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Tyryshkin et al., (unpublished ?)• 29Si nuclei in purified 28Si: T2 = 25 s at RT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?