Josephson-Junction Qubits

Transcription

1 Josephson-Junction Qubits John Martinis Kristine Lang, Ray Simmonds, Robert McDermott, Sae Woo Nam, Jose Aumentado, Dustin Hite, Dave Pappas, NST Boulder Cristian Urbina, (CNRS/CEA Saclay) Qubit 8µm Atom based on nonlinear microwave resonator Scalable system using C fabrication

2 Main Concepts Challenge: Coupling vs. Decoherence Qubits from E&M modes non-linearity essential Josephson effect and non-linear inductance: Qubit fromed by non-linear LC Noise model of decoherence (equivalent to Spin-Boson) Decoupling of qubits through impedance transformer circuits Promising experimental results (circuits work!) New decoherence mechanism: Need to improve on junction fabrication

3 Operations for Quantum Computation (DiVincenzo criteria) Classical Computation: nitialize state Logic not and Output result > > > > > > Logic errors: Error correction possible Quantum Computation: nitialize state Ψ i =..> Logic via series of operations: State Manipulation ( qubit) Controlled not ( qubit) bit > > > > > > > > ( >+ >)/ / > > > } > > > > > > > > > control Final state measurement Measure qubits of state Ψ f + linear superposition Coherence: τ coherence / τ logic ~ number logic operations

4 Challenge: Coupling vs. Decoherence control measure control measure control measure qubit qubit qubit noise & dissipation Experimental challenge: Couple qubits to each other, control, & measure, not noise and dissipation

5 Experimental Systems Atoms Feynmann (985): it seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms, and quantum behavior holds sway. mesoscopic atomic ons Neutral Atoms NMR Spin Semiconductor spin Quantum Dot

6 Experimental Systems Feynmann (985): it seems that the laws of physics present no barrier to reducing the size of computers until bits are the size of atoms, and quantum behavior holds sway. microscopic mesoscopic atomic Atoms EM modes Photons ons Neutral Atoms NMR Spin Semiconductor spin Quantum Dot Coherence easier Superconductor SET -Degenerate SSET 3 Junction SQUD RF SQUD Josephson Junctions (this work) Coupling easier Charge (NEC, Chalmers) (Saclay) (Delft) (MT) Flux Phase (NST, Kansas)

7 Superconductivity k y k Phase degree of freedom: φ Cooper pairs states of zero momentum Only remaining degree of freedom is φ Supercurrent: φ Energy gap of excitations: j s No dissipation for f < 7 GHz (Nb) Flux quantization Ψ.76 kt c = k k f k x iφ + + ( u k + vke ck c k) k Josephson effect Ψ Tc kl c kl c + kr c + kr Ψ c. c. φ L φ R iφ L iφr e e c.c. = sin( φ φ ) L R nm φ L φ L + e h Vdt Large gives little/no dissipation in junction from fabrication imperfections

8 Josephson-Junction Physics (classical) sc insul. sc j = sin V = ( Φ /π ) & V R C = V/R + sin + CV U = J Vdt Φ = π Φ = π d sin dt cos dt U() Φ C π Φ && + R π & + Φ π cos mass damping potential U() =.9 Φ π = =.3 zero voltage state voltage state V sc ~3mV

9 Qubit: Nonlinear LC resonator U() E E R L J <V> = γ C = sin Φ V = & π Γ n+ Γ Γ n ~ Γ & = cos & j (/L J ) V L J = Φ /π cos nonlinear inductor E U ω p ω p 4 = 3 π = Φ Φ π [ / / ] 3/ [ ] / 4 C U γ RC Lifetime of state > Γ <V> ~ mv ( E ) n E n / h ω + p ω ω 5 U hω p ω 3

10 Circuit elements: Qubit Taxonomy L C Quantum mechanics: [ Φ ˆ, Q ˆ ] = ih Hamiltonian: Φ π cos Φ π ( Φ ) L e C q [ ˆ, qˆ] = i Non-linearity E E J C Φ / e / C π Phase Flux Charge L J L Bloch states in = 4 Area (µm ): -.-. Potential & wavefunction Engineering Z J =/ω C Ω 3 Ω 5 Ω

11 Josephson-Junction Qubit State Preparation Wait t > /γ for decay to > Qubit logic with bias control = dc + dc (t) + µwc (t)cosω t + µws (t)sinω t State Measurement > : zero voltage > : voltage <V> = H () = s + s + s x y z <V> = mv potential ( µwc h / ω C ( / ω C µws dc (t) ( E / dc > > / ) / h / ) / )/ (Junction acts as photomultiplier ) With Γ i /Γ i- ~ phase Fidelity > 99%!!. ω microwave pulse. pulse (lower barrier)

12 Effective Hamiltonian s B [ ] Φ + Φ + = + Φ + = ˆ ˆ ˆ ˆ ˆ ˆ), ( ˆ ˆ ) ( ˆ cos ˆ π ω π π H q C q C H dc cubic dc h Solve numerically Perturbation = t i e V ω ) ( ~ V V i HV V H t = + + h Basis transform (rotating frame): ( ) + + Φ + + Φ + = ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ sin cos ~ π ω ω π ω ω i i e e t t H uws uwc dc t i t i uws uwc dc Rotating wave approximation (neglect off resonant terms): s x s z s y

13 Qubit Logic with Capacitive Coupling C x 3 C C C stray ~.5-.5 ff/µm H int C σ x q y y Turn-off interaction with single qubit operations (eg. NMR) σ q Modulate interaction by de-tuning resonance frequencies ω Theory (Maryland): CNOT & Phase gates ω no interaction τ logic ~3 cycles no interaction time Experiment (Maryland) Level splittings Large junctions C stray unimportant Coupling to more qubits, lower crosstalk

14 Josephson Junction Decoherence C ω H c z () = + s + s z s x y z (t) ( h / µwc ω C / ) / h ω C / ) / µws(t) ( / dc (t) ( E / dc )/ θ φ c y y + i + x c x Ψ = cos θ i φ + sin θ e

15 Josephson Junction Decoherence S (ω) Z(ω) Environment C ω H c z () = + s + s z s x y z (t) ( h / µwc ω C / ) / h ω C / ) / µws(t) ( / dc (t) ( E / dc Spontaneous Emission (decay ->) )/ Noise at ω (σ x, σ y op. s) Noise at LF (σ z op. s) Decoherence arises from noise and dissipation θ φ c y y x c x

16 Decoherence from Dissipation (Spontaneous Emission) Z(ω) Environment C ω γ = { ω } Re / Z ( ) C Dissipation at ω

17 Noise and Spectral Density Bandpass filter at f, Hz wide Measure S (f) = df S ( f ) ( t) () = df S ( f )cos(πft) Resistor R (low f) S =4kT/R

18 Decoherence from Noise at ω (Stimulated Emission) S (ω) Z(ω) Environment C ω γ s = S (ω /π) h ω C Noise at ω γ s = RC exp( hω / kt ) Exponentially small! (Thermal eq. non-trival assumption)

19 Decoherence from LF Noise (Phase Noise) S (ω) Z(ω) Environment φ n C ω ( ) t t = ω dt ( t n ) S (ω/π) Noise at ω<<ω Equivalent to Spin-Boson calculations (PRB 67, 945) φ n ( t) ω = = ω ω = t dt t' df S df S dt' ( f ) n ( t) t dt n t' ( t') sin ( πft) ( f ) ( πf ) dt'cos[πf ( t t')] High frequency cutoff at f~/t

20 Josephson Junction Decoherence 4kT R Z(ω)=R C γ φ n Environment µs coherence times R ~ 6 Ω at 5- GHz ~ 5 Ω at - GHz Spontaneous Emission Ψ( φ n t i ω t) = + e = γ = / RC Phase Noise (No energy exchange!) + e i ω t+ iφ [(t)] dt n ( t) 4kT sin ( π ω ft) ( t) = df R ( πf ) kt = t 3 U RC

21 Problem: Wire mpedances ~ Ω Z L ~ Z vac = 377Ω meter Embed `Josephson junction atom in an anti-cavity!

22 Broadband mpedance Transformers φ R φ ~ 5 Ω L M Thevenin equivalent φ M L L R φ M L Key concept: Power matching R ~ constant

23 Qubit operation Measurement Readout φ s U() fast decay ~5 states Φ

24 Qubit operation Measurement Readout φ s U() fast decay ~5 states Amplifier (and its dissipation!) turned on & off with s - Adjustable T - Switching current µα Φ s ~ Balanced: no flux coupling R ~ 9 Ω SQUD flux s large mbalanced: flux measured R ~ 4 Ω Qubit Cycle φ s Measure p time Qubit Op Meas Amp Reset flux

25 C Fabrication Qubit µwave s µm φ Al junction process & optical lithography qp trap for BE AuCu Al via junction Al qp trap for CE SiO AuCu Al Si/SiO substrate

26 Sequencer & Timer switch fiber optics µwaves V source ~ppm noise V source ~ppm noise db x kω rf filters V s φ kω 3K 4K db 3 mk µw filters mk Experimental Apparatus db 6 mk db

27 Spectroscopy φ ω ω p ω/π (GHz)

28 Energy Levels vs. Bias Current w ω/π (GHz) w p = : blue p = : red ncreasing (arb. Units)

29 Rabi Oscillations t r ω ω 3

30 t relax Energy Relaxation ω ω 3 p t relax (ns) Comparable with other experiments with Al junctions Energy relaxation time probably limited by flux bias R

31 Flux Qubit (Delft) meas. qubit

32 Charge Qubit (Saclay) Large /f charge noise requires operation at degeneracy point (dω/dq = )

33 p = : blue p = : red Resonances & Rabi Oscillations Rabi oscillations disappear at spurious resonances Decrease in coherence amplitude, not coherence time

34 Amplitude Decoherence Need to report amplitude of oscillations (visibility) barrier coh. time coh. ampl. Saclay AlOx µs (3%) Delft AlOx 5 ns 5% NST AlOx 4 ns 3% NST NbAlOx ns 5% Kansas AlN 5 µs % All experiments have reduced amplitude p 3% Spurious resonances observed in Saclay, Delft experiments (Small junctions: fewer resonances, but greater effect) Decoherence likely from small (unobserved) resonances Resonances are major source of decoherence?! t rabi

35 What causes extra resonances? Level-repulsion: Uncontrolled Coupled qubit Frequency shifts rule-out macroscopic EM modes ω/π (arb units) const. H Model as modulation of from resonant defect motion int Φ = π Φ + π = A B cos h ω C cos Ψ A Ψ B Ψ A Ψ B ( e g + g e ) e g = / = / ( Ψ Ψ ) A ( Ψ + Ψ ) A g ω B B e g time (8 hour run) defect position: e A ω AB B Explains energy-level repulsion

36 Current (µa) Al AlOx Al G G.. E-3 N S e = τ h i e τ h (b) V s : Decoherence : /f Noise 6 nm Tunnel channels i E Voltage (µv) i i e = h τ ~ 4x -3 N ch ~ 3 /τ N ch τ ( < ev < ) channel / (6nm) / ~/N ch ~ 7x -6 ω/π (GHz) (arb. units) 3 µm junction 5 MHz splittings ( ~6x -5 ) resonance / 6MHz 8 channels / res. 5 res. / dec.-fr. - µm Resonances and /f noise are same phenomenon See individual traps in sub-micron junctions (Savo, Wellstood, Clarke):. µm junction (Van Harlingen) ~ -4 ( /N ch ~ 3x -3 ) trap / decade freq. Assuming resonances & traps turn on/off channels:.4 channels / trap traps / dec.-fr. - µm

37 Decoherence & Materials All oxide tunnel barriers give similar /f noise (Van Harlingen et al) S / (Hz)A / / (µm pa/hz / /µa) Al-AlOx-Al (~5) Nb-AlOx-Nb 7- Nb-Ox-Pbn 7- Nb-NbOx-PbnAu 8 Pbn-Ox-Pb 5 NbN-AlN-NbN (epi) Need Materials Research Qubits have vastly different requirements Past Research: High T c, Q <, low leakage junctions Qubits: T c > K, leakage tolerated low fluctuations, low dielectric loss Our research directions: Barrier uniformity, barrier materials (AlN), epitaxial growth Large-area qubits are ideal test circuits

38 Si()-(7 7) Flash anneal 5 C Ordered MBE Growth of Al on Si() Si() LEED Auger Electron Spectroscopy Al seed layer 5 Å Evap. K Anneal 45 K Ordered Al homo-epitaxy Evap. 3K Anneal 45 K Ordered 56 ev 36 ev 36 ev Silicon Aluminum (5 Å) Aluminum ( Å) Oxidation T, min, 3K Amorphous AES ntensity No LEED Pattern AlO x Al AlO x (~ 5 Å) Oxygen Al counter-electrode Amorphous... Dosage of O (Torr s) Aluminum (6 Å) Auger Electron Energy (ev) 6

39 Main Concepts Challenge: Coupling vs. Decoherence Qubits from E&M modes non-linearity essential Josephson effect and non-linear inductance: Qubit fromed by non-linear LC Noise model of decoherence (equivalent to Spin-Boson) Decoupling of qubits through impedance transformer circuits Promising experimental results (circuits work!) New decoherence mechanism: Need to improve on junction fabrication

40 Measurement X-coupling C x x C U() fast decay ~5 states Classical V x Drives transitions t t Φ P error ~ (C x /C) What is quantum S x (ω )? Quantum x Rounding from tunneling, decay t

41 Why is Quantum Computing Useful? n possible input states S = S = S 3 = n bits Classical logic gates n computations (serial) spans all input states output Ψ i = > Superposition + > + > of n states + n qubits Quantum logic gates Superposition of states Measure State output computation (parallel) Parallel computation of exponentially-large states Factorization of large numbers (Shor) Exponential speedup of algorithm Fast search algorithms (Grover) { n / vs. n } Adiabatic algorithms for minimization (Farhi) Simulation of quantum systems (Feynman) Other? (Quantum nformation Theory)