Transcript Outline - uni

Two electron spin qubits in
GaAs quantum dots
Hendrik Bluhm
Harvard University
Experimental results presented mostly reflect work in the Yacoby and Marcus groups at Harvard.
Quantum computing – the goal
Principles of quantum mechanics
 Built-in parallelism
 Exponential speedup (for some problems)
Classical bits
Quantum bits
0 or 1
|0 + b |1
N bits => 2N states 0, 1, …, 2N-1
N qubits: 2N dimensional
Hilbert space
|0, |1, …, |2N-1
2
The case for spin qubits
Quantum computing needs two level systems
Spins natural choice
Compatible with semiconductor technology
 Potential for scalability
Why not charge?
• Charge couples to phonons, photons,
other charges, cell phones, …
Now: Intel Pentium i7-980X
Future: Quantum i2
• Spins are very weakly coupled to other things
e.g.: Electric vs. magnetic dipole transitions
(Reason: lack of a magnetic monopole)
Reason for weak coupling
• Time reversal symmetry enforces degeneracy at B = 0
(Kramer’s doublets)
=> no dephasing from electric fields
• Matrix elements for decoherence cancel to lowest order
(Van Vleck cancellation)
Decoherence times (bulk)
• P- donor electrons in 28Si: T2 = 600 ms
Tyryshkin et al., (unpublished ?)
• 29Si nuclei in purified 28Si: T2 = 25 s at RT
Ladd et al., PRB 71, 014401 (2005)
Problem: Single spins difficult to control
Two electron spin qubits
Idea: use two spins for one qubit
 Electrically controllable exchange interaction
• Tunable electric coupling
• Fast, convenient manipulation
• Relies on same techniques as single-spin
GaAs qubits in quantum dots (Lars Schreiber)
Longest coherence time of all electrically
controllable solid state qubits.
Outline
Lecture I
• Motivation
• Encoded qubits
• Physical realization in double quantum dots
• Principles of qubit operation
• Single shot readout
Lecture II
• Decoherence
• Hyperfine interaction with nuclear spins
• Recent progress on extending coherence
Outline
Motivation
Encoded qubits
Physical realization in double quantum dots
Principles of qubit operation
Single shot readout
Requirements for qubits
DiVincenzo Criteria for a viable qubit
1. Well-defined qubit
2. Initialization
3. Universal gates
4. Readout
5. Coherence
Encoded qubits
• Qubit = coherent two level system
=> single spin ½ most natural qubit
1
0
• Any 2D subspace of a quantum system can
serve as a qubit.
Qubit
subspace
Advantages
+ Wider choice of physical qubits
+ Decoherence “free” subspace – choose states that are
decoupled from certain perturbations
+ Reduced control requirements – choose subspace with
convenient knobs.
Caveats:
- Leakage out of logical subspace can cause additional errors.
- More complex control sequences.
S-T0 qubit using two spins
Idea: Encode logical qubit in two spins
All spin states:




S 


T0


1
  
2
1

  
2
T   , T  
m=0
logical
subspace
m = 1
Decoherence “free” subspace (DFS)
m = 0 for both logical states
 no coupling to homogeneous magnetic field
 insensitive to fluctuations
Simplified operation
Use exchange coupling between two spins
=> no need for single spin rotations.
Theoretical proposal: J. Levy, PRL 89, 147902 (2002)
Bloch Sphere
1
  0  b 1
0 i 1
2
0 1
0 1
2
2
0 i1
Mixed states are
statistical mixtures
of pure states and
can be inside the
Bloch sphere.
1 1

2
0
• Any pure state of a qubit corresponds
to a point on the surface of a sphere.
• They can be identified with the
direction of a spin ½.
0 0
  1/ 4 0 0  3 / 4 0 0
Single qubit operations
• Unitary transformations are rotations on the Bloch sphere
• Universal quantum computing requires arbitrary rotations, which
can be composed from rotations around two different axis.
1


z
x
0
 x  i y 
1  z

H   iˆ i  
 z 
2   x  i y
i x, y , z
Standard Rabi control
• Modulate x resonant with z.
(e.g. AC magnetic field for spins)
• Changing phase of AC signal changes
rotation axis in the rotating frame.
Gate operations
1) In field gradient:
=>

and

g B DBz
H
z
2
acquire relative phase
B2
B1
DBz = B1 – B2
2) Exchange: H 
J
J
s1  s1   x
2
2
=> mixing between

and
J


DBz
T0
J

S
Single spin vs. S-T0
Single spin qubit
Two-spin encoded qubit


DBz
Bz
Bx

• Typically uses resonant
modulation of Bx.
• Bx can be an effective field
(e.g. spin-orbit).
T0
J
S

Typically relies on switching of J
Two-qubit gates
SWAP
• Quantum computing requires (at least) one
entangling gate between two (or more)
qubits (cNOT, cPHASE, SWAP ).

• Single spins: p/2 exchange
provides SWAP
J
  i 
• Encoded qubits: construct gates
from several steps.
• S-T0: Construction of nAND gate,
equivalent to cNOT, cPHASE
• In practice, can also use Coulomb interaction
to implement cPHASE gate directly.
2

nAND gate for S-T0 qubit
Evolve in field
gradient (p/2)
Evolve in field
SWAP inner
spins (exchange) gradient (p/2)
Spin 1A
B1
B1
Spin 1B
B2
B2
Spin B1
B1
B1
Spin B2
B2
B2
SWAP inner
spins
Qubit A
Qubit B
Principle of operation:
0 1   
0 0   
Initial state
Acquire phase
 
Acquires phase
 
 
No phase acquired
 
Outside logical subspace!
Return to subspace
Exchange-only with three spins ½
Idea: use m = ½ subspace.
J1(t)
J2(t)
Single qubit:
4 steps
Two qubit:
27 steps
• No magnetic field required.
• Uses only exchange.
DiVincenzo et al. Nature 408, p. 339 (2000)
• Experimental status: Suitable samples developed,
but no coherent control yet. (Gaudreau et al. arxiv)
J1(t)
J2(t)
Tradeoffs summary
Encoding a qubit in several spins reduces control
requirements at the expense of complexity.
Spins/qubit
1
2
Static control
requirement
Magnetic field Magnetic field None
difference
AC control
requirement
(effective)
Exchange
transverse
magnetic field
Exchange
Mechanism
for 2-qubit
gate
Exchange (or
dipolar)
Exchange
(or Coulomb)
Exchange
# of steps in
2-qubit gate
1
3-6
19
(experimentally most difficult step in red)
3
Outline
Motivation
Encoded qubits
Physical realization in double quantum dots
Principles of qubit operation
• Theory of operation
• Experimental procedures
Single shot readout
2D-electron gas (2DEG)
Wafer surface
GaAs heterostructure
conduction band
edge
Dopants induce
electric field
Step at material
interface
• Structure grown layer by layer with
Molecular Beam Epitaxy (MBE)
• Atomically smooth transitions
• Ultra-high purity
Electrons in triangular
confining potential occupy
lowest subband.
Device fabrication
Fabrication
+
Negative gate voltage pushes electrons away.
2DEG
500 nm
Metal gate
Goal:
trap two
electrons
- +
V
Graphics: Thesis L. Willems van Beveren, TU Delft
Understanding a complex system
Metal gates
90 nm
+
+ + + + + + + + + + + +
Dopants, defects
and impurities
cause disorder
2D electron gas
(Fermi-sea)
Individual
confined
electrons
Conduction band edge
Electrostatic
potential from
gates
First realization and overview of experimental toolbox: Petta et al., Science 309, p. 2180 (2005)
Charge control
E
500 nm
- +
-V
S(0, 2)
Metal gate
2DEG
- +
+V
- +
V0
e<0
0
e>0
(1, 1)
V(x)
e
(0, 2)
x
e  E1, 1E0, 2  V
-V
+V
Charge control
E
500 nm
- +
V
- +
V
e<0
(1, 1)
(0, 2)
0 
e / 2

H  
 0  e / 2
0
e>0
(1, 1)
V(x)
e
(0, 2)
x
e  E1, 1E0, 2  V
-V
+V
Singlet-Triplet splitting in (0,2)
First excited state 0 E
S-T splitt.
Ground state 0
0
(1, 1)
S(0, 2)
T(0, 2)
0
0
e / 2



H  0
e / 2
0

 0

0

e
/
2




e
(0, 2) states:
Spin singlet: (x1, x2) = 0(x1) 0(x2)|S>
Spin triplet:
(x1, x2) = (0(x1) 1(x2)-0(x2) 1(x1))|T>
(0, 2) Triplet has higher energy
than (0, 2) Singlet.
Tunnel coupling
E
S(0, 2)
Tunnel coupling
J(e)
T(1, 1)
S(1, 1)
S(0, 2)
0
0 
e / 2


H  0 e /2
tc 
 0

t

e
/
2
c


T
->
S
 J (t ) 0 


 0 0
0
e
Tunnel coupling
Avoided crossing for singlet
Triplet crossing at larger e can be ignored.
Conveniently described in terms of J(e)
Zeeman splitting
T0

1
  
2
1

  
2
S 



E
S(0, 2)
m=0
T  
m=1
T  
m = -1
Ez = g B Bext
0
H Z  g * B Bz Sˆz
Bz ~ 10 mT to 1 T
e
Qubit states
S 


T0


1
  
2
1

  
2
T   , T  
E
S(0, 2)
Ez = g B Bext

T0
S

0
e
Qubit dynamics with field gradients
E
S(0, 2)
Transitions between
S and T+ driven by
DB.
J(e)
0
e << 0: Free precession
BextDBz/2

e
DBz
e ~< 0: Coherent exchange
T0
J

S
Effective Hamiltonians
T0
S
 J (t ) DBz / 2 

J , DBz << Bext : H  
0 
 DBz / 2
In logical subspace:
S
T-
All spin states:
H=
Coish and Loss, PRB 72, 125337
T0
T+
Outline
Motivation
Encoded qubits
Physical realization in double quantum dots
Principles of qubit operation
• Theory of operation
• Experimental procedures
Single shot readout
Isolating two electrons
# electrons in each dot
G
(1, 0)
(nL, nR)=(1, 1)
VL
Gqpc
Gqpc
- +
VL
- +
VR
(0, 0)
10 mV
Conductance depends on
electric field from electrons
VR
(1, 1)
V(x)
(0, 2)
(0, 1)
(0, 2)
2 mV
(1, 2)
VL
(1, 1)
x
VL
VR
VL
VR G
qpc
(0, 2)
(0, 1)
VR
Tuning the tunnel coupling
Measure current through double dot
10
VL
VSD =0.4 mV
0
I
20
2 mV
VGateR
Isd (pA)
VL
Gqpc
- +
VL
- +
VR
VL
(1, 2)
(1, 1)
(0, 2)
(0, 1)
VR
2 mV
VGateR
Gqpc
Magnitude and variation of current and
charge signal reveal tunnel couplings.
Target: tc ~ 20 eV
Tunneling rate to leads ~ 100 MHz
Pulsed Measurements
S
(1, 1)
M
R
(0, 2)
Gqpc
1 ns gate control
Typical pulse cycle for qubit operation
1) Initialize S at reload point R.
(1, 1)
V(x)
(0, 2)
2) Manipulate (nearly) separated
electrons (S)
x
Q
3) Return to M for measurement.
Readout
E
S(0, 2)
Goal: distinguish S and T
state of separated electrons.
Mechanism:
•Increase e.
•(1, 1)S adiabatically
transitions to (0, 2).
e
0
•Life time long enough to
detect charge signal.
X
S
•T stays in (1, 1)
(metastable).
T0
Q
Q
Johnson et al., Nature 435, p. 925 (2005)
Readout region and Initialization
S
(1, 1)
Region in which (1, 1)T
is long lived
(Spin Blockade)
e
(1, 1)
M
(0, 2)
R
Outside blocked region, (1, 1)
can decay to lead.
E
(0, 2)
Gqpc
Initialization of S at reload point R after
a measurement:
S(0,
2)
If in (0, 2)S, nothing happens.
(1, 1)T -> (0, 1) -> (0, 2)S via exchange with leads.
0
e
Duration ~ 100 ns.
High fidelity due to large S-T splitting
Outline
Motivation
Encoded qubits
Physical realization in double quantum dots
Principles of qubit operation
Single shot readout
Single shot readout
For many experiments, can average signal over many pulses.
• No high readout bandwidth required.
• Reduce noise by long averaging.
=> Can use standard low-freq lock-in measurement with roomtemperature amplification to measure GQPC.
Minimum averaging: 30 ms, 3000 pulses.
Single shot readout
Determine qubit state after each single pulse with high fidelity.
Benefits and applications:
• Quantum error correction.
• Verify entanglement through correlations and Bell inequalities.
• Fundamental studies (e.g. projective measurement)
• Fast and accurate data acquisition.
RF-reflectometry
Demodulation
Goal: increase bandwidth and sensitivity of
charge readout with RF lock-in technique.
Reilly et al., APL 91, 162101 (2007)
Reflected signal
RF components 50
W, sensor 50 kW
=> Impedance
matching with LC
resonator.
Excitation
Low noise cryogenic
amplifier
Single shot readout
Sensor signal
Histogram of cycle-averages
Reinitialization and
manipulation of qubit
=> random new state
Averaging window
(s scale)
Need to distinguish state before the
metastable triplet can decay (s scale).
Barthel et al., PRL 103 160503 (2009)
• Each peak corresponds to
one qubit state.
• Broadening due to
(amplifier) readout noise.
Improvement with quantum dot sensor
Quantum point contact
Quantum dot
(single electron transistor)
Qubit state modulates
single tunnel barrier.
Modulation of ability to add
electron to island
Quantum dot
Factor 3 increase in
sensitivity
=> factor 10 reduction
in averaging time.
Peaks need to be well
separated to distinguish
states.
Barthel et al., PRB 81 161308(R), 2010
QPC
Readout summary
• Qubit is read out by
spin-to-charge conversion
utilizing spin blockade.
• State is read using a
charge sensor before the
metastable (1, 1)T decays.
X
S
T0
Q
• RF reflectometry allows single shot readout
• Fidelity > 90 %
Q
Measuring coherent exchange
(gate voltage)
e
Exchange pulse
initialize
evolve
readout
(0, 2)
(1, 1)
t
t
Petta et al., Science 2005
E

S
T0
J(e)
J
S(0, 2)
e

Decay reflects dephasing due
to electric noise.
Exchange echo
(gate voltage)
e

initialize
evolve
t/2
readout
(0, 2)
t/2  Dt
t
p
p
DBz
(1, 1)
DBz - rotation
Echo signal
T0
S
J

T2 = 1.6 s
t = 2 s
Coherence times
x CPMG
 Hahn-echo
All data fitted with ~1 nV/Hz1/2 white noise with
3 MHz cutoff.
Consistent with expected Johnson noise in DC
wires => improvement with filtering.
Outline
Lecture I
• Conceptual and theoretical background
• Physical realization and principles of qubit operation
• Single shot readout
Lecture II
• Decoherence
• Hyperfine interaction with nuclear spins
• Recent progress on extending coherence
Main results
• Used qubit as quantum
feedback loop to suppress
nuclear fluctuations and
enhance T2*.
• Detailed picture of bath
dynamics and decoherence
from echo experiments.
• T2  200 s achieved with
quantum decoupling.
• Universal control.
Outline
Background
• Error correction
• Decoherence
• Hyperfine interaction
Measuring and manipulating the nuclear hyperfine field
Universal control
Reduction of nuclear fluctuations via 1-qubit feedback loop
Coherence with echo and dynamic decoupling
Decoherence vs. control – the challenge
• Qubits are analog
=> small errors matter
• Using phase
=> Uncertainty relation forbids any leakage of information
However:
• Need to manipulate qubit
• Qubits have to interact
• Eventually want to measure qubit
 need extremely tight control over interactions.
Impossible? – not quite.
“Only” need ~102 - 106 coherent operations per error
with quantum error correction.
Threshold theorem
Small enough error probability per gate operation
=> error correction can make QC fault tolerant without
exponential overhead.
Basic idea:
• Encode logical qubits redundantly in several physical qubits,
e.g. |1L = |111, |0L = |000.
• Can detect errors that leave the logical subspace
=> encoded information is not extracted.
• Correct errors if detected.
Hurdle: Error correction operations will be subject to errors themselves.
Solution:
• (Error probability) x #(physical gate operations per logical gate) < 1
=> reduce error by hierarchically concatenating error correction codes (i.e. using
the logical qubits of on level as the physical qubits of the next higher level).
Steane Code
1
 0000000  1010101 0110011 1100110  0001111 1011010  0111100  1101001
8
1
 1111111 0101010  1001100  0011001 1110000  0100101 1000011 0010110
1L 
8
0L 
Ancilla
qubits
7 physical
qubits
encoding
a logical
qubits
(from Nielsen and Chuang)
Measurement
indicating if
and what error
occurred.
Decoherence
Decoherence = loss of information stored in a qubit.
Classical picture of environment: Fluctuation of Hamiltonian
Quantum mechanical picture: Entanglement with environment.
1
Decoherence turns pure states
into mixed states
=>  goes into Bloch sphere.
0
Energy relaxation
1
1
E01
• Corresponds to classical bit flip error
• Due to noise at f = E01/h
• Timescale T1
0
0
•Practically not important for spins in GaAs
•Measured T1 in GaAs up to 1 s (Amasha et al., PRL 100, 046803 (2008))
Dephasing
= Loss of phase information due to variation of E01.
T2: true decoherence from fast,
uncorrelated noise. Needs to be weak
enough to enable error correction.
1
T2* : broadening from slow fluctuations
(or ensemble measurements).
Long temporal correlations
help to remove it.
0
Rough measure of error probablility:
Duration of operation/Coherence time.
(exact only for exponential decay from Markovian (unstructured)
bath, otherwise misleading.)
Noise sources
Noise limits measurements and causes decoherence
and gate errors.
Local environment
Fluctuating spins (electron, nuclear)
Phonons
Charge traps
Superconducting vortices.
Relevance for GaAs spin qubits
Dominant source of decoherence
?
Wafer dependent
None
Electrical noise
Pulse generator, voltage sources
Interference
Johnson noise from resistors
Generally avoidable
(but devil in the details).
Some work to be done.
Hyperfine basics
Confined s-band electron in GaAs
 (x )
50 nm
N ~ 106 nuclei
B


m  N I
m=nIA
B = n I / L = m/V
 m (xj)
2


2
H   B( x)  s  ( x)
 
2
 A I j  s  ( x j )
j
 
  Aj I j  s
j
Electron feels an effective magnetic field.
Typical magnitude = A / N1/2 ~ 2 mT.
Fluctuations of this field cause decoherence.
Nuclear dynamics
Flip-flops: 100 s
(Dipolar interaction)
Bext
Spin diffusion: 1 s – 1 min
=> Slow enough for real time probing, manipulation
Larmor precession: 0.1 – 1 s.
Dephasing : ~100 s
Bext
Outline
Background
Measuring and manipulating the nuclear hyperfine field
Universal control
Reduction of nuclear fluctuations via 1-qubit feedback loop
Coherence with echo and dynamic decoupling
Probing DBz

Bext+Bnuc,
DBz
z
T0
 t 
Sensor signal  cos2  S 
 2 
S
  g * B DBz / 
N ~106 nuclei

  
DB  BL  BR
Q
10 mT
Q (e)
 1/DBz
Typical time trace of hyperfine gradient
DBz
Data
0.55 s of data:
Fit
Manipulating Bnuc
E
T+ -> S
S(0, 2)
T  
e
T+-loading
Dmz = -1
 S 
S-loading
Dmz = +1
Quantities of interest
• Average polarization of both dots (Petta et al., Reilly et al.)
• Bi-directional real time control of gradient.

1
  
2

Effect of pumping on DBz
Apply pump pulses between measurements (typically ~106 cycles)
Real time control of DBz
S-loading
pump
T+-loading
pump
0
500
Time (s)
Steady state
when relaxation
compensates
pumping.
1000
Outline
Background
Summary of device operation
• Measure nuclear field gradient reflected in S-T0 mixing frequency
every second.
• Manipulate gradient by nuclear polarization between
measurements.
Use of gradient control
• Universal qubit control
• Reduction of nuclear fluctuations by operating
qubit as a feedback loop
Coherence with echo and dynamic decoupling
Universal single qubit gates

• Fully electrical
•• Nuclei
turned
into
Nanosecond
gate time
Foletti et al., Nature Physics 5, p. 903 (2009)
E
resource DBz
tc
• Fast (ns gate times)T
S
J
DBz 
 J
J(e)
• Fully

H   electrical
0 
 DBz
• Extrapolated
fidelity of 99.99 % at QEC threshold
0
in S , T0 basis.
Adiabatic preparation

e
Evolution
S
S



T0
S(0, 2)
Data
Data
Model
Model
T0
Dephasing due to nuclear fluctuations
Fluctuation of DBz over time
Q (e)
Precession in “instantaneous” DBz
(0.55 s acquisition time)
Q (e)
Time - average
Preparing the bath via feedback
Control and measurement faster than bath dynamics
=> Software feedback – adjust pump rate to keep DBz stable.
Fixed pumping
Feedback
200
z
gD B /h (MHz)
250
150
100
0
500
1000
1500
t (s)
2000
• Qubit measures the nuclear bath
• Qubit manipulates bath
=> let it do all the feedback!
2500
3000
Pulses with built-in feedback
smaller DBz => more pumping
=> DBz increases
E
larger DBz => less pumping
=> DBz decreases
intermediate DBz
=> stable fixpoint
S(0, 2)
Ez
e

DBz
T0
S
Singlet prob.
Fixed precession time
1
0

t
T2* enhancement and narrowing
pDBz)
Q (e)
No feedback
Q (e)
Qubit feedback
pDBz)
Operated qubit as a complete feedback loop stabilizing
its own environment and enhancing coherence.
HB et al., arxiv:1003.4031
Outline
Background
Measuring and manipulating the nuclear hyperfine field
Universal control
Reduction of fluctuations via feedback
So far: Averaging over slow fluctuations (T2*)
Coherence time and short time dynamics (T2)
• Hahn echo
• Nuclear dynamics and model
• 200 s coherence time with
Carr-Purcell-Meiboom-Gill (CPMG) decoupling
Hahn echo

DBz
T0
J
S
• Perfect refocussing for static DBz
• Decoherence reveals bath dynamics.

Dephasing during free precession
Bext+Bnuc,z
p – pulse via coherent exchange
Experiment
Data
Fits
Bext  400 mT:

Echo exp  (t / 30s)4

Mostly dipolar spin diffusion
Normalization:
1: complete refocussing, no decoherence
0: fully dephased, mixed state
Experiment
Data
Fits
Bext  400 mT:

Echo exp  (t / 30s)4

Mostly dipolar spin diffusion
Lower fields:
Periodic collapses and revivals
due to Larmor precession.
Decoherence model
Predicted by Cywiński, Das Sarma et al., (PRL,PRB 2009)
based on quantum treatment.
Intuitive picture: Yao et al., PRB 2006, PRL 2007
Classical model

Bnuc

2
B
(
t
)
nuc
Hˆ (t )  Bext Sˆ z  B (t )Sˆ z  
Sˆ z
2Bext
z
nuc
B
z
nuc
Btot 
Bext
2
z
nuc
B

Bnuc
 Bext 
2 Bext
Spin diffusion :
field independent decay

exp  (t / 35s) 4

(e.g. Witzel et al. PRB 2006)
z
Bnuc
Origin of revivals

Bnuc
2

BIsotope
Abundance
Gyromag.
ratio
nuc oscillates
due to relative
Larmor
75As
50 %
7 MHz/T
precession.
71Ga
20 %
13 MHz/T
69Ga
30 %
10 MHz/T
71Ga
69Ga
Bext
Total phase = 0 when evolving over
whole period
 Revivals
75As

Bnuc
2
Random phase otherwise
 Collapses
t/2
Dephasing of Larmor precession
(dipolar, quadrupolar shifts)
=> faster low-field envelope decay
t
Echo revivals
Fit model: average over
initial conditions. Exactly
reproduces quantum results.
Field independent fit
parameters:
#nuclei = 4.4 x 106
Spread of Larmor fields = 3 G
Spin diffusion decay time = 37 s
Data
Fits
Carr-Purcell-Meiboom-Gill (CPMG)
Hahn echo
CPMG
Init
t/2
t/2n
p
Read
t/2
t/n
…
= concatenation of Hahn echo sequences.
Prediction: Witzel et al., PRL 2007
t/n
t/2n
CPMG - data
normalized echo amplitude
B = 0.4 T
Subtracted mixed-state reference
(no p-pulses), normalize by t = 0
data.
Initial linear decay may reflect
single-spin relaxation.
Linear fit extrapolates to
t = 276 s
HB et al., arxiv:1005.2995
Summary
• Semiclassical model provides detailed understanding of
Hahn echo decay.
• Dynamic decoupling highly effective.
Figures of merit for qubit
• Memory time T2  200 s, sub-ns gates .
=> Exceeding 105 operations within T2.
• Extrapolated gate error from nuclear fluctuations ~10-4.
Future directions
Quantum computing
•
•
•
•
•
Two-qubit gates.
High fidelity gates.
Decoupled gates.
Multi-qubit devices.
Materials improvement.
Nuclear bath physics
• Interplay with spin orbit coupling
• Short time polarization dynamics
• Ultimate limit of (nuclear) decoherence?