Quantum Computing Architectures! Budapest University of Technology and Economics 2018 Fall. Lecture 3 Qubits based on the electron spin

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1 Quantum Computing Architectures! Budapest University of Technology and Economics 2018 Fall Lecture 3 Qubits based on the electron spin!! From Lecture 3 Physical system Microsopic Hamiltonian Effective Hamiltonian Effective dynamics Quantum logic gates Lectures 1 & 2

2 Schedule of this course Szerda augusztus Regisztrációs hét -- szeptember 5. szeptember 12. szeptember 19. szeptember 26. TTK Dékáni szünet október 3. október 10. október 17. október 24. október 31. november 7. november 14. TDK konferencia november 21. november 28. december 5. lecture 01 lecture 02 lecture 03 (today) lecture 04 lecture 05 lecture 06 lecture 07 lecture 08 lecture 09 lecture 10 Introduction Spin qubits (electron spin) Superconducting qubits (transmon)

3 Qubit Checklist 1. make a few qubits 2. initialize 3. control (1-qubit gate, 2-qubit gate) 4. read out 5. understand and reduce information loss today Qubits based on the electron spin! (Spin qubits) review papers: Hanson et al., Rev. Mod. Phys. (2007), Zwanenburg et al., Rev. Mod. Phys. (2013)

4 Make a qubit QUANTUM Gate voltage tunes the number of electrons in the quantum dot

5 Workhorse material: GaAs crystal structure: zinc-blende electronic band structure conduction-band effective mass: m =0.063 m e conduction-band effective mass: static dielectric constant: r = 12.9 effective g-factor g = 0.4

6 A double quantum dot in a semiconductor heterostructure 2DEG (2D electron gas) confined in GaAs at the GaAs/AlGaAs interface AlGaAs layer height ~ 30 nm, gate features ~ 50 nm

7 Top view of the gate structure single dot double dot P P P P: Plunger gates: tune (mostly) the on-site potential energy QPC: Quantum Point Contact; used as a charge sensor

8 Energy scales confinement energy, charging energy, thermal energy Assume circular confinement: H = p2 x + p 2 y 2m m!2 0(x 2 + y 2 ) Energy spectrum: E n,m r= ~! 0 (n + m + 1) Orbital level spacing: (a.k.a. confinement energy) E orb = ~! 0 r Spatial extension of ground state: (a.k.a. oscillator length) (a.k.a. confinement length) ` = r ~ r m! 0 Homework: assume 50 nm confinement length in GaAs; then, what is the confinement energy? It is ~0.5 mev.

9 Energy scales confinement energy, charging energy, thermal energy e e charging energy estimate: U e2 4 0 r` for GaAs, with ` = 50 nm, U 2.2 mev thermal energy at 300 K: k B T 30 mev thermal energy at 100 mk: k B T 10 µev thermal E << confinement E, charging E required to confine a single electron on a single level Experiments are done at T ~ 100 mk

10 Readout of a spin qubit 1 2 :tunnelrate Zeeman splitting: E # E " = g µ B B 230µeV for B = 10 T Step 1: load an electron (up or down) Goal = Readout = Distinguish between up and down Step 2: spin is converted to charge `Elzerman-style spin readout : Elzerman et al., Nature (2004)

11 Dot charge is detected by Quantum Point Contact conductance or current through QPC QPC current is sensitive to! - its own gate voltage Q - any other gate voltage M, P, R, T - number of electrons in the dot QPC gate voltage (Q)

12 Exercise: estimatethe tunnel rate from the data. Spin Down: blip in the current (dashed).! Spin Up: no blip (solid). Data from the experiment: # "

13 Readout is wrong if spin relaxation is too fast wait and 5 mv during t read (which is 0.5 ms for all measurements). b, Schematic fast spin relaxation: spin 2 Two-level pulse technique used to inject a single electron and measure its spin ion. a, Shape of the voltage pulse applied to gate P. The pulse level is 10 mv spin relaxes before tunneling 2 Two-levelFigure pulse technique 2 Two-level used pulse to inject technique a single used electron to inject and a single measure electron its spin and measure it ion. a, Shape orientation. of the voltage a, Shape pulseofapplied the voltage to gate pulse P. The applied pulse tolevel gateis P. 10 The mvpulse level is 10 int wait and 5 mv during during t wait t read and(which 5 mv during is 0.5 ms t read for(which all measurements). is 0.5 ms for all b, measurements). Schematic b, Sche lse-response t, Q dot, QPC if the pulse-response injected electron if the has injected spin-" (solid electron line) has or spin-" spin-# (solid (dotted line) line; orthe spin-# (dotted lin ce pe, with thedifference solid line iswith onlythe seen soliduring line isthe only read-out seen during stage). the Arrows read-out indicate stage). Arrows indic used ent an in electron the moment tunnelsan into electron or out tunnels of the quantum into or out dot. ofc, the Schematic quantumenergy dot. c, Schematic ene sgaas/ for spin-" diagrams (E " ) and for spin-# spin-" (E # )(E during " ) andthe spin-# different (E # ) during stagesthe of the different pulse. stages Blackof the pulse. B Figure Figure 2 Two-level 2 Two-level pulse pulse technique used used to to inject aa single electron and measure its

14 Elzerman-style spin readout is rather slow Readout time scale: millisecond. Control time scale (q-gates): microsecond.

15 A basic application: measurement of spin relaxation time waiting time spin relaxation: exponential decay, P # (t wait ) 1 2 e t wait T spin = 1 2 e spint wait For increasing B-field, spin relaxation gets faster.

16 condensed-matter physics (5, 6). ture grown by molecular beam epitaxy with a V An attractive candidate for a solid-state two-dimensional electron gas 100 nm below relaxation: spinquantum dots, te thefast spin qubit is based on semiconductor Spin-to-charge conversion in a double dot which allow controlled coupling of one or m the data. : tunnel rate more electrons, using rapidly switchable voltt ages applied to electrostatic gates (7 9). Retwait n: Two-site cent experiments suggest that spin in quantum Hubbard model: spin spin relaxation: exponential decay, P (t ) 1 e Tspin = 1 e spin twait # wait 2 2 dots may be a particularly promising holder of quantum information, because the spin relaxtime (T1) can approach of millihhubbardation =H +H HCoulomb on-site tun +tens twait seconds (10 13). Although gallium arsenide t = 0 H 1 1 Tspin spin twait Hon-site = "LisPna#Ldemonstrated +wait "R)n (GaAs) material Rexceptional xponential decay, (t e = e 2 2 quantum dots, for fabricating it has the po confined inhtuntential = thdrawback a L" athat + a aelectrons R" R# + h.c. L# teract with on the order of 106 spin-3/2 nuclei th = 0.1U through thel" hyperfine HCoulomb = U (n nl# +interaction. nr" nr#there )H =we0 twait 1 Tspin twait system (logical present a1quantum two-level spin ) e = e ait 2 2 on two-electron qubit) based spin states detuning parameter: "=" = " (14) Fig. 1. (A) Scanning electron micrograph of a s R 1 2 (0,2) 1 t =0 n t = a a, etc. energy [U] energy [U] 3 Department of Physics, Harvard University, Cambridge, MA 02138, USA. 2Department of Condensed 3 Matter Physics, Weizmann Institute H of Science, 3 H Department, UniRehovotL" 76100,(2,0) Israel. 2 L" L" Materials versity of California at Santa Barbara, Santa Barbara, 1 CA 93106, USA. = 0.1U zoom-in electrostatic gates on the surface of a two-dim control the number of electrons in the left an tunnel coupling. The quantum point contact con electrons in0.04 the right dot. (B) gs measured as a f stability diagram 0.02 (a background slope has been m is the number of electrons the left dot and T in(x3) 0.00 charge state gives a distinct reading of gs. energy [U] L S (x1) (1,1) x4control QUESTIONS SEPTEMBER 2005 VOL 309 SCIENCE EXERCISES, -1-1 th = 0.1U detuning, [U] detuning, [U] detuning, [U] formance of quantum computing could exceed that of classical computing. 0 0

17 Spin-to-charge conversion in a double dot energy [U] add magnetic field: T + (1, 1) T (1, 1) detuning, [U] i = S(1, 1)i + T 0 (1, 1)i Task: do a measurement in the S-T 0 basis.

18 Spin-to-charge conversion in a double dot energy [U] add magnetic field: T + (1, 1) T (1, 1) detuning, [U] Solution: sweep slowly and then measure charge in right dot. f = S(0, 2)i + e i' T 0 (1, 1)i

19 Summary of key results 1. a spin qubit can be defined in a quantum dot 2. Elzerman readout of a spin qubit 3. the relaxation of a spin qubit can be measured 4. two electrons can be used to define a singlet-triplet qubit 5. Pauli blockade readout of a singlet-triplet qubit Potential extensions 1. Pauli blockade: thermal << Zeeman not required 2. Pauli blockade readout for a spin qubit 3. readout based on gate reflectometry 4. ways to reduce the readout time 5. how to control the singlet-triplet qubit