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piq quipe de recherche en Physique de lInformation Quantique Superconducting qubits Alexandre Blais Universit de Sherbrooke, Qubec, Canada Which quantum computer is right for you? Quantum information processing: the challenge


  1. piq Équipe de recherche en Physique de l’Information Quantique Superconducting qubits Alexandre Blais Université de Sherbrooke, Québec, Canada

  2. Which quantum computer is right for you?

  3. Quantum information processing: the challenge Two-qubit Qubits: entangling gates Two-level systems Qubit readout | 1 i | 1 i | 1 i 0 1 | 0 i | 0 i | 0 i Single-qubit control Conflicting requirements: long-lived quantum effects, fast control and readout D. DiVincenzo, Fortschritte der Physik 48 , 771 (2000)

  4. Outline • Artificial atoms • Physics 101: Harmonic oscillators and basic electrical circuits • Superconductivity and Josephson junctions • Circuit QED: a possible QC architecture • Recent realizations and challenges

  5. ‘Atomic atoms’ Good «two-level Energy | 2 � � atom» approximation | 1 � E 01 = E 1 − E 0 = ~ ω 01 ω 01 | 0 � Control by shining laser tuned at the desired transition frequency Hyperfine levels of 9 Be + have long relaxation and dephasing times T 1 ∼ a few years T 2 & 10 seconds T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Nature 464 , 45 (2010)

  6. Relaxation and dephasing times T 1 : Relaxation = amplitude damping channel ≠ bit flip channel Probability of qubit | 1 � | 1 � having relaxed at time t: � e − t/T 1 = e − γ 1 t | 0 � | 0 � (Energy is conserved) T 2 : Dephasing = phase damping channel = phase flip channel Probability of phase decay at time t: 1 e − t/T 2 ✓ | c 0 | 2 ◆ c 0 c ∗ | ψ i = c 0 | 0 i + c 1 | 1 i → ρ = 0 c 1 e − t/T 2 | c 1 | 2 c ∗ e − t/T 2 = e − γ 2 t

  7. ‘Atomic atoms’ Good «two-level Energy | 2 � � atom» approximation | 1 � E 01 = E 1 − E 0 = ~ ω 01 | 0 � Control by shining laser tuned at the desired transition frequency Reasonably short gate time Hyperfine levels of 9 Be + have long relaxation and dephasing times T not ∼ 5 µ s Low error per gates: ~ 0.48% T 1 ∼ a few years T 2 & 10 seconds T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Nature 464 , 45 (2010)

  8. Artificial atoms • Based on microfabricated circuit elements Capacitor Standard toolkit • Well defined energy levels Inductor • Nonlinear distribution of energy levels Resistor • Maximize numbers of thumbs up!

  9. Avoiding dissipation: superconductivity Normal metals dissipate energy Dilution fridge No resistance in superconducting temperature Resistance ~ 10 mK state ⇒ no dissipation Superconductivity is a (macroscopic) Normal metal (Copper, Gold, …) quantum effect Superconductor (Aluminium, Niobium, …) T c of Aluminum Sherbrooke Sydney Temperature Light bulb filament ~ -272 o C (~1K) A good starting point for a quantum ~ 3000 o C device…

  10. Basic circuit elements (classical version) Inductor: Capacitor: - A non-resistive wire - Two metal plates separated - Relates voltage to change of by an insulator Capacitor (C) Standard toolkit current - Relates voltage to charge Q Inductor (L) V I V ✗ Resistor (R) V = LdI Q = CV dt Φ = LI Z t dt 0 V ( t 0 ) Φ = �1

  11. Basic circuit elements (classical version) Inductor: Capacitor: - A non-resistive wire - Two metal plates separated - Relates voltage to change of by an insulator Capacitor (C) Standard toolkit current - Relates voltage to charge Q Inductor (L) V I V ✗ Resistor (R) V = LdI Q = CV dt Current: Change of charge in time I = dQ dt

  12. Basic circuit elements (classical version) Inductor: Capacitor: - A non-resistive wire - Two metal plates separated - Relates voltage to change of by an insulator Capacitor (C) Standard toolkit current - Relates voltage to charge Q Inductor (L) V I V ✗ Resistor (R) V = LdI Q = CV dt Voltage is the same across L and C: C = Ld 2 Q Q V = Q V = LdI dt 2 C dt 1 Q ( t ) = Q (0) cos( ω LC t ) ω LC = √ LC LC oscillator

  13. Basic circuit elements (classical version) Oscillations of the charge: +Q o Q ( t ) = Q (0) cos( ω LC t ) Charge 1 0 ω LC = √ LC -Q o Time One out of countless examples of harmonic oscillator Height Time

  14. Classical harmonic oscillator E = 1 2 kx 2 Height, z max max = p 2 E = 1 max 2 mv 2 ( p = mv ) z max 2 m Momentum x max Position (x) H = p 2 2 m + 1 2 kx 2 Energy at arbitrary x : = Hamiltonian p Time k/m Frequency of oscillation: ω = Mass at rest Mass at rest Maximal velocity

  15. Quantum harmonic oscillator Height, z p 2 H = ˆ 2 m + 1 ˆ x 2 Energy at arbitrary x : 2 k ˆ = Hamiltonian Position, x Heisenberg uncertainty principle: Impossible to know precisely both x and p Classical variables are promoted to hermitian operator acting on Hilbert space [ˆ x, ˆ p ] = i ~ p → ˆ x → ˆ p x

  16. Quantum harmonic oscillator p 2 Height, z H = ˆ 2 m + 1 ˆ x 2 Energy at arbitrary x : 2 k ˆ Classical variables are promoted to hermitian operator acting on Hilbert space [ˆ x, ˆ p ] = i ~ p → ˆ x → ˆ p x Position, x Useful to introduce: Commutation relation: ◆ 1 / 4 ✓ ✓ mk ◆ p ˆ a † ] = 1 [ˆ a, ˆ [ˆ x, ˆ p ] = i ~ a = x + i ˆ ˆ 4 ~ 2 √ mk ˆ a † ˆ H = ~ ω ˆ a = ~ ω ˆ ◆ 1 / 4 ✓ n ✓ mk ◆ p ˆ a † = x − i ˆ ˆ p √ ( ω = k/m ) 4 ~ 2 mk

  17. Quantum harmonic oscillator a | n i =? ˆ ˆ a † ˆ a † ] = 1 n | n i = n | n i [ˆ a, ˆ H = ~ ω ˆ a = ~ ω ˆ ˆ n a † | n i =? ˆ a † ˆ ˆ What is the action of and on the eigenstates of ? ˆ a n a † ] = ˆ a † [ˆ n, ˆ First observation: [ˆ n, ˆ a ] = − ˆ and a n (ˆ ˆ a | n i ) = ˆ a ˆ n | n i � ˆ a | n i = ( n � 1)ˆ a | n i ⇒ a | n i / | n � 1 i ˆ a | n i || 2 = h n | ˆ a † ˆ || ˆ a | n i = h n | ˆ n | n i = n n ∈ N 0 Second observation: ⇒ p a | n i = p n | n � 1 i a † | n i = n + 1 | n + 1 i ˆ ˆ

  18. Quantum harmonic oscillator a | n i = p n | n � 1 i ˆ ˆ a † ˆ n | n i = n | n i ˆ n ≥ 0 H = ~ ω ˆ a = ~ ω ˆ p n a † | n i = n + 1 | n + 1 i ˆ p 2 H = ˆ 2 m + 1 p ˆ | 3 � x 2 Energy k/m 2 k ˆ ω = a † ~ ω ˆ ˆ a | 2 � a † ~ ω ˆ ˆ a | 1 � ˆ ˆ Q 2 Φ 2 1 a † ˆ ~ ω ω LC = ˆ ˆ H = 2 C + a √ LC | 0 � 2 L Z Flux, ! Flux: Φ = dtV ( t ) ◆ 1 / 4 ! ˆ ✓ C Q a † a † = ˆ adds a photon to the LC circuit ˆ Φ − i ˆ 4 L ~ 2 p C/L n = number of photons stored in the LC circuit (Magnetic field) (Electric field)

  19. Artificial atom | 3 � Energy = V 0 cos ω 01 t | 2 � V ( t ) | 1 � 1 | 0 � ω LC = √ LC Flux, ! Initialization to ground state is simple Aluminum on Sapphire 3 0 0 K √ ω 01 = 1 / LC ∼ 10 GHz 1 0 m K ∼ 0 . 5 K Not a good «two-level» atom, not a qubit…

  20. Josephson junction Superconductor (Al) Capacitor (C) Standard toolkit Insulator (AlO x ) Inductor (L) Superconductor (Al) ✗ Resistor (R) Josephson junctions

  21. Josephson junction E J Capacitor (C) Standard toolkit S I S Inductor (L) ✗ Resistor (R) Josephson junctions 100 nm

  22. Artificial atom toolkit Capacitor: Inductor: Artificial atom toolkit - Two metal plates separated - A non-resistive wire by an insulator Capacitor (C) - Relates current to flux - Relates voltage to charge Q Inductor (L) V I V Josephson junction V = LdI Q = CV dt Φ = LI Josephson junction: Z t - Two superconductors separated by an insulator dt 0 V ( t 0 ) - Relates current to flux Φ = I �1 V I = I 0 sin(2 π Φ / Φ 0 )

  23. Superconducting artificial atom Energy = V 0 cos ω 01 t | 3 � V ( t ) | 2 � � | 1 � | 0 � Flux, ! Very short π -pulse time 100,00 1 000,0 T π ∼ 4 − 20 ns 10,00 100,0 T 2 [µs] T 1 [µs] 1,00 10,0 Big improvements in relaxation 0,10 1,0 and dephasing times in last 10 0,01 0,1 years 0,00 0,0 1999 2002 2007 2012 Error per gates of 0.2%, similar Year to trapped ion results Low error per gates: E. Magesan et al , Phys. Rev. Lett. 109 , 080505 (2012) Long T 1 and T 2 : H. Paik et al , Phys. Rev. Lett. 107 , 240501 (2011)

  24. Superconducting transmon qubits ~ 300 µm | 3 � | 2 � V(t) | 1 � | 0 � J. Koch et al. Phys. Rev. A 76 , 042319 (2007)

  25. Superconducting qubits, a family tree Charge Phase Flux Delft, 1999 NEC, Saclay, 1999 NIST 2002 Quantronium Fluxonium Superinductor: array of junctions a n n e t n A Phase-slip Φ ext junction Tunable coupling a n n junctions (SQUIDs) e t n A Saclay, 2002 5 xmon μ m Yale, 2009 Transmon UCSB, 2013 = Yale, 2007

  26. Circuit QED = V 0 cos ω 01 t V ( t ) ˆ V p | 3 � | 3 � | 2 � | 2 � | 1 � | 1 � | 0 � | 0 �

  27. Circuit QED = V 0 cos ω 01 t V ( t ) V ( t ) | 3 � | 3 � | 2 � | 2 � | 1 � | 1 � | 0 � | 0 �

  28. Circuit QED: Multi-qubit architecture = V 0 cos ω 01 t V ( t ) V ( t ) = V 0 cos ω 01 t V ( t )

  29. Circuit QED: Resonant and dispersive regimes | 3 � | 3 � | 3 � | 3 � | 2 � | 2 � | 2 � | 2 � | 1 � | 1 � | 1 � | 1 � | 0 � | 0 � | 0 � | 0 � Resonant regime: Dispersive regime: • Identical 0-1 transition frequencies • Different 0-1 transition frequencies • Energy exchange between qubits • No energy exchange between qubits and oscillator and oscillator • Oscillator acts as quantum bus for • Qubit-state dependent oscillator entangling qubits frequency leads allows qubit readout

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