1.0 10 0.8 01 switching probability 0.6 0.4 0.2 00 Electrical Quantum Engineering with Superconducting Circuits R. Heeres & P. Bertet SPEC, CEA Saclay (France), Quantronics group 11 0.0 0 100 200 300 400 swap duration (ns)
Outline Lecture 1: Basics of superconducting qubits Lecture 2: Qubit readout and circuit quantum electrodynamics 1) Readout using a resonator 2) Amplification & Feedback 3) Quantum state engineering Lecture 3: Multi-qubit gates and algorithms Lecture 4: Introduction to Hybrid Quantum Devices
Ideal Qubit Readout a 0>+b 1> 1 0 a + b 0? 1 t meas << T 1 1 0 1 0 a + b a + b 0? 1 0? 1 1 0 1 1 0 0 relax. 1 1 0 0 0 0 0 0 Hi-Fid Quantum Non Demolition (QND) BUT HOW??? SURPRISINGLY DIFFICULT AND INTERESTING QUESTION FOR SUPERCONDUCTING QUBITS
The Readout Problem 1) Readout should be FAST: t meas T 1 ~40μs for high fidelity (F 1 t meas T 1 ) Ideally t meas < few hundred ns 2) Readout should be NON-INVASIVE (Quantum Non-Destructive, QND): Unwanted transition caused by readout process errors (if we find 0 as the answer, the state should remain 0 ) 3) Readout should be COMPLETELY OFF during manipulations: (i.e. have no interaction with environment decoherence)
Measurement of Photonic Qubits Measure a photon: CLICK! (there was a photon) Measure polarization: PBS CLICK! (photon was H)
Readout in Different Systems PhD Thesis, A. Burrell (2010)
Cavity Quantum Electrodynamics Interaction between a cavity and an atom Key work by Haroche (Nobel 2012) State-selective detector (Measure P(e) by selective ionization)
Qubit Resonator Interaction 2D geometry Microwave Fabri-Perot V in C c L = nl n / 2 C c V out Superconductor B E Si/SiO2 Coplanar Waveguide (CPW) : TEM mode
Qubit Resonator Interaction 2D geometry CPW resonator Superconducting artificial atom R. Schoelkopf, 2004 A. Blais et al., Phys. Rev. A 69, 062320 (2004) A. Walraff et al., Nature 431, 162 (2004) I. Chiorescu et al., Nature 431, 159 (2004) or: CIRCUIT QUANTUM ELECTRODYNAMICS
Qubit Resonator Interaction 3D geometry Storage Cavity Transmon Meas. Cavity Input/Output still: CIRCUIT QUANTUM ELECTRODYNAMICS
A. Blais et al., PRA 69, 062320 (2004) CPB Coupled to CPW Resonator ˆ V V aˆ aˆ V g 0 Cg gext V ext c ˆ ˆn Hˆ E cos ˆ 4E nˆ nˆ aˆ aˆ 2 tot J C g c 2 Hˆ E cos ˆ 4E nˆ n aˆ aˆ 8( C V E / 2 e) nˆ ( a a ) tot J C gext c g 0 C Hˆ q Hˆ cav 2-level approximation + Rotating Wave Approximation Ĥint H tot = ω ge 2 σ z + ω c a a + g(σ + a + σ a ) Jaynes-Cummings Hamiltonian
Strong Coupling Regime g 2 ev ( C / C) 0 n 1 ˆ 0 g GEOMETRICAL dependence of g Easily tuned by circuit design Can be made very large! (Typically : 0 200MHz) g 200 MHz, 100 500kHz Strong coupling condition naturally fulfilled with superconducting circuits (Q=100 enough for strong coupling!!)
The Jaynes-Cummings Model g,3> g,2> g,1> g,0> gn, 1 en, d d e,3> e,2> e,1> e,0> ge HJ C z c( a a 1/ 2) g( a a ) 2 HJ C Restriction of H J-C to g,n+1>, e,n> gn, 1 couples only level doublets g,n+1>, e,n> Exact diagonalization possible en, ( n 1) c / 2 g n 1 g n 1 ( n 1) c / 2 ge c Note : g,0> state is left unchanged by H J-C with E g,0 =-/2
The Jaynes-Cummings Model g,3> g,2> g,1> g,0> d d e,3> e,2> e,1> e,0> ge HJ C z c( a a 1/ 2) g( a a ) 2 HJ C Coupled states couples only level doublets g,n+1>, e,n> Exact diagonalization possible, n cos e, n sin g, n 1 n, n sin e, n cos g, n 1 n n n 1 2 1 n tan 2 2 2 E, n ( n 1) c 4 g ( n 1) 2 2 2 E, n ( n 1) c 4 g ( n 1) 2 1 g n
The Jaynes-Cummings model 1.08 en, 1.04 E/(h c ) 1.00 gn, 1 gn, 1 0.96 en, 0.92-5 0 5 /g ge c
The Jaynes-Cummings model 1.08,n en, 1.04 E/(h c ) 1.00 gn, 1 2g n1,n gn, 1 0.96 en, 0.92-5 0 5 /g ge c
Two Interesting Limits 1.08,n en, 1.04 E/(h c ) 1.00 gn, 1 2g n1,n gn, 1 0.96 0.92 en, RESONANT REGIME (=0) -5 0 5 /g ge c
Two Interesting Limits 1.08,n en, 1.04 E/(h c ) 1.00 gn, 1 2g n1,n gn, 1 0.96 0.92 en, DISPERSIVE REGIME ( >>g) RESONANT REGIME (=0) -5 0 5 /g DISPERSIVE REGIME ( >>g)
Two Interesting Limits 1.08,n en, 1.04 E/(h c ) 1.00 gn, 1 2g n1 QUANTUM STATE ENGINEERING,n gn, 1 0.96 QUBIT STATE en, READOUT 0.92 DISPERSIVE REGIME ( >>g) RESONANT REGIME (=0) -5 0 5 /g QUBIT STATE READOUT DISPERSIVE REGIME ( >>g)
JC: Dispersive Regime (δ g) ge ge 2 ( aa1/ 2) HJ C / z ( c z ) a a 0 z ca a 2 2 2 g with the dispersive coupling constant 1) Qubit state-dependent shift of the cavity frequency c c z Cavity can probe the qubit state non-destructively 2) Light shift of the qubit transition in the presence of n photons ge 2n Field in the resonator causes qubit frequency shift and decoherence (if fluctuating)
Dispersive Readout Principle iωct Ve iωct Ve Ve iω t + c L.O ω + χσ c z a 1 a 0 0> or 1>?? = c p or 0??? 1 2 0> p 1> 0,96 1,00 1,04 d / c
Typical 2D Implementation 5 mm (f 0 =6.5GHz) Q=700 80mm 40mm g 45MHz (optical+e-beam lithography) 2mm
Typical Setup MW drive MW meas LO I Q 20dB 50MHz Fast Digitizer A(t) (t) COIL V c db G=56dB 300K 20dB G=40dB T N =2.5K DC-8 GHz 50 4K 30dB 600mK 1.4-20 GHz 20dB 4-8 GHz 50 18mK
Typical Setup Wet fridge Dry fridge Last 5-10 years - Much easier to operate (no liquid He) - Much more space
JC Hamiltonian in Action Signature for strong coupling: Placing a single resonant atom inside the cavity leads to splitting of transmission peak Bishop et al., Nat. Phys. (2009) observed in: vacuum Rabi splitting cavity QED R.J. Thompson et al., PRL 68, 1132 (1992) I. Schuster et al. Nature Physics 4, 382-385 (2008) atom off-resonance on resonance circuit QED A. Wallraff et al., Nature 431, 162 (2004) quantum dot systems J.P. Reithmaier et al., Nature 432, 197 (2004) T. Yoshie et al., Nature 432, 200 (2004) 25 A. Wallraff et al., Nature 431, 162 (2004)
Qubit Spectroscopy -120 MW drv Pump TLS -122 g MW meas Probe resonator phase ( ) -124-126 Some e p 5,25 5,30 5,35 Drive freq (GHz) e g p 0.95 1.00 1.05 / c
Detecting Rabi Oscillations MW drv MW meas Δt Variable-length drive Projective measurement x 10000 Ensemble averaging -108 0-111 T 2R =316 ns Y ( ) X 1-114 -117 0 200 400 600 800 1000 t (ns)
PART II: AMPLIFICATION & FEEDBACK
Readout: Signal-to-Noise iωct Ve iωct Ve Ve iω t + c ω + χσ c a 1 z 0> or 1>?? Ideal amplifier L.O a 0 = c p 0 2 or??? 1 0> p 1> 0,96 1,00 1,04 d / c
Readout: Signal-to-Noise iωct Ve iωct Ve Ve iω t + c ω + χσ c z a 1 0> or 1>?? Real amplifier T N =5K L.O a 0 = c 0 or 1??? No discrimination in 1 shot p 2 0> p 1> 0,96 1,00 1,04 d / c
Readout: Signal-to-Noise iωct Ve iωct Ve QUANTUM- LIMITED AMPLIFIER?? Ve iω t + c ω + χσ c a 1 z 0> or 1>?? Real amplifier T N =5K L.O a 0 = c 0 or 1 in one single-shot?? p 2 0> p 1> 0,96 1,00 1,04 d / c
Parametric Amplification Castellanos, PhD thesis (2010)
Parametric Amplification (Slide by Michel Devoret)
Parametric Amplification Josephson pot. ~sin (φ) Gives all odd terms! Josephson pot. ~cos (φ) Gives all even terms! (Slide by Michel Devoret)
Parametric Amplification (Slide by Michel Devoret)
Parametric Amplification JPA / JPC + Very simple to make - Narrow band + Ultimate efficiency TWPA - Complicated:1000s JJ - Slightly less efficient now + Broadband, directional Quantumcircuits.com Macklin et al., Science (2015)
Signal-to-Noise with Par-amp M. Castellanos-Beltran, K. Lehnert, APL (2007) (quantum limit on how good an amplifier can be : Caves theorem) Actually reached in several experiments : quantum limited measurement
Typical Measurement Chain OUTPUT INPUT DRIVE (courtesy I. Siddiqi)
Pulsed Readout readout off readout on R. Vijay, D.H. Slichter, and I. Siddiqi, PRL 106, 110502 (2011) (courtesy I. Siddiqi)
Single-Shot Histograms No Par-amp g> e> g> e> With Par-amp g> e> g> e> Single-shot discrimination of qubit state Abdo et al., PRL (2014)
Cont. Readout: Quantum Jumps You can observe transitions in real-time! (in this case for a fluxonium qubit) Vool et al., PRL (2014)
Single-Shot Readout Feedback
Real-Time Feedback Many use FPGAs for real-time feedback Ofek et al., Nature 2016
Quantum Error Correction The ultimate quantum feedback machine Fowler, PRA (2012)
PART III: QUANTUM STATE ENGINEERING
Two Interesting Limits 1.08,n en, 1.04 E/(h c ) 1.00 0.96 en, 0.92 gn, 1 QUBIT STATE READOUT DISPERSIVE REGIME ( >>g) QUANTUM 2g n STATE 1 ENGINEERING RESONANT REGIME (=0) -5 0 5 /g,n QUBIT STATE READOUT DISPERSIVE REGIME ( >>g) gn, 1
Resonant JC: Vacuum Rabi Osc., n e, n g, n 1 / 2, n e, n g, n 1 / 2 COMPLETE ATOM-PHOTON MIXING Uncoupled sytem prepared in e,n> COUPLING SUDDENLY ON at t=0 P(e) 1.0 0.5 P( e) 1/ 2(1 cos2g n 1 t) g,n+1> 0.0 0.0 0.5 1.0 1.5 2.0 e, n i g, n 1 / 2 g n0 t/2p 1/ t p
Phase Qubit Coupled to Resonator I saturate meas. Coupling turned on & off via qubit bias = 40 MHz Current bias (courtesy J. Martinis)
Generating Fock States: Pumping Photons One by One (As done for ion traps) 4 3 1 0 g e 4 3 1 0 (courtesy J. Martinis)
Stimulated Emission Swap Oscillations Depend on n (courtesy J. Martinis)
Swap Oscillations Depend on n (courtesy J. Martinis)
Swap Oscillations Depend on n COUPLING SUDDENLY ON at t=0 1,0 - Atom prepared in g> - Oscillator in unknown state P(g) 0,8 0,6 0,4 0,2 P( g) 1/ 2 1 p( n)cos 2g nt n 0,0 0 2 4 6 8 gt/2p Measurement of p(n)
Generating Arbitrary States 0.577 0 1 i 2 Desired final state g Law and Eberly, PRL (1996) Reverse-engineer final state by building pulse sequence backwards arrow indicates complex amplitude of corresponding basis state 2g 1g 0g ub phae ubit roup 1e 0e zero amplitude Energy level representation of desired state ub (courtesy J. Martinis)
State Tomo. of Harmonic Oscillator (courtesy J. Martinis)
State Tomo. of Harmonic Oscillator P ( ) p n n (courtesy J. Martinis)
State Tomo. of Harmonic Oscillator P ( ) p n n (courtesy J. Martinis)
Wigner Tomography of Fock State 7 Theory Experiment ub phae ubit roup ub (courtesy J. Martinis)
Wigner Tomography, 0 N states: Hofheinz, Nature (2009)
Two Interesting Limits 1.08,n en, 1.04 E/(h c ) 1.00 0.96 QUBIT STATE en, READOUT 0.92 gn, 1 QUANTUM STATE ENGINEERING! DISPERSIVE REGIME ( >>g) 2g n1 QUANTUM STATE ENGINEERING RESONANT REGIME (=0) -5 0 5 /g,n QUANTUM STATE ENGINEERING! DISPERSIVE REGIME ( >>g) gn, 1 QUBIT STATE READOUT
Dispersive State Manipulation We can displace the oscillator state Creates coherent state from vacuum Closest analog to a classical state Poisson photon number distribution Average photon number n = α 2 α = c n n 2 1 n =2 5 4 3 0 0 1 2 3 4 5
Dispersive State Manipulation The dispersive Hamiltonian allows selective qubit rotations +π/2 0 φ 2 1 5 4 3 0
Dispersive State Manipulation Operation library: Qubit rotations Cavity displacements Conditional qubit rotations Conditional cavity phase Cat state Entangling gates: τ 1 χ Theoretically universal (using SNAP gate) Krastanov et al., PRA 2015, Heeres et al., PRL 2016 Vlastakis et al., Science 2013
Dispersive State Manipulation How to account for additional terms? How to do non-coherent state operations? Faster by making control-knobs more flexible? H C,I = a + a H C,Q = i(a a ) H T,I = b + b H T,Q = i(b b )
GRadient Ascent Pulse Engineering Numerical search for pulse implementing desired operation GRAPE Khaneja et al., J. Mag. Res. (2005)
Example Application Realize 6 0 to make Fock state 6? The effect of the pulse is a unitary operation We have only specified what happens to 0! (many unitaries satisfy our request ) Heeres et al., arxiv
Conclusions Circuit Quantum Electrodynamics: Allows QND readout of qubit state in the dispersive regime Using parametric amplifiers high-fidelity singleshot readout is possible In both the resonant and the dispersive regime oscillator control and tomography is possible