two electron spin qubits in gaas quantum dots hendrik bluhm harvard university
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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 - PowerPoint PPT PresentationTRANSCRIPT
Two electron spin qubits in GaAs quantum dots
Hendrik BluhmHarvard University
Experimental results presented mostly reflect work in the Yacoby and Marcus groups at Harvard.
2
Quantum computing – the goal
Principles of quantum mechanics Þ Built-in parallelism Þ Exponential speedup (for some problems)
Classical bits 0 or 1
N bits => 2N states 0, 1, …, 2N-1
Quantum bitsa|0 + b |1
N qubits: 2N dimensional Hilbert space|0, |1, …, |2N-1
The case for spin qubits
Quantum computing needs two level systemsÞSpins natural choice
Compatible with semiconductor technologyÞ Potential for scalability
Why not charge?Now: Intel Pentium i7-980XFuture: Quantum i2
• Charge couples to phonons, photons, other charges, cell phones, …
• 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Tyryshkin et al., (unpublished ?)• 29Si nuclei in purified 28Si: T2 = 25 s at RT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
Þ 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 electricallycontrollable 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
• Any 2D subspace of a quantum system can serve as a qubit.
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.
1
0
Qubit subspace
S-T0 qubit using two spins
Idea: Encode logical qubit in two spins
All spin states:
TT
T
S
,
2
12
1
0
Theoretical proposal: J. Levy, PRL 89, 147902 (2002)
Decoherence “free” subspace (DFS)m = 0 for both logical states Þ no coupling to homogeneous magnetic fieldÞ insensitive to fluctuations
Simplified operationUse exchange coupling between two spins => no need for single spin rotations.
m = 0 logical subspace
m = 1
Bloch Sphere
2
10
0
1
2
10
10
2
10 i
2
10 i
• 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.
Mixed states are statistical mixtures of pure states andcan be inside the Bloch sphere.
004/3004/1
11
00
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.
zyxi zyx
yxzii i
iH
,, 2
1ˆ
0
1
x
z
Standard Rabi control• Modulate wx resonant with wz.
(e.g. AC magnetic field for spins)• Changing phase of AC signal changes
rotation axis in the rotating frame.
Gate operations
DBz
S0TJ
1) In field gradient:
=> and acquire relative phase
zzB Bg
H 2
B1B2
DBz = B1 – B2
x
JJH
22 11 ss
2) Exchange:
=> mixing between and
J
Single spin vs. S-T0
DBz
S0T
J
Bz
Bx
Single spin qubit Two-spin encoded qubit
•Typically uses resonant modulation of Bx.
•Bx can be an effective field (e.g. spin-orbit).
Typically relies on switching of J
Two-qubit gates
• Quantum computing requires (at least) one entangling gate between two (or more) qubits (cNOT, cPHASE, ).
• Single spins: p/2 exchange provides
• 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.
SWAP
SWAP
J
2
i
SWAP
nAND gate for S-T0 qubit
Qubit A
Qubit B
Spin 1A
Spin 1B
Spin B1
Spin B2
Evolve in field gradient (p/2)
Evolve in field gradient (p/2)
SWAP inner spins (exchange)
SWAP inner spins
B1
B2
B1
B2
B1
B2
B1
B2
00
10
Initial state
No phase acquired
Acquires phase
Outside logical subspace! Return to subspace
Acquire phase
Principle of operation:
Exchange-only with three spins ½
• 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)J2(t)
J1(t)Single qubit:4 steps
Two qubit:27 steps
Idea: use m = ½ subspace.
Tradeoffs summary
Spins/qubit 1 2 3
Static control requirement
Magnetic field Magnetic field difference
None
AC control requirement
(effective) transverse magnetic field
Exchange 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)
Encoding a qubit in several spins reduces control requirements at the expense of complexity.
Outline
Motivation
Encoded qubits
Physical realization in double quantum dotsPrinciples of qubit operation
• Theory of operation• Experimental procedures
Single shot readout
2D-electron gas (2DEG)
conduction band edgeWafer surface
• Structure grown layer by layer with Molecular Beam Epitaxy (MBE)
• Atomically smooth transitions• Ultra-high purity
GaAs heterostructure
Dopants induceelectric field
Step at material interface
Electrons in triangular confining potential occupy lowest subband.
Device fabrication
-+
Graphics: Thesis L. Willems van Beveren, TU Delft
Fabrication
Goal: trap two electrons
500 nm
- + V
2DEG
Met
al g
ate
Negative gate voltage pushes electrons away.
Understanding a complex system
GaAs
Al 0.3Ga 0.7
As
Si doping
+ + + + + + + + + + + +-+
Metal gates
Individual confined electrons
90 nm
2D electron gas(Fermi-sea)
Conduction band edge
Dopants, defects and impurities cause disorder
Electrostatic potential from gates
First realization and overview of experimental toolbox: Petta et al., Science 309, p. 2180 (2005)
Charge control
500 nm
V(x)
x
(1, 1) (0, 2)
S(0, 2)
T 0(1, 1
)
S(0, 2)
e
E
S(1, 1)
T +(1, 1
)
T -(1, 1
)
0 > 0 e < 0 e
= e E(1, 1)-E(0, 2) V-V +V
- + +V
- + -V
2DEG
Met
al g
ate
- + V0
Charge control
500 nm
V(x)
x
(1, 1) (0, 2)
(0, 2)
(1, 1)
e
E
0 > 0 e < 0 e
= e E(1, 1)-E(0, 2) V-V +V
- + V
- + V
2/0
02/
H
(1, 1) (0, 2)
Singlet-Triplet splitting in (0,2)
Ground state 0
First excited state 0
(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>
Þ(0, 2) Triplet has higher energy than (0, 2) Singlet.
S(0, 2)
(1, 1)
e
E
0
T(0, 2)
S-T splitt.
2/00
02/0
002/
H
(1, 1) S(0, 2) T(0, 2)
Tunnel coupling
T(1, 1)
e
E
0
Tunnel coupling
S(0, 2)
S(0, 2)
e
E
S(1, 1)
0
T(0, 2)
tunnel
coupl.
2/0
2/0
002/
c
c
t
tH
T(1, 1) S(1, 1) S(0, 2)
Tunnel couplingÞAvoided crossing for singlet
Triplet crossing at larger e can be ignored.
Conveniently described in terms of J(e)
J(e)
00
0)(tJST
->
Zeeman splitting
S(0, 2)
T 0(1, 1
)
S(0, 2)
e
E
S(1, 1)
T +(1, 1
)
T -(1, 1
)
0
Ez = g mB Bext
T
T
T
S
2
12
1
0
m = 0
m = 1
m = -1
zzBZ SBgH ˆ*
Bz ~ 10 mT to 1 T
Qubit states
TT
T
S
,
2
12
1
0
S0T
S(0, 2)
T 0(1, 1
)
S(0, 2)
e
E
S(1, 1)
T +(1, 1
)
T -(1, 1
)
tunnel
coupl.Ez = g mB Bext
0
DBz
S0TJ
Qubit dynamics with field gradients
BextBz/2
e << 0: Free precession e ~< 0: Coherent exchange
S(0, 2)
T 0(1, 1
)
S(0, 2)
e
E
S(1, 1)
T +(1, 1
)
T -(1, 1
)
J(e)
0
Transitions between S and T+ driven by DB.
Effective Hamiltonians
S T- T0 T+
H =
Coish and Loss, PRB 72, 125337
All spin states:
In logical subspace:
02/
2/)(:,
z
zextz B
BtJHBBJ
ST0
Outline
Motivation
Encoded qubits
Physical realization in double quantum dotsPrinciples of qubit operation
• Theory of operation• Experimental procedures
Single shot readout
VR
V L
10 mV
(nL, nR)=(1, 1)
(0, 0)
(1, 0)
(0, 1)(0, 2)
Gqpc
Isolating two electrons
2 mV
(0, 1) (0, 2)
(1, 1)(1, 2)
V(x)
x
(1, 1) (0, 2)
Gqpc
V L
VR
VL VRVL VR
# electrons in each dot
Conductance depends on electric field from electrons
Gqpc
- + VR
- + VL
G
Tuning the tunnel coupling
Gqpc
2 mV
V L
VGateR
Isd (pA)
2 mV
0
10
20
V L
VGateR
Gqpc
- + VR
- + VL
VSD =0.4 mV
Measure current through double dot
I
(0, 1) (0, 2)
(1, 1)(1, 2)
VR
V L
Magnitude and variation of current and charge signal reveal tunnel couplings.Target: tc ~ 20 meVTunneling rate to leads ~ 100 MHz
Pulsed Measurements
1 ns gate control
(0, 2)
(1, 1)
R
M
S
Typical pulse cycle for qubit operation1) Initialize S at reload point R.
2) Manipulate (nearly) separated electrons (S)
3) Return to M for measurement.
V(x)
x
(1, 1) (0, 2)
Q
Gqpc
Readout
S(0, 2)
T 0(1, 1
)
S(0, 2)
e
E
S(1, 1)
T +(1, 1
)
T -(1, 1
)
0
QS
Q
T0
X
Goal: distinguish S and T state of separated electrons.
Mechanism:•Increase e.
•(1, 1)S adiabatically transitions to (0, 2).
•T stays in (1, 1) (metastable).
•Life time long enough to detect charge signal.
Johnson et al., Nature 435, p. 925 (2005)
Readout region and Initialization
(0, 2)
(1, 1)
Region in which (1, 1)T is long lived (Spin Blockade)
e
Outside blocked region, (1, 1) can decay to lead.
(0, 2)
(1, 1)
R
M
S
Initialization of S at reload point R aftera measurement:
If in (0, 2)S, nothing happens.
(1, 1)T -> (0, 1) -> (0, 2)S via exchange with leads.
- Duration ~ 100 ns.- High fidelity due to large S-T splitting
S(0, 2)
T 0(1, 1
)S(0, 2)
e
E
S(1, 1)
T +(1, 1
)T -(1
, 1)
0
Gqpc
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 room-temperature amplification to measure GQPC.Minimum averaging: 30 ms, 3000 pulses.
Single shot readoutDetermine 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
Goal: increase bandwidth and sensitivity ofcharge readout with RF lock-in technique.Reilly et al., APL 91, 162101 (2007)
RF components 50 W, sensor 50 kW=> Impedance matching with LC resonator.
Low noise cryogenic amplifier
Exci
tatio
n
Refle
cted
sig
nal
Demodulation
Single shot readout
Barthel et al., PRL 103 160503 (2009)
Sensor signal
Reinitialization and manipulation of qubit=> random new state
Averaging window(ms scale)
Histogram of cycle-averages
• Each peak corresponds to one qubit state.
• Broadening due to (amplifier) readout noise.
Need to distinguish state before the metastable triplet can decay (ms scale).
Improvement with quantum dot sensor
QPC
Quantum dot
Quantum point contactQubit state modulates single tunnel barrier.
Quantum dot(single electron transistor)Modulation of ability to add electron to island
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
Readout summary
QS
Q
T0
X
• 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.
• RF reflectometry allows single shot readout
• Fidelity > 90 %
Measuring coherent exchange
t
initialize readout
t
(gat
e vo
ltage
)
e
(1, 1)
(0, 2)evolve
JS0T
Petta et al., Science 2005
Exchange pulse
Decay reflects dephasing due to electric noise.
S(0, 2)
T 0(1, 1
)
e
E
S(1, 1)
J(e)
Exchange echo
/2 + tDt
/2t
initialize readout
p t(gat
e vo
ltage
)e
(1, 1)
(0, 2)evolve
J
DBz
S0T
DBz - rotation
p
Echo signal
t = 2 ms
T2 = 1.6 ms
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 ms 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
Þ 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
(from Nielsen and Chuang)
001011010000110100101111000000110011001100010101011111118
11
110100101111001011010000111111001100110011101010100000008
10
L
L
7 physical qubits encoding a logical qubits
Ancilla qubits
Measurement indicating if and what error occurred.
Decoherence
Decoherence = loss of information stored in a qubit.
Classical picture of environment: Fluctuation of HamiltonianQuantum mechanical picture: Entanglement with environment.
1
0
Decoherence turns pure states into mixed states=> Y goes into Bloch sphere.
Energy relaxation
• Corresponds to classical bit flip error• Due to noise at f = E01/h • Timescale T1
1
0
1
0
E01
•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.
T2* : broadening from slow fluctuations
(or ensemble measurements). Long temporal correlations help to remove it.
Rough measure of error probablility:Duration of operation/Coherence time. (exact only for exponential decay from Markovian (unstructured)
bath, otherwise misleading.)
1
0
Noise sources
Noise limits measurements and causes decoherence and gate errors.
Local environmentFluctuating spins (electron, nuclear)PhononsCharge trapsSuperconducting vortices.
Electrical noisePulse generator, voltage sourcesInterferenceJohnson noise from resistors
Generally avoidable(but devil in the details).Some work to be done.
Relevance for GaAs spin qubitsDominant source of decoherence?Wafer dependentNone
Hyperfine basics
50 nmN ~ 106 nuclei
2)(x
Confined s-band electron in GaAs
Im N
m = n I AB = n I / L = m/V m d(xj)
jjj
jjj
sIA
xsIA
xsxBH
2
2
)(
)()(
B
Electron feels an effective magnetic field. Typical magnitude = A / N1/2 ~ 2 mT.Fluctuations of this field cause decoherence.
Nuclear dynamics
Flip-flops: 100 ms(Dipolar interaction)
Spin diffusion: 1 s – 1 min=> Slow enough for real time probing, manipulation
Larmor precession: 0.1 – 1 ms. Dephasing : ~100 ms
Bext
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
0.55 s of data:
N ~106 nuclei
RL BBB
2cossignalSensor 2 S
DataFit
Bext+Bnuc,z
Q
/* zB Bg
DBz
S0T
DB
z10
mT
Typical time trace of hyperfine gradient
Q (e
)
1/DBz
Manipulating Bnuc
S-loadingDmz = +1
T+-loadingDmz = -1
S(0, 2)
e
E
S(1, 1)
T +(1, 1
)
Quantities of interest• Average polarization of both dots (Petta et al., Reilly et al.)• Bi-directional real time control of gradient.
2
1S
T
T+ -> S
Effect of pumping on DBz
Apply pump pulses between measurements (typically ~106 cycles)
Steady state when relaxation compensates pumping.
1000
S-loadingpump
T+-loadingpump
0 500Time (s)
Real time control of DBz
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
DataModel
Universal single qubit gates
S(0, 2)
T 0(1, 1
)
e
E
S(1, 1)
tc
J(e)
0z
z
B
BJH
in basis.0, TS
• Fully electrical• Nanosecond gate time
J
DBz
S0T
• Nuclei turned into resource• Fast (ns gate times)• Fully electrical• Extrapolated fidelity of 99.99 % at QEC threshold
DataModel
SS
0T 0T
EvolutionAdiabatic preparation
Foletti et al., Nature Physics 5, p. 903 (2009)
Dephasing due to nuclear fluctuations
Precession in “instantaneous” DBz
(0.55 s acquisition time)
Fluctuation of DBz over time
Time - average
Q (e
)
Q (e
)
Preparing the bath via feedback
Control and measurement faster than bath dynamics => Software feedback – adjust pump rate to keep DBz stable.
0 500 1000 1500 2000 2500 3000100
150
200
250
t (s)
gB
z/h (
MH
z)
Fixed pumping Feedback
• Qubit measures the nuclear bath• Qubit manipulates bath=> let it do all the feedback!
Pulses with built-in feedback
S(0, 2)
e
E
S(1, 1)
T +(1, 1
)
Ez
DBz
S0T
smaller DBz => more pumping => DBz increases
larger DBz => less pumping => DBz decreases
intermediate DBz
=> stable fixpoint
0
1Si
ngle
t pro
b.
Fixed precession time
t
T2* enhancement and narrowing
No feedback
Qubit feedback
Q (e
)Q
(e)
p(DBz)
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 ms coherence time with
Carr-Purcell-Meiboom-Gill (CPMG) decoupling
DBz
S0TJ
Hahn echo
Bext+Bnuc,z
Dephasing during free precession p – pulse via coherent exchange
• Perfect refocussing for static DBz
• Decoherence reveals bath dynamics.
Experiment
Bext 400 mT:
Mostly dipolar spin diffusion
4)30/(expEcho s
DataFits
Normalization:1: complete refocussing, no decoherence0: fully dephased, mixed state
Experiment
Bext 400 mT:
Mostly dipolar spin diffusion
4)30/(expEcho s
DataFits
Lower fields:Periodic collapses and revivals due to Larmor precession.
Decoherence model
nucB
znucB
extB
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
ext
nucext
znuc
tot
B
BBB
B
2
2
zext
nucz
znuczext S
B
tBStBSBtH ˆ
2
)(ˆ)(ˆ)(ˆ2
znucBSpin diffusion :
field independent decay 4)35/(exp s
(e.g. Witzel et al. PRB 2006)
Origin of revivals
t
75As71Ga69Ga
nucB
2nucB
Bext
t/2
oscillates due to relative Larmor precession.
2nucB
Total phase = 0 when evolving over whole periodÞ Revivals
Random phase otherwiseÞ Collapses
Dephasing of Larmor precession (dipolar, quadrupolar shifts) => faster low-field envelope decay
Isotope Abundance Gyromag. ratio75As 50 % 7 MHz/T71Ga 20 % 13 MHz/T69Ga 30 % 10 MHz/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 GSpin diffusion decay time = 37 ms
DataFits
Carr-Purcell-Meiboom-Gill (CPMG)
Prediction: Witzel et al., PRL 2007
/t n/2t n /2t n/t n…
/2t/2tInit ReadpHahn echo
CPMG
= concatenation of Hahn echo sequences.
CPMG - data
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 ms
B = 0.4 T
norm
alize
d ec
ho a
mpl
itude
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 ms, 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?