Quantum Control of Superconducting Circuits

Transcription

1 Quantum Control of Superconducting Circuits Liang Jiang Yale University, Applied Physics Victor Albert, Stefan Krastanov, Chao Shen, Changling Zou Brian Vlastakis, Matt Reagor, Andrei Petrenko, Steven Touzard, Zaki Leghtas, Reinier Heeres, Wolfgang Pfaff Mazyar Mirrahimi, Michel Devoret, Rob Schoelkopf City Tech Physics Seminar Supported by: DARPA, ARO, AFOSR, Sloan, Packard.

2 Superconduc*ng Cavity- Qubit System 45mm storage cavity transmon qubit readout cavity coupler port Effec*ve Hamiltonian: H = ω s a a + ω q e e χ s a a e e Interac9on strength dominates photon loss & qubit decoherence QND readout of the qubit has been demonstrated in many groups (e.g., F>99.5% in 300 ns) Long lived SC cavity (T>10 ms >> T qubit ~100us) κ s γ q χ s κ s,γ q : photon decay rate : qubit decoh. rate

3 Manipulate the cavity storage cavity H = ( ω s χ s e e )a a + ω q e e +ε ( t)a e iω s t + h.c. Strong Drive: Uncondi9onal cavity displacement D β 0 β indep. of qubit state Phase- space diagram Weak Drive: Condi9onal cavity displacement D 0, g β, g D 0, e 0,e g β g β Phase- space diagram quadrature β θ Quantum circuit cavity D β quadrature 0,e β,g Quantum circuit cavity qubit g D β in- phase in- phase Create superposi4on of cavity state depending on the qubit.

4 Manipulate the qubit storage cavity H = ω s a a + ( ω q χ s a a) e e +Ω ( m t) e g e i ( ω q mχ s )t + h.c. Integrated Signal Qubit Spectroscopy χ s Strong Drive: Uncondi9onal qubit rota9on: Y π n,g n,e Y π m for all n Weak Drive: Condi9onal qubit rota9on: m,g m,e Quantum circuit qubit cavity Y π Spectroscopy frequency (GHz) Y π m n,g n m n,g qubit Y π m Query the qubit: Are there m photons in the cavity? Johnson B.R. et al. Nature Phys. 6, Kirchmair G. et al. Nature

5 Determinis)c qubit- cavity mapping Transfer arbitrary state from qubit to cavity Measure Wigner func*on (based on QND parity meas) Storage 0 D β C π D β D β qubit 0 ag + Ybe π /2 Y π - Can we prepare the cavity in arbitrary superpositon of photon number states? - Can we robustly encode quantum informaton in cavity? Theory: Leghtas et. al PRA 87, (2013) a β + b β g W Tr D D ( α) = ( 1) aa ( α) ρ ( α) Experiment: Vlastakis et. al Science 342, 6158 (2013)

6 Theory of Cat Codes Mo*va*ons: 1. How to construct robust quantum memory? a) Overcome cavity dephasing errors b) Overcome cavity loss errors 2. How to control such robust quantum memory? a) Gates over single logical qubits b) Gates between two logical qubits

7 Dominant Errors in Cavity Quantum Memory Dephasing Error 1 1 κφddephaseρ = κ φ nρn n ρ ρn Photon Loss Error 1 1 κ ρ κ ρ ρ ρ 2 2 1D1 loss = 1 a a a a a a No Change in photon number! (Env. is probing photon number) Strategy: quickly shuffle photon number Reduce photon number by one! (Env. is stealing our photons one by one) Strategy: monitor photon parity

8 Driven damped harmonic oscillator generator LC oscillator Q Dissipation to Transmission line Driving + DissipaTon à Pure Steady State à Suppress dephasing noise! I d dt ρ = [εa ε * a,ρ]+κ 1 D a ρ d dt ρ = κ D a α 1 ρ D a ρ = aρa 1 2 a aρ 1 2 ρa a Steady state: ρ = α α,α = 2ε /κ

9 Mul*- photon Driven & Mul*- photon Damped Oscillator Storage Cavity 45mm Vertical Transmon Qubit Case d=2 : 2-dim steady state subspace! { c, 2 / } 2 2 ρ ± α α = ε κ ± Q Pump & Readout Cavity I d * [ d( a d ) d a d, ] d d a dt ρ = ε ε ρ + κ D ρ Ongoing experiment in Devoret s group Coupler Port d dt ρ = κ D d ad d α ρ ( ) 1/d α = 2ε d /κ d Case d=4 : 4-dim steady state subspace! { c 1/4 () i () i, (2 4/ 4) } ρ ± ± α α = ε κ Q Mirrahimi, Leghtas, Albert, et al., NJP 16, (2014) Q I I

10 Steady states dependent on ini*al states Case d=2 : Two dimensional steady state subspace! { c, 2 / } 2 2 ρ ± ± α α = ε κ Q Q I I i α e δφ α n = 0 N( α + α ) = c2n 2n

11 Choice of memory basis ( n=0> and n=1>) +z = 0 Z = C + α Y x = 0 1 X α +x = X α z = 1 Z = C α + Z = n = 0 = n = 1 Z Vulnerable to both - Dephasing Errors - Photon Loss Errors

12 Choice of qubit basis (2- photon process) +z = α + α Z = C + α +1 Y Parity x = α X α + +x = +α X α -1 z = α α Z = C α + = C + = N( α + α ) = c 2n Z α Z = C α = N( α α ) = c2n+ 1 2n+ 1 2n cn = e 2 α 2 n α n!

13 Choice of qubit basis (2- photon process) + Z = C + α Y x = α X α + +x = +α X α Z = C α + x = c n n= 0 n= 0 ( ) x = 1 c n n n n with cn = e 2 α 2 n α n!

14 Choice of qubit basis (2- photon process) + Z = C + α Y x = α X α + +x = +α X α Z = C α Key property: Difference in average photon number <n> is exponen-ally small, + for any superpositon state of C α and C α e α 2

15 Choice of qubit basis (4- photon process) + Z = C (0mod4) α Zaki Leghtas +1 Y Parity X C + iα + X C + α -1 Z = C (2mod4) α ( ) + ( iα + iα ) ( ) ( iα + iα ) ( ) = c 4n ( ) = c 4n+2 + Z (0mod 4) = C α = N α + α 4n Z (2mod 4) = C α = N α + α 4n + 2 Key property: Difference in average photon number <n> is exponen-ally small, ( 0mod 4) ( 2mod 4) for any superpositon state of C α and C α e α 2 /2 Leghtas, et al., PRL 111, (2012).

16 Effect of photon dephasing (in presence of driven dissipa*ve process) κ 1 1 φd ρ κ φ ρ ρ ρ 2 2 [ ] 2 2 n = n n n n No change in photon number (modulo 2 or 4) è Constant populaton of ± Z è No logical bit- flip errors

17 Suppression of cavity dephasing error - - by mul*- photon driven dissipa*ve process d dt -2gbit-flip êkf (a) Two-photon [ ] d * d d ρ = [ εd( a ) εd a, ρ] + κdd a ρ+ κφd n ρ »a 10-6 MulT- photon driven dissipaton »a 2 Pert. Thry. Num: k f êk 2 ph =1ê200 Num: k f êk 2 ph =1ê20 = -2gbit-flip êkf Dephasing Two Dimensionless Para: κφ / κd ( 2 / ) 1/ α = εd κ d (b) Four-photon Induced phase- flip rate: ExponenTally suppressed with the cat size 2 α. 1 Pert. Thry. Num: k f êk 2 ph =1ê200 Num: k f êk 2 ph =1ê »a 2 Victor Albert d Num: k f êk 2 ph =1ê200 Num: k f êk 2 ph =1ê20

18 Dominant Errors in Cavity Quantum Memory Dephasing Error 1 1 κφddephaseρ = κ φ nρn n ρ ρn Photon Loss Error 1 1 κ ρ κ ρ ρ ρ 2 2 1D1 loss = 1 a a a a a a No Change in photon number! (Env. is probing photon number) Strategy: quickly shuffle photon number Reduce photon number by one! (Env. is stealing our photons one by one) Strategy: monitor photon parity

19 QND Measurement of Qubit! Results from Devoret group, Yale: Hatridge et al., Science 2013* dispersive circuit QED readout + JJ paramp 5 0 Readout fidelity > 99.5% in ~ 300 nsec Time (ms) Many groups now working with JJ paramps & feedback, including: Berkeley, Delft, JILA, ENS/Paris, IBM, Wisc., Saclay, UCSB, *First jumps: R. Vijay et al., (UCB)

20 QND Measurement of Photon Number Parity H = χ qs a a e e U T = π χ qs = g g + e e for even n g g e e for odd n Sun, Petrenko, et al, Nature 511, 444 (2014)

21 Effect of photon loss (in presence of driven dissipa*ve process) For two- photon process: photon loss à logical bit- flip error a C α + Cα and a Cα Cα + For four- photon process: photon loss à tractable by parity measurement a C α (0mod 4) a C α (2mod 4) C α (3mod 4) C α (1mod 4) (( ) + i( iα iα )) (( ) i( iα iα )) = N α α = N α α

22 Summary on Quantum Memory Two- photon process: 1. Logical qubit basis of 2. Photon dephasing induces phase- flip errors whose rate is exp. suppressed by the cat size. 3. Photon loss induces bit- flip errors. Four- photon process: 1. Logical qubit basis of 2. Photon dephasing induces phase- flip errors whose rate is exp. suppressed by the cat size. 3. Photon loss induces errors that are tractable by photon- number parity measurements. How to achieve universal gates on encoded quantum memory?

23 X θ gates (2- ph process) d ( 2) 2 2 i X [ a a, ] 2-ph[ a a, ] 2-ph [ a ] dt ρ = ε + ρ + ε ρ + κ D ρ Quantum Zeno dynamics for X 2-1. Resonant drive è Small displacement D(iε) C + α = N ( e iεα α + iε + e iεα α + iε ) 2. Two- photon process è ProjecTon on to logical space N ( e iεα α + iε + e iεα α + iε ) cos(εα) C + α + isin(εα) Cα ε << κ ph Steady State Subspace C + α, Cα { } ε X Π + Cα (a + a )Π, C + α Cα = (α + α )ε (, C X C + Cα α + Cα Cα + ) X L α

24 Universal Gates on Quantum Memory Two-photon process: Decay operator κ 2- ph a 2 Driving Hamiltonian iε 2- ph (a* 2 - a 2 ) Arbitrary rot. around X ε X (a*+a) π/2- rotaton around Z - χ Kerr (a*a) 2 Two- qubit entanglement ε XX (a 1 *a 2 +a 2 *a 1 ) Mazyar Mirrahimi Four-photon process: Decay operator κ 4- ph a 4 Driving Hamiltonian iε 4- ph (a* 4 - a 4 ) Arbitrary rot. around X ε X (a* 2 +a 2 ) π/2- rotaton around Z - χ Kerr (a*a) 2 Two- qubit entanglement ε XX (a 1 * 2 a 22 +a 2 * 2 a 12 ) Mirrahimi, Leghtas, Albert, et al., NJP 16, (2014)

25 Summary of Cat Codes 1. How to construct robust quantum memory? a) Overcome cavity dephasing errors b) Overcome cavity loss errors Autonomous QEC of cat code Q Q I I 2. How to control such robust quantum memory? a) Single qubit gates b) Two qubit gates Quantum Zeno Dynamics Steady State Subspace

26 Symmetry & Conserved Quan**es in Lindblad Systems Q: What informaton from inital state is preserved as infinite Tme? For open system d dt ρ = Lρ =κ D 2 a2 2 α which is c ++ c + c + c in the steady state basis C ρ, initial state ρ init involves into ρ = elt ρ init t, { +, C α α }. Each degree c jk is associated with a conserved quantity J jk, such that c jk = Tr J jk ρ init. The corresponding quantities can be calculated: Victor Albert which satisfies d dt J jk = L J jk 0 with no evolution (conserved). Key result: #(conserved quanttes) = #(degrees in steady state density matrix) Conserved quantities: (1) efficient tool to compute steady state, (2) extract stored quantum information, (3) perturbative calculation for other decoherences, Albert, L J, PRA 89, (2014); Mirrahimi, et al., NJP 16, (2014)

27 Outlook: Two- Qubit Quantum Modules Design of Two- Qubit Quantum Modules 1. Memory Qubit (m) Long coherence Tme 2. CommunicaTon Qubit (c) IniTalizaTon Measurement Entanglement generaton 3. Local Two- Qubit Unitary Gates m c

28 Outlook: Scalable QC DiVincenzo s Criteria A scalable physical system with well characterized qubits The ability to ini*alize the state of the qubits The ability to measure specific single qubits A universal set of quantum gates Long relevant decoherence *me, much longer than the gate opera*on *mes Use cavity mode for long- lived quantum memory!

29 Outlook: Scalable QC DiVincenzo s Criteria A scalable physical system with well characterized qubits The ability to ini*alize the state of the qubits The ability to measure specific single qubits A universal set of quantum gates (esp., focusing on the remote gates) Long relevant decoherence *me, much longer than the gate opera*on *mes Switchable Router

30 Acknowledgement Victor Albert Reinier Heeres Mazyar Mirrahimi Stefan Krastanov Brian Vlastakis Michel Devoret Chao Shen Zaki Leghtas Rob Schoelkopf