Principle Concepts behind IBM Q D. Stancil, Quantum Computing Seminar, 15 Feb 2018

Transcription

1 Priniple Conepts behind IBM Q D. Stanil, Quantum Computing Seminar, 15 Feb 018 V. Bouhiat, et al, Quantum Coherene with a Single Cooper Pair, Physia Sripta, Vol. T76, (1998) A. Blais, et al, Cavity quantum eletrodynamis for superonduting eletrial iruits: An arhiteture for quantum omputation, Phys. Rev. A, Vol. 9, 0630 (004) J. Koh, et al, Charge-insensitive qubit design derived from the Cooper Pair Box, Phys. Rev. A Vol. 76, (007). J. Clarke & F. Wilhelm, Superonduting quantum bits, Nature, Vol. 453, 19 June 008, p J. Gambetta, et al, Building logial qubits in a superonduting quantum omputing system, npj Quantum Information, 13 Jan. 017

2 Outline Cavity Quantum Eletrodynamis Rabi osillations Cooper pairs and superondutivity Josephson juntion SQUID Cooper Pair Box Transmon Entangled transmons IBM examples

3 Cavity Quantum Eletrodynamis EM field reation & annihilation operators: a, a Atomi energy level raising and + lowering operators: σ, σ Hamiltonian of oupled system (Jaynes-Cummings Hamiltonian): Blais, et al 1 Ω = ( + ) κ γ z H ωr aa σ g aσ σ a H H EM field atomi hamiltonian Coupling between atom and EM field Cavity energy loss Radiative energy loss

4 Rabi Osillations When a two-level system is oupled to a driving field at preisely the frequeny orresponding to the energy differene between the states, the system will osillate between the two states at the Rabi frequeny ψ = ()0 t + ()1 t 0 1 if (0) = 0, (0) = 1, then P0() t = 0() t = 1 osωrt 1 Pt 1() = 1() t = 1+ osωrt ( ) ( ) Ω R t Ω R = 1 H I 0

5 Changing States with Pulses π-pulse : Ω t = π R / π/-pulse : Ω t = π R (Hadamard gate) inverts the state reates equal superposition of states Key point: you an flip a state or reate a superposition state by ontrolling the pulse length Ω R t

6 Cooper Pairs and Superondutivity Spin ½ partiles are Fermions Fermions obey the Pauli exlusion priniple: no two an be in the same state Eletrons are Fermions Spin 1 partiles are Bosons Bosons do not obey the Pauli exlusion priniple: you an have as many in a state as you want Photons are Bosons In a superondutor, an effetive attrative interation between eletrons auses them to be loosely bound together and at like a single spin 1 partile: Cooper Pair Sine Cooper pairs are spin 1, they at like Bosons, and you an have multiple Cooper pairs in the same state

7 Cooper Pairs are the result of the Eletron-Phonon interation in the theory of Bardeen, Cooper, and Shreifer (BCS Theory) Eletrons normally repel one another, but are attrated to ions in the rystal lattie If the ions are pulled slightly toward an eletron, from a distane it an appear as though there is a net positive harge, attrating another eletron Image from Quora

8 Josephson tunnel juntion Looks like a non-linear indutor: origin of anharmoniity: spaing between energy levels is not the same Enables the individual addressing of a single pair of states In ontrast, in the harmoni osillator, all states are equally spaed Ph.D. thesis: Vratislav Mihal di = I dt = I V = = dϕ osϕ dt πv osϕ Φ 0 0 = = π I 1 ( I / I) dt 0 Φ0 di πi osϕ dt πi 1 sin Φ Φ di ϕ dt Clarke & Wilhelm di L eff di ( I) dt

9 Superonduting Quantum Interferene Devie (SQUID) Parallel Josephson Juntions Current depends on applied magneti field Preview: enables qubits to be tuned by an external magneti field πφ πφ πφ i = I os sin φ1+, ϕ ϕ1 = πn+ Φ0 Φ0 Φ0 LI πφ πφ φ = φext + sin os φ1 + Φ0 Φ0 φ = BA ext Area A (transendental equation for φ) MIT Leture 1

10 Cooper Pair Box Energy in a apaitor: For an eletron: 1 = = Q E = C E CU, Q CU Q= e, e E =, CΣ = C + C C g j Σ E is the harge energy assoiated with a single eletron Total harge energy with n C-Ps: Q = en E q = 4En Bouhiat, et al

11 Cooper Pair Box Hamiltonion The harge stored on a box with an applied voltage U to the gate : Q = C U = en The Hamiltonian of an isolated box an be written Additional C-Ps an tunnel in & out through the Josephson juntion, desribed by the oupling Hamiltonian E This interation opens gaps of width E J at the rossing points n g g CU g = e ( ) H = 4E n n n n Q g n ( 1 1 ) J HJ = n n+ + n+ n n g sweet spot Bouhiat, et al.

12 Two lowest states of CPB H ψ = E ψ EJ 4E ( n ng) n n ( n n+ 1 + n+ 1 n ) ψ = E ψ n n Consider the lowest two states only, 0, 1 : EJ 4En g 0 1 = E0 EJ 4E( 1 ng) 1 0 = E 1 EJ E0 0 1 = E 0 EJ E1 1 0 = E 1 Look for Eigenvalues and Eigenvetors: E0 EJ / 0 0 E EJ / E 1 = 1 1 E / EJ / 0 0 E EJ / E / = 1 1 E 1 0 E 0 1 J 0 0 E = 1 1 E x E σ J σ z 0 0 E = 1 1 H = sh, 1 s= σ, h= xe ˆ J + ze ˆ Bouhiat, et al.

13 Ciruit Quantum Eletrodynamis Co-planar mirostrip resonator formed by gaps in enter ondutor Isomorphi with Cavity Quantum Eletrodynamis used in trapped ion quantum omputers Express Hamiltonian in up & down states along the field h 1 Ω = ( + ) z H ωr aa σ g aσ σ a EM field CPB hamiltonian Coupling between CPB and EM field in oplanar transmission line Blais, et al

14 Control and Read-out When the CPB ouples to the mirostrip resonator and both are tuned to the same frequeny, there is a splitting into two modes Sending in a pulse near ω r enables you to read-out the state either from the phase, or the amplitude Sending in a pulse detuned from the resonane rotates the state, but does not make a measurement (there is no information about the state in the refleted signal) Blais, et al

15 Control Pulse simulation Control signal turned on for 7 pi pulses then turned off The state rotates between the exited and ground states The avity photon level is small sine the avity is detuned from resonane Blais, et al

16 Cooper Pair Box vs Transmon Quantities ruial to operation of CPB: Anharmoniity (energy levels not equally spaed results from JJ nonlinearities) Charge dispersion (variation of energy levels to flutuations in offset harge and gate voltage redues oherene time) Key ratio: E J /E C Small values give larger anharmoniity and lean CPB harge states Large values redue anharmoniity but also redue sensitivity to harge flutuations Anharmoniity redues algebraially with inreasing E J /E C while harge dispersion redues exponentially with inreasing E J /E C Transmon: similar to CPB but operates at a large E J /E C to inrease oherene time while maintaining adequate anharmoniity

17 Transmon Adds extra apaitane by adding λ/0 transmission lines to either side of a pair of Josephson Juntions forming a SQUID Inreases the ratio E J /E by reduing E : e E =, CΣ = C + C + C C J g B Σ Koh, et al (not to sale)

18 Reduing the Sensitivity to bias voltage with E J /E In the large E J /E limit, the energy differene between the ground and exited states beomes insensitive to the bias voltage Suffiient differene in energy levels is maintained Koh, et al.

19 Mehanial Analogy Large E J /E means gravity dominates the effet of the harge & B field Look at exitations near ϕ = 0 Charge noise stability the result of exponential dependene of tunneling on barrier width Anharmoniity is the result of the sinusoidal potential energy n E E n J g = = = L z mgl / 8ml qb l 0 Koh, et al.

20 Entangling multiple qubits Qubits oupled through the photon field Multiple qubits ould be plaed in the same o-planar mirostrip resonator Qubits in separate resonators an be oupled via a mirostirp bus

21 Example Devies Fabriated at IBM qubits 1 bus readout resonators (-qubit gates) 3 qubits bus 3 readout resonators (demonstrated parity Measurement) 4 qubits 4 bus 4 readout resonators (demonstrated [,0,] ode) 8 qubits 4 bus 8 readout resonators (study both Z and X Parity heks) Gambetta, et al Mirograph of individual transmon qubit