable• New breakthroughs:
Improved fidelityUniversal gates, with tomography
• 50 qubit – easy to couple
Nadav Katz
Work done while at UCSB with Prof. John Martinis and group.
Contact: [email protected]: 84133
Racah Institute of Physics Colloquium, Nov. 2007
Experimental Quantum Information Processing (QIP)
a perplexing explosion of different systems
Experimental QIP – a guide for the perplexed
Smaller
IonsNeutral AtomsNMR
Semiconductor SpinsQuantum Dots
Superconducting Circuits
Easier to isolate Easier to couple & construct
Bigger
• NMR: 2 to 7 qubits; scalability?• Ions: up to 8 qubits & scalable
• Dots: LONG T1 (T2?)• Coherent Oscillations • No dissipation
• Pretty good coherence times• Coupled qubits• Decoherence??
Goal - reach the fault tolerant threshold – F > 99.95%
The Josephson JunctionSC
SC
~1nm barrier
Silicon or sapphire substrate
Al top electrode
Al bottom electrode
AlOx tunnel barrier
Josephson junction
“Josephson Phase”2
2ie
1 2
0 / 2V
0 sin( )JI I
0 / 2h e
11
ie
Electrical notation
Idc
The Qubit (phase)
IdcI RC
Idc + C V + V / R = I.
Kirchoff’s Laws:
V
equation of motion
Controllable
“kinetic” energy potential energy
0 / 2V
0 sin( )JI I
1 2 0 / 2h e
2 2
0 0 0 00
1cos( ) 0
2 2 2 2 dcC I IR
damping
Transform to Hamiltonian rep. Quantize ( is an operator)…
Superconducting Qubits
Phase Flux Charge
104 102 1
Area (µm2): 10-100 (1) 0.1-1 0.01
Potential &wavefunction
EngineeringZJ=1/10C 30 103 105
Yale, Saclay, NEC, Chalmers
Delft, IBM, Berkeley
UCSB, NIST, Maryland, Wisconsin, Jerusalem
0 02
/ 2
/ 2J
C
E I
E e C
Our Qubit
microwave drive
Junction
Flux bias
SQUID
~ 100 microns
Idc
Qubit
Flux bias
VSQ
SQUID
Iµw
inductor
Operation of the Phase Qubit
Qubit basis states |0, |1
Tune qubit state energies E10 with dc current Idc
Control qubit states with microwave current Iµw at 10
Measure state occupation by selective tunneling
Minimize fluctuations and dissipation for qubit coherence
Idc
Qubit
Flux bias
VSQ
SQUID
Iµw
10|0
|1
01
0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
Measure Pulse amplitude (V)
Sw
itchi
ng p
roba
bilit
y
|0
|1
(1) State Preparation Wait t > 1/10 for decay to |0>
Josephson-junction qubit
dt
e 01
|0>
|1>
I = Idc + dIp(t) + Imwc(t)cosw10t + Imws(t)sinw10t
phase
pote
ntia
l
pulse height of dIp
Pro
b. T
unne
l
|0> : no tunnel
|1> : tunnel
|0>|1>
3 ns Gaussian pulse96%
mwcI
mwsI
dIp(t)(2) Qubit logic with current bias
(3) State Measurement: U(Idc+dIp) Fast single shot – high fidelity
GHz) 6 ; ;mK (20 10kT
U
ExperimentalApparatus
V source
20dB 4K
20mK
300K
30dB
I-Q switch
Sequencer & Timer
mwaves
IsIfVs
fiber optics rf filters
mw filters
~10ppm noise
V source~10ppm noise
20dB
20dB
Z, measure
X, Y
Ip
Imw
Is
Iftime
Reset Compute Meas. Readout
Ip
Imw
Vs
0 1
X Y
Z
Repeat 1000xProbability 0,1
10ns
3ns
~5 ns pulses
GHz DAC Electronics
Old analog system:
time (ns)
mw
ave
amp
litu
de
I Q
mw
14 bits, 2x Gs/sFPGA memory, ~2k$
-8 -6 -4 -2 0 2 4 6 8
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
measured waveform
Spectroscopy
Bias current I (au)
10/ U
saturate
Ip
Imw
meas.
10(I)
26
P1 = grayscale
Qubit Characterization
T2 ~350ns
Meas.time
T1 ~450ns
0 100 200 300 400 500 600
time [ns]
T~100ns
Rabi
time
x/2
time
x/2
x/2 x/2y
Ramsey
Echo
time
xlifetime
P1
0
1
0
1
Standard State Tomography (Z, Y, X meas.)
time (ns)
P1
I,X,Y
I
XY
0
1
10 10 i
Z
Y/2
State prep.
0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
Measure Pulse amplitude (V)
Sw
itchi
ng p
roba
bilit
y
|0
|1
Measurement in detail
Idc
Imw pulse
What is the quantum state after a partial measurement (p<1) ?
Question:
Full measurement (p=1) projects to either or 1 0
12sin02cos 000
2cos2sin 02
002
1 pp
p~1
p=0.5
Partial measurement evolution
Theory: A. Korotkov, UCRFollowing Dalibard et al. PRL
68, 580 (1992).
N
pe Mi 110M
Prob. = p/2tunnel out
Prob. = 1-p/2
Apply state tomography to test theory
2/tan1tan2 01 pM
2
10
Answer:
2sin
2sin112/sin102/cos
02
0200
0
pwellqubitofouttunnel
pN
pe Mi
M
M
M
2sin
2sin112/sin102/cos
02
0200
0
pwellqubitofouttunnel
pN
pe Mi
M
1
0
0 0.2 0.4 0.6 0.8 10
50
100
Partial measurement probability
Pol
ar a
ngle
M
(D
eg)
Partial measurement - results
2/tan1tan2 01 pM
0 0.5 10
500
1000
1500
2000
Measure pulse amplitude (V)A
zim
utha
l rot
atio
n
(Deg
)
But can the effect of such a partial measurement be undone?
High fidelity zrotations
Quantum erasure
Partialmeasure
Erasure
Erasure(0.9)
01
tomography & final measure
statepreparation
7 ns
partial measure p
Iw
Iz
p
t10 ns
partial measure p
p
10 ns 7 ns
x
0 1
2
i 0 1
2
Probablistic recovery of quantum state even with strong measurement
Nontrivial sequence – Very good control
Process tomography ofthe erasure (~85% fidelity)
Im Re
0.05 0.7p
Coupled Qubits
Cc
C
0110100110 C
CH c
coupling
0 0
1 0 0 1
1 1
On Resonance:
Straightforward to implement: simple coupling tunable fast readout simultaneous measurement
eg. UMaryland
Cc
0 100 200 300 4000
0.2
0.4
0.6
0.8
1
osc
(ns)
Pro
babi
lity
P10
P01
P11
0 100 200 300 4000
0.2
0.4
0.6
0.8
1
osc
(ns)
Pro
babi
lity
P10
P01
P11
0 100 200 300 4000
0.2
0.4
0.6
0.8
1
osc
(ns)
Pro
babi
lity
P10
P01
P11
Simultaneous Measure of Coupled Qubits: i-SWAP gate
)2sin(01)2cos(10
01100110
oscosc
21
21 osc
SiS
e iS
0 0
p
1 0 0 1
1 1S
i-SWAP gate
p PABA
B
tosc
z-gate
/2 z-gate
P10
P01
P11
0110 i0110 i
0110 0110 i
Eigenstate, Bell singlet
Tomography: Direct Proof of Entanglement
pA
B
p/2
state tomographyI,X,Y
I,X,Y
01 10 i
Re Im
00
10
01
11 0010
0111
00
10
01
11 0010
0111
) 0101 1010 )(2/1(incoh
) 1001 0110
0101 1010 )(2/1(
)0110)(0110(coh
ii
ii
fidelity = 0.86expect = 0.87
Process Tomography
y tomographstateswap01 i
10,10,1,010,10,1,0 ii
4 initial states / qubit
pA
B
i-swap
state tomographyI,X,Y
I,X,Y
Samples Bloch sphere enough to describe gate for ANY initial state
(i-swap)1/2 is a universal gate
16 Density Matrices:16 Density Matrices:Data Data (3 min.)(3 min.)
Process Tomography
DATA
T1 = 450nsCM = 8% CuW= 5%vis = 85%g/ππ = 20MHz
Re [] Im [] Preliminary Data
SIM
Fidelity:Tr( thy exp) = 0.427
Qubit Coherence: Where’s the Problem?
Ene
rgy
D. of States
Inductors & Junctions
Capacitors
Superconductors: Gap protects from dissipation X-tal or amorphous metal Protected from magnetic defects
2D~4Tc
eV
Circuits
Good circuit design (uwave eng.)
resonator
(X-tal) (amorphous)
Many low-E statesOnly see at low T
Qubit Improvements(dielectric loss)
P1
(p
roba
bili
ty)
1st gen.
T1 = 500 ns
tRabi (ns)
2nd gen.
3rd gen.
T1 = 40 ns
T1 = 110 ns
40%
60%
90%
No Si waferSiO2 -> SiNx
Small junction+ shunting C
(loss of SiNx limits T1)
60 m
SiNx capacitor
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
T1 = 474 ns
Wave Delay (ns)
P1
New Qubit DataP
1 (p
roba
bili
ty)
tRabi (ns)
4th gen.
T1 = 470 ns
90%
Interdigitated C – (topologically protected)
sapphire dielectric (radiation from large size?)
T ~ 300 ns
Optimistic for further dramatic improvements • We know crystals are “superinsulators”• How to fabricate?
5th gen.a-Si:H dielectric (Q ~ 40000)
T1 = 450 ns
time16 ns
X
12 ns 12 ns
swap swaphold
time16 ns
TLS
X interact with TLS
1610 1614 1618 1622
6.8
7
7.2
Flux Bias [mV]
Fre
qu
en
cy [G
Hz]
0 1 2 3 4 0
0.5
1
time [s]
0 0.5 10
0.5
1
time [s]
T1,TLS ~ 1.2s
0 50 1000
1
time [ns]
Tswap ~ 12ns
• Strong interaction with TLS (S = 40MHz)• Long-lived TLS is quantum memory
P1
P1
excite qubit off-resonancez-pulse into resonance
“on”
“off”
measure
offon
TLSoffon
Bias
Fre
quen
cyTLS Resonance – not a bug, a feature…
• On-Off coupling with change in bias
8%
Quantum Memory with Process Tomography
)Im(
)Re(
16ns16 ns
TLS
init
12 ns 12 ns
store loadmem
1 2 3
1 – InitializeCreate states over the entire Bloch sphere.
2 – StoreSwap state into TLS. Qubit now in ground state.
3 – LoadAfter holding for 16ns, swap again to retrieve state from TLS.
Process tomography:identity operation dominates process
Fidelity:Tr(thmeas)
= 79%
New Frontier: 50 atoms
• “Atom” with 50 W impedance|Zqubit| =1/w10C
Zqubit ()
11K1M
phasequbitQ F
atoms
377 50
Z mismatch makes coherence easier
Z match makes coupling easier
Error threshold
Unlimited range 10-3 – 10-4
2D lattice nearest-neighbor 10-5
1D lattice nearest-neighbor 10-8
Architecture
• 50 enables long distance coupling Much better error threshold !
Future Prospects
•Demonstrated basic qubit operations
Initialize, gate operations, controlled measurement
10 to 100 logic operations
Tomography conclusively demonstrates entanglement
•Decoherence mechanism understood
Optimize dielectrics, expect future improvements
Problem is NOT (only) T1 !!
•Future: tunable coupling, CNOT gate with process tomography
•New designs and regimes (cavity QED and microbridges)
•Scale-up infrastructure designed (“brute force” to ~40 qubits)
Very optimistic about 4 -10 qubit quantum computer