Electron Spin Qubits Steve Lyon Electrical Engineering Department Princeton University
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1 Electron Spin Qubits Steve Lyon Electrical Engineering Department Princeton University Review of quantum dots (mostly GaAs/AlGaAs), with many references: Hanson, Kouwenhoven, Petta, Tarucha, Vandersypen, Rev. Mod. Phys. 79, 1217 (2007). Overview of older silicon spin qubit work: Morton, McCamey, Eriksson, Lyon, Nature, 479, 345 (2011).
2 Outline Background on Si and electrons Spin resonance measurements of spin coherence in Si GaAs quantum dot spin experiments Si quantum dot spin experiments Si donor spin experiments Other schemes (electrons on helium, )
3 Why Electron Spins? ~10 9 Qubits for full-scale Quantum Computer Need small qubits Need fast qubits Recent Ion Trap proposal (shuttle ions) 100 x 100 m 2 [Lekitsch, Sci. Adv. 3, e (2017).] = 2.5 acres
4 Surface Code (Error Correction) Every cycle 1. CNOT gates between the black dots and 4 neighbor white dots colored yellow 2. Measure state of black dot 3. Repeat 1 & 2 for green colored regions Turn off these operations in regions to define logical qubits Move (braid) these logical qubits to perform logical gates Fowler, Phys. Rev. A 86, (2012)
5 Electron spins as quantum bits Natural 2-level system (for S = ½) Spins interact weakly (magnetic dipole) long coherence (in principle) But, weak interaction difficult to manipulate spin difficult to measure spin difficult to make 2-qubit gates
6 Energies and transitions Zeeman Hamiltonian H = g B B S = E Z Often g ~ 2, and S = ½ E Z ~ 10 GHz at B=0.35T ~ 0.5K Drive spins with microwaves Typically 5 50 GHz h Driven by magnetic field of microwaves (weak) Can initialize by cooling to mk Conversions: 10 GHz ~ 450 mk ~ 41 ev E Z 1 0
7 Electron spins in semiconductors Why semiconductors? Electrons readily moved and controlled in semiconductors Huge industry exists which can manufacture chips with > devices Data from 1950 s showed long spin lifetimes (T 1 ) Semiconductors can be extremely pure So, maybe long spin coherence (T 2 )
8 Decoherence on the Bloch Sphere 0 0 T 1 = bit flip time T X 1 Z Y T 1 X 1 Z Y T 2 = phase flip time 0 Z 0 Z T X Y T 2 X Y
9 What do we know about electron spin coherence in Si? (in ~2000 assumed maximum coherence ~ 1 ms)
10 Pulsed electron spin resonance (ESR = EPR) Ensemble technique (measure ~ ) spins at once a spatial ensemble Need many spins for enough signal Often will not do time ensemble Some sort of ensemble required for coherence Often simpler sample preparation than devices (sometimes just cutting to the right size)
11 Donor impurities in silicon Si Si Si Si Si Si Si Si Si e - Si Si Si + P Si Si Si Si Si Si Si Si Si Si Si Si Ionization energy (phosphorus) ~ 50 mev Impurity Bohr radius ~ 2 nm Lowest excited state ~ 10 mev (not hydrogenic) Nuclear spin (phosphorus) = ½
12 Spin relaxation times, T 1 and T 2 (s) Spin qubits T 1 T 2 T 2 (Gordon, 1958) (Feher, 1958) T 1 ~ e -E/kT Temperature (K) DiVincenzo, Science 270, 255 (1995)
13 Electron Spin Resonance (ESR) the Laboratory Frame B 1 x z Magnet (B 0 ) y B 0 Si g B B 0 Microwaves (B 1 ) Adjust magnetic field (B 0 ) so that g B B 0 ~ h and measure microwave absorption and emission.
14 EPR Intensity Electron Spin Resonance (ESR) Put spins into magnetic field, B 0 Si g B B 0 Microwaves at h =g B B <.1% 29 Si P/cm 3 12K 29 Si 31 P 29 Si 0 29 Si e - 29 Si 29 Si -3 Natural Si 31 P 29 Si e - 29 Si 29 Si Magnetic Field (G) 31 P 29 Si e -
15 The Rotating Frame and a /2 Pulse The electrons magnetic moment is precessing about B 0 at the Larmor frequency (~10GHz) Reference frame rotating at g B B 0 spins nearly stationary In rotating frame a /2-pulse (B 1 ) rotates the magnetization 90 around the x or y axis. The B 1 pulse must rotate at the Larmor frequency
16 Spin echo (Hahn echo) Decoherence: accidents happen T 1 = spins relax (thermalize) T 2 = spins lose phase information T 2 Microwave pulses pulse = reverse /2 pulse = start Microwave pulses pulse = reverse /2 pulse = start T 1 Echo = emission of microwaves Echo = emission of microwaves
17 Spin Echo Animation
18 Magnetic Resonance Jargon /2 FID = Free Induction Decay = initial emisson T 2 * = Dephasing Time = time for different spins to go out of phase with one another in FID Refocusing pulse = the -pulse T 2 = decay time of the echo = coherence time = spin-spin relaxation time T 1 = Relaxation time = Spin flip time = spinlattice relaxation time
19 First Microwave Spin Echo Experiment Si:P Gordon & Bowers, Phys. Rev. Lett. 1, 369 (1958) Measured T 2 ~ 520 s with 28 Si:P Longest electron spin T 2 until ~2003 (?)
20 Hahn Echo Signal Hahn Echo Signal Electron Spin Echoes from Donors in Silicon Si:P 3x10 4 K 16 /cm 3 T = 4K 0.2 Si:P 3x10 16 /cm 3 T = 14K Time ( sec) Natural silicon Time ( sec) Isotopically enriched silicon
21 T 1, T 2 (sec) Donor electron spin qubits Doping ~10 16 /cm 3 Isotopically purified 28 Si:P (800ppm 29 Si) T 1 (Feher et al, 1958) T1 T (Gordon, 1958) 0.1% 29 Si T 2 Exponential: E ~ 10 mev = first excited state 1x Temperature (K)
22 Spin Echo Intensity Gordon & Bowers measured T 2 ~ 520 s with 28 Si:P Now measured it with fancier (= more expensive) equipment and get a similar T 2 What was limiting T 2? 1 8.1K Si:P Natural Si:P t (ms)
23 Global Magnetic Field Noise Single-shot Hahn echo, quadrature detection, 3 different 1-shot experiments In-phase signal Quadrature signal Strong echo signals Spins stay in phase with one another Global magnetic field fluctuations Randomized echo phase
24 2-Pulse ESE Intensity Magnitude Detection 1 28 Si:P (magnitude detection) Si:P (quadrature detection) 8.1K Natural Si:P (magnitude detection) t (ms)
25 Spin relaxation times, T 1 and T 2 (s) Coherence without Instantaneous Diffusion T 1 (Feher, 1958) Extrapolated Hahn Echo T T 2 (Gordon, 1958) ~ e -E/kT Temperature (K)
26 1/T 2 (ms 1 ) Eliminating Instantaneous Diffusion Vary power of second pulse probability to flip spin without flipping neighbors Si:P T 2 14ms T 1 = 18.5ms T 2 60ms T 1 = 280ms T = 8.1 K T = 6.9 K sin 2 ( 2 /2) /2 < -pulse echo /2 -pulse echo
27 Avogadro Project Silicon Very highly enriched (~50ppm 29 Si) Very high chemical purity (boron, phosphorus ~10 12 /cm 3 ) Large pieces of bulk Si
28 Echo Signal Intensity 30x More Lightly Doped 28 Si:P 4x10 12 P/cm 3, 8x10 12 B/cm 3, 100 ppm 29 Si K Time (s) Echo e t / T2 ( t e / T SD ) n T SD = 0.7 sec; n = 2 T 2 = 0.8 sec
29 Spin relaxation times, T 1 and T 2 (s) T 1 (Feher, 1958) Extrapolated Hahn Echo (Gordon, 1958) T 2 ~ e -E/kT Temperature (K)
30 ESR Signal Intensity 1/T 2 (s -1 ) Suppressing Donor Flip-Flops with a Magnetic Field Gradient take P/cm 3, 2K ESR lineshape with gradient Magnetic Field (Gauss) sin 2 ( 2 /2) No gradient T 2 = sec Gradient (0.3Hz/nm) T 2 = 12 7 sec
31 T 2 (s) Electron Spin Coherence in Ultra-pure Silicon - Isolated Donor T 2 in 28 Si E-3 1E-4 Physical process 0.5 ms (1958) Instantaneous diffusion 60 ms (2003) T 1 = 2000 s at 2K Donor T 1 -flips 0.6 s (2009) 10 s (2010) (2012) Donor flip-flops Material 1000ppm 29 Si 4x10 16 /cm 3 800ppm 29 Si 1x10 15 /cm 3 100ppm 29 Si 1.4x10 14 /cm 3 100ppm 29 Si 1.2x10 14 /cm 3 A.M. Tyryshkin, et al., Nature Materials, 11, 143 (2012).
32 Frequency (GHz) Clock Transitions for Si:Bi 7 GHz clock transition 5 GHz clock transition Magnetic field (Gauss)
33 Hahn Echo Intensity Electron T 2 for 28 Si Bi/cm 3, 50 ppm 29 Si f = GHz B = G T 2 = 2.7 s 4.3 K 10 T = 4.3 K Time (s)
34 Dynamical Decoupling Hahn Echo ( /2 - t - - t - echo) removes static and slow phase decoherence (frequencies < 1/t) So, just repeat Carr-Purcell = /2 - t - - t t - echo Filters noise, only allowing noise at ~ 1/t [Biercuk J.Phys.B. 44, (2011)] But, need to worry about pulse errors CP = x /2 - t - x - t - x - - x - t - echo Pulse errors echo dies off quickly CPMG = Carr-Purcell-Meiboom-Gill (fix some errors) = x /2 - t - y - t - y - - y - t - echo
35 Carr-Purcell-Meiboom-Gill Sequence Refocus spins every 130 s (train of -pulses) decouples B-noise retain coherence for > 6 msec 28 Si:P /cm 3 T 2 = 6.5 msec Train of refocused echoes: Pulse train: /2 -pulses time (msec)
36 CPMG State Fidelities Dynamical Decoupling with CPMG? Easy, just decouple noise with CPMG? No, there s no free lunch CPMG preserves only one state Y-state X-state Z-state Number of CPMG Pulses
37 Echo Intensity Better Dynamical Decoupling Sequences Sequences made from XYXY are good (for spins): Especially concatenated versions: K. Khodjasteh, PRL 95, (2005) &Wang, PRB 85, (2012) Concatenated XZXZ pulse sequence p 1 (4 pulses) p 2 (14 pulses) p 3 (60 pulses) p 4 (242 pulses) p 5 (972 pulses) T 2 = 15ms Initial State Sensitivity: XYXY sequence Time (ms)
38 Dynamical Decoupling Where are we? Dynamical decoupling is good for non-markovian noise Like 1/f noise Errors that standard quantum error correction is bad at Dynamically makes a decoherence-free subspace If spin is up and down half the time insensitive to B 0 Not known, in general, how to interleave DD and gates Gates can be performed between repetitions of sequences Must wait until concatenated sequence is complete CPMG is not useful dynamical decoupling for qubits Does not protect general state Pulse errors + dipolar interactions cause odd effects Is useful for extending coherence of single QB s, but unclear what physics is being averaged out
39 What is the limit for donor electron spin We don t know T 2 in Si? Our measurements are still limited by donor-donor interactions and residual 29 Si. In most QC architectures, the donor-donor disance would be larger than we have Expect T 2 >> 10 s. Do we need it longer? Bigger is better? Many other effects in devices surfaces and interfaces, gate voltages,
40 Nuclear Echo Intensity Nuclear T 2 in highly enriched silicon (~100 ppm 29 Si ) x10 14 P/cm K 50 T 2n = 15 s Time (s) T 2n is limited by electron spin flip-flops. Ionized donor nuclei T 2n ~ 3 hr. Useful? Bigger is better?
41 Recovered FID Intensity 2-Pulse Echo Intensity Mobile qubits? Free 2D Electron Spin Relaxation Times in Si/SiGe Heterostructure Electron Spin Echoes T = 4.2K T 1 = 1.95 ±0.14 sec Inversion Recovery T = 4.2k 0.5 T 2 = 3.06 ± 0.12 sec Time ( sec) Time ( sec)
42 Origin of the Spin Relaxation in 2D T 2 > T 1 requires anisotropic relaxation mechanism Abstractly, fluctuating B-fields cause relaxation Appears that fluctuating fields arise from Rashba effect (spin-orbit interaction from broken symmetry at interface): H R ( k E) Sˆ B R Use correlation time, t c = 10 psec (from mobility) In-plane fields (B x, B y ) = 10 Gauss e - Perpendicular (B z ) = 5 Gauss k Ē z B R k B R k
43 Outline Background on Si and electrons Spin resonance measurements of spin coherence in Si GaAs quantum dot spin experiments Si quantum dot spin experiments Si donor spin experiments Other schemes (electrons on helium, )
44 1998: Two proposals 1. Loss & DiVincenzo, PRA 57, 120 (1998) - Quantum dots 2. Kane, Nature 393, 133 (1998) - Donor impurities in Si
45 Electrons in quantum dots Gates (white) hold electrons (red) in dots Electrons assumed to be held at interface between light and dark materials (unspecified) Use gate voltages to cause 1 and 2 to overlap exchange interaction (J) B ext 1 2
46 Exchange gates No overlap + - Overlap T - T 0 T + S Split by B Z H J ~ JS 1 S 2 In B Z S X, S Y mix 0 & 1 Mixture oscillates at Z H J ~ JS 1z S 2z
47 Exchange CPHASE H J ~ JS 1z S 2z
48 Quantum dot jargon Solution-grown nanocrystal quantum dots Typically less than 10 nm in size Coated with organic ligands Self-assembled quantum dots Typically less than 20 nm across, and a few nm thick Formed during growth by Molecular Beam Epitaxy Vertical quantum dots Few x 100 nm pillars etched into multilayer GaAs/AlGaAs Have been used to study electron spins Lateral or gate-defined quantum dots Typically 100 s of nm across Start with 2D electron heterostructure These are the main dots we want to use
49 GaAs/AlGaAs Heterostructures Conduction band 2D electrons Valence band Fermi Energy GaAs Cap AlGaAs GaAs Substrate
50 Properties of GaAs/AlGaAs Nearly defect-free interfaces Reduce electron density to ~10 10 /cm 2 before charges trap ( 100nm between electrons) Single conduction band valley Small effective mass (m * ~ 0.07 m 0 ) Small mass means large energy splitting (for spatial part of ), even in a large device Approximately harmonic potential E ~ 1/ m * Little leakage for Au gates on GaAs cap Charge motion in doping layer can cause noise No stable isotopes with zero nuclear spin As: 100% 75 As; Ga: 60% 69 Ga, 40% 71 Ga; all with I = 3/2
51 Single quantum dot structures Vertical dot GaAs Metal depletion gates 450 nm Dot electron Kouwenhoven, Rep. Prog. Phys. 64, 701 (2001) For Both Ciorga, PRB 61, (2000) Size of dot can be controlled by side gate(s) Must pass current through dots to measure
52 Quantum point contacts (QPC) Narrow channel acts as electron waveguide. Step increase in current when new mode becomes available van Wees, PRL 60, 848 (1988) A QPC forms a very high-gain detector of electric fields (or charge). M. Field showed that single electrons can be detected - Field, PRL 70, 1311 (1993)
53 Single dot spin measurement Elzerman, Nature 430, 431 (2004) Need large spin splitting (g B B >> kt e )
54 Spin blockade in double quantum dot Singlet-Triplet splitting can be >> kt, meaning electron can only move from left to right dot if the two form a singlet Koppens, et al. Nature 442, 766 (2006)
55 Single spin electron spin resonance (ESR) Quantum dot covered with large co-planar strip microwave guide. Shorting waveguide gives maximum B and mininum E at the dots. Rotating a spin with the microwaves can lift the blockade by converting triplets into singlets (middle two lines) Koppens, et al. Nature 442, 766 (2006)
56 Single spin coherence Current increases, as triplet probability increases, then decreases as spin rotates past the optimal point to form singlets. Oscillations decay nonexponentially, but conclude T 2 ~ 500 ns. Koppens, et al. Nature 442, 766 (2006)
57 Double Dot Spin Coherence nonlocal, since two electrons are in different dots
58 Energy V L (mv) Spin state measurement (spin-to-charge conversion) (1,1) (1,2) G QPC (0,1) (0,2) (1,1) (0,2) T V R (mv) Singlet measurement T - T O T + S Petta, Science 309, 2180 (2005) S
59 Energy V L (mv) Spin state measurement (spin-to-charge conversion) (1,1) (1,2) G QPC (0,1) (0,2) (1,1) (0,2) T V R (mv) Triplet measurement T - T O T + S Petta, Science 309, 2180 (2005) S
60 Different effective nuclear fields in the two dots (T 2 * not T 2 )
61 Singlet/Triplet Qubits use 2 electrons to make 1 qubit Petta et al. Science 309, 2180 (2005)
62 Extending coherence dynamical decoupling Red is reference. Difference represents the loss of coherence Coherence can be extended to > 200 s with CPMG Bluhm, Nat. Phys. 7, 109 (2011)
63 Where are GaAs/AlGaAs dots? Qubits 1 and 2 qubit gates Decoherence Initialization Measurement Singlet-Triplet pairs in double QD (DQD) Two DQD s dots have been coupled ~200 s with decoupling Just freeze into ground state Use QPC and spin blockade
64 What about Si QD s? Low density of non-zero spin nuclei (4.75% 29 Si), and isotopically enriched 28 Si is available Medium m * ~ 0.2 need smaller dots than for GaAs Multiple conduction band valleys could cause complications Si/SiGe heterostructures are analogous to GaAs/AlGaAs structures, but more difficult to grow and still somewhat lower mobility (if that matters?)
65 Spin measurement with Si/SiGe QD Here the spin was measured single shot rather than signal averaging over many electrons Simmons, PRL 106, (2011)
66 Driving Spin Electrically in nat Si/SiGe Microwave B-fields often (low-power) spead out ~ Electric fields better confined Address individual dot/electron Results: T Hahn 2 ~ 70 s, T XY8x16 2 ~ 400 s (extra charge noise) Gate fidelity ~ 98 99% Kawakami, PNAS 113, (2016)
67 28 Si MOS Quantum Dots Metal-oxide-Si (MOS) is close to Si industry 28 Si eliminates decoherence from nuclear spins Rabi oscillations Results: T * 2 ~ 120 s (Ramsey fringe) T Hahn 2 ~ 1.2 ms (Ramsey echo) T CPMGx500 2 ~ 28 ms Gate fidelities ~ 99.2% % Veldhorst, Nat Nano 9, 981 (2014)
68 Undoped Si/SiGe double quantum dot Maune, Nature, 481, 344 (2012)
69 T 2 * in nat Si double quantum dot Find T 2 * ~ 350 ns Couldn t make J small enough for whole Bloch Sphere
70 Do we need singlet-triplet qubits in Si? No T 2 with just Hahn echo is longer than GaAs S-T qubit World s best 28 Si (50 ppm) and similarly enriched Ge might make T 2 * ~ 10 s need dynamical decoupling and limited by gate noise Do we want singlet-triplet qubits in Si? Maybe Inherently insensitive to B (decoherence-free subspace) One fast 1-qubit operation (exchange) Other 1-qubit operation difficult, but Need micromagnet for B?
71 28 Si-MOS 2-Electron Double Dot CPHASE between 2 electrons CNOT Now working on improving fidelity Veldhorst, Nature 526, 410 (2015)
72 Donors in Si the Kane Scheme 1-Qubit 2-Qubit
73 e 2 /C Single electron transistor (SET) Plunger Angus, Nano Lett. 7, 2051 (2007) The island, length d, is small enough that the simple Coulomb energy to add an electron (½ e 2 /C) is non-negligible. The current has spikes as a function of plunger-gate voltage, V p. I I 0 V p
74 e 2 /C Single electron transistor (SET) Plunger Angus, Nano Lett. 7, 2051 (2007) The island, length d, is small enough that the simple Coulomb energy to add an electron (½ e 2 /C) is non-negligible. The current has spikes as a function of plunger-gate voltage, V p. I I 0 V p
75 e 2 /C Single electron transistor (SET) Plunger Angus, Nano Lett. 7, 2051 (2007) The island, length d, is small enough that the simple Coulomb energy to add an electron (½ e 2 /C) is non-negligible. The current has spikes as a function of plunger-gate voltage, V p. I I 0 V p
76 Measuring T 1 with the SET The measurement proceeds like the quantum dot measurements of single electron, with the donor being the dot. Obtained T 1 ~ 6 s ~ dozen donors Morello, Nature, 467, 687 (2010)
77 Measuring T 2 of a single donor electron Send microwaves down shorted transmission line, as with the dots Rabi oscillations ~ dozen donors Results: T 2 * ~ 200 s (Ramsey fringe) T 2 Hahn ~ 1 ms (Ramsey echo) T 2 CPMGx8000 ~ 0.5 s Nuclear spin T 2 CPMG ~ 35 s Muhonen, Nat. Nano, 9, 986 (2014)
78 Issues with Si qubits Stark tuning donor spin requires high fields for MHz tuning Donor pitch too tight Donor wavefunction ~ 2nm Donor exchange oscillations Exchange interaction between two donors is not monotonic in the distance between them (varies by 10x, or more, while donor moves 1 lattice constant)
79 Linewidth: Every spin is the same, right? Assume surface code cycle = 1 s Allowed phase error/cycle = 0.1% 2x10-3 Maximum linewidth = 1kHz (10-3 cycles in 1 s) Bulk 28 Si:P few khz to 10 s khz (not really understood) Epitaxial 28 Si:P/ nat Si 100 s khz Can we individually tune 10 9 qubits? Large local strains from oxide & gate metals Probably contributes to donor linewidth Seems to be much bigger in SOI
80 Who Cares About Defects? Defects (thermodynamics says there are some)? ~10 10 /cm 2 with depth 3meV at MOS interface ~100nm between traps Are they a problem? Distort donor-dot energies Just calibration issue? Extra trapped electrons Exchange with dot electrons or transiting electrons? Mess up microwave coupling?
81 Conduction band valleys in Si Donor exchange oscillations Valley splitting ~ Spin splitting Get valley states mixing with spin states in dots bad! Interface details (atomic steps?) valley splitting
82 Coupling Donor to Quantum Dot Singlet-triplet qubit with one electron on/off phosphorus Electron on donor rotate between S & T 0 (vertically) Electron off donor just exchange (rotate horizontally) Fit oscillations, including J 0 with electron on donor Used 4 electrons not 2, because small valley splitting Filled lower valley states, so only consider upper pair
83 Si quantum dot vs. SRAM
84 Alternate Schemes Pica, PRB 93, (2016) Tosi, ArXiv: (2105) Morse, arxiv: (2016) Hill, Sci Adv 1, e (2015)
85 It s the economy, stupid. * materials * James Carville
86 Experimental System LHe Vacuum ~1eV Image E 0 0 Electron 0.7meV + ~115Å Electron bound to image in liquid He z = radial hydrogenic wave function x,y = plane waves Submerged gates can modify binding Extremely mobile electrons (~100 x 10 6 cm 2 /V-s) Low density 2D system ( /cm 2 )
87 Spin Orbit Interaction e - ~7.5 nm He He He He He He He He He He He He He He Overlap of electron above surface with helium 2P-band?
88 Spin Orbit in Electron Bubble He He He He He He He He He He e - He He He He He He He He He He g/g < 3x10-7 vs. Si with g/g ~ 10-3 so 10 7 less decoherence Spin coherence for mobile electrons > 1 second Reichert and Jarosik, PRB 27, 2710 (1983).
89 Helium-Filled Channels Helium depth set by thickness of deposited layer Easy to make precise depths of nm to a few m
90 UV Electrons Channels Zn Van der Waals He film Bulk He Cu cell
91 Potential Energy 3-phase CCD Potential Underlying gates Helium electron - +
92 Potential Energy 3-phase CCD Clocking Electron has moved one pixel (3 gates) to the right
93 George Smith, Nobel Prize Lecture on CCDs, 2009
94 Electrons per channel Horizontal Clocking Efficiency Clock (pixel) rate = 240kHz 120 parallel channels 1.6 K Pixels transferred No measurable errors ~ 70 min. ~ 9 km.
95 One Sheep per Clock
96 Device Structure 78 parallel channels Turnstile Narrow CCD Sense Store Reservoir 2.3μm Electrons deposited in reservoir T 5 T 4 T 3 T 2 T 1 0.8μm 1.2μm deep Clock into channel and measure Go through narrow CCD Extras back to reservoir Clock into turnstile region ~20 electrons/channel
97 Electron signal (a.u.) e - /channel Plateau (measuring 78 channels in parallel) K 78 parallel channels Offset voltage (volts)
98 Conclusions Electron spin qubits in Si have long enough coherence for quantum computing 1 and 2 qubit gates have been demonstrated using electrons in quantum dots and electrons bound to donors Devices can be very small and fast Individual devices can have high fidelity Scaling to a large quantum computer will require solving some difficult materials issues People working on various approaches to dealing with these issues
99 Race to a Quantum Computer 200,000,000 inches (Qubits) 0 inches (Qubits) N R I A I and R have a big lead. N seems to have hit a brick wall, H is just starting, S says he s coming on fast, and T says he knows a shortcut S L T H
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