piq Équipe de recherche en Physique de l’Information Quantique Superconducting qubits Alexandre Blais Université de Sherbrooke, Québec, Canada
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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)
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
‘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)
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
‘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)
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!
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…
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
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
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
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
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
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
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
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
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)
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…
Josephson junction Superconductor (Al) Capacitor (C) Standard toolkit Insulator (AlO x ) Inductor (L) Superconductor (Al) ✗ Resistor (R) Josephson junctions
Josephson junction E J Capacitor (C) Standard toolkit S I S Inductor (L) ✗ Resistor (R) Josephson junctions 100 nm
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 )
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)
Superconducting transmon qubits ~ 300 µm | 3 � | 2 � V(t) | 1 � | 0 � J. Koch et al. Phys. Rev. A 76 , 042319 (2007)
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
Circuit QED = V 0 cos ω 01 t V ( t ) ˆ V p | 3 � | 3 � | 2 � | 2 � | 1 � | 1 � | 0 � | 0 �
Circuit QED = V 0 cos ω 01 t V ( t ) V ( t ) | 3 � | 3 � | 2 � | 2 � | 1 � | 1 � | 0 � | 0 �
Circuit QED: Multi-qubit architecture = V 0 cos ω 01 t V ( t ) V ( t ) = V 0 cos ω 01 t V ( t )
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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