Entanglement Dynamics of Two Superconducting Qubits

1. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Entanglement Dynamics of Two Superconducting Qubits Subject to Random Telegraph Noise Marta Agati Università degl Studi di Catania Dipartimento di Fisica e Astronomia Corso di Laurea in Fisica Matis CNR-IMM UOS Catania Centro Siciliano Fisica Nucleare e Struttura della Materia (CSFNSM) QUINN QUantum INformation and Nanonsystems group Relatore Prof.ssa Elisabetta Paladino Correlatore Prof. Giuseppe Falci Dott. Antonio D’Arrigo July 16, 2013 Marta Agati Entanglement Dynamics

2. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Contents 1 Quantum Computation Quantum Computing and Quantum Mechanics 2 Superconducting Qubits Charge Qubit 3 Noise in Josephson Qubits Methods 4 Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling 5 Conclusions Marta Agati Entanglement Dynamics

3. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Quantum Computing and Quantum Mechanics Introduction to Quantum Computation Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010 Marta Agati Entanglement Dynamics

4. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Quantum Computing and Quantum Mechanics Contents 1 Quantum Computation Quantum Computing and Quantum Mechanics 2 Superconducting Qubits Charge Qubit 3 Noise in Josephson Qubits Methods 4 Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling 5 Conclusions Marta Agati Entanglement Dynamics

5. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Quantum Computing and Quantum Mechanics Unit of Quantum Information Quantum bit or Qubit Quantum bit or Qubit ψ = α0|0 + α1|1 Superposition Principle Multiple-Qubit state Two qubits ψ = α00|00 + α01|01 + α10|10 + α11|11 Product State ψS = |01 +|11 √ 2 = |0 +|1 √ 2 ⊗ |1 Entangled State (Bell State) ψE = |00 +|11 √ 2 Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010 Marta Agati Entanglement Dynamics

8. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Quantum Computing and Quantum Mechanics Entanglement Quantiﬁers ρ ≡ Two-Qubit Density Matrix =⇒ ˜ρ = (σy ⊗ σy ) ρ (σy ⊗ σy ) Wootters Concurrence C(t) = 2Max 0, √ λ1 − √ λ2 − √ λ3 − √ λ4 λi , i = {1, . . . , 4}, eigenvalues of the matrix ρ˜ρ arranged in decreasing order. Maximally Entangled States C=1 Product States C=0 Invariance for Local Unitary Transformations. W. K. Wotters, Entanglement of Formation of an Arbitrary State of two Qubits, Phys. Rev. Lett., 80, 10,( 1998) Marta Agati Entanglement Dynamics

9. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Quantum Computing and Quantum Mechanics Entanglement Quantiﬁers ρ ≡ Two-Qubit Density Matrix =⇒ ˜ρ = (σy ⊗ σy ) ρ (σy ⊗ σy ) Wootters Concurrence C(t) = 2Max 0, √ λ1 − √ λ2 − √ λ3 − √ λ4 λi , i = {1, . . . , 4}, eigenvalues of the matrix ρ˜ρ arranged in decreasing order. Maximally Entangled States C=1 Product States C=0 Invariance for Local Unitary Transformations. W. K. Wotters, Entanglement of Formation of an Arbitrary State of two Qubits, Phys. Rev. Lett., 80, 10,( 1998) Marta Agati Entanglement Dynamics

10. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Quantum Computing and Quantum Mechanics Quantum Gates Universary set of Quantum Gates Any multiple qubits logic gate may be composed of single qubit gates and at least one entanglement-generating two-qubit gate. CNot Gate (|0 + |1 ) |0 √ 2 ⇒ |00 + |11 √ 2 Motivation for our study on the sensitivity of the entanglement to external inﬂuences (environment) Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010 Marta Agati Entanglement Dynamics

11. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Quantum Computing and Quantum Mechanics Quantum Gates Universary set of Quantum Gates Any multiple qubits logic gate may be composed of single qubit gates and at least one entanglement-generating two-qubit gate. CNot Gate (|0 + |1 ) |0 √ 2 ⇒ |00 + |11 √ 2 Motivation for our study on the sensitivity of the entanglement to external inﬂuences (environment) Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010 Marta Agati Entanglement Dynamics

12. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Quantum Computing and Quantum Mechanics Quantum Computers Implementations G. Chen, D. A. Church, B.G. Englert, C. Henkel, B. Ronwedder, M. O. Scully, M. Zubairy, Quantum Computing Devices: principles, Designs and Analysis, Chapman et Hall/CRC, 2007 Marta Agati Entanglement Dynamics

13. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Charge Qubit Superconducting materials and Josephson junctions Characteristics of Superconducting Materials Hallmarks: Perfect Conductivity Perfect Diamagnetism (Meissner Effect) Cooper pairs Josephson Effect Josephson Equations I = IC sin φ Stationary Josephson Effect: a current ﬂows at 0 Voltage. V(t) = 2e ∂ ∂t φ A.C. Josephson Effect Michael Tinkham, Introduction to Superconductivity, McGRAW-HILL EDITIONS, 1996 Marta Agati Entanglement Dynamics

14. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Charge Qubit Superconducting materials and Josephson junctions Characteristics of Superconducting Materials Hallmarks: Perfect Conductivity Perfect Diamagnetism (Meissner Effect) Cooper pairs Josephson Effect Josephson Equations I = IC sin φ Stationary Josephson Effect: a current ﬂows at 0 Voltage. V(t) = 2e ∂ ∂t φ A.C. Josephson Effect Michael Tinkham, Introduction to Superconductivity, McGRAW-HILL EDITIONS, 1996 Marta Agati Entanglement Dynamics

15. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Charge Qubit Superconducting Qubits Charge Qubit Phase Qubit Other qubits based on Cooper Pair Box: Quantronium and Trasmon Flux Qubit Appunti del Corso di ﬁsica dei Nanosistemi, Giuseppe Falci, AA 2012-2013 Marta Agati Entanglement Dynamics

16. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Charge Qubit Superconducting Qubits Charge Qubit Phase Qubit Other qubits based on Cooper Pair Box: Quantronium and Trasmon Flux Qubit Appunti del Corso di ﬁsica dei Nanosistemi, Giuseppe Falci, AA 2012-2013 Marta Agati Entanglement Dynamics

17. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Charge Qubit Contents 1 Quantum Computation Quantum Computing and Quantum Mechanics 2 Superconducting Qubits Charge Qubit 3 Noise in Josephson Qubits Methods 4 Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling 5 Conclusions Marta Agati Entanglement Dynamics

18. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Charge Qubit Network Equations for Josephson Circuits (Lagrangian form) Electrostatic Energy K = CΣ 2 2e ˙φ + Cg CΣ Vg 2 CΣ ≡ (C + Cg) Magnetic Energy UJ (φ) = t 0 dt I(t ) ˙Φ(t ) = EJ (1 − cosφ) EJ ≡ 2e Ic Lagrangian L(2e φ, 2e ˙φ) = K( ˙φ) − U(φ) Classical Hamiltonian H(Q, 2e φ) = 1 2CΣ 2e (Q − CgVg)2 + EJ (1 − cos φ) Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013 Marta Agati Entanglement Dynamics

21. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Charge Qubit Charge Qubit Hamiltonian |n , |n + 1 ≡ Eigenstates of the charge in the island. Quantum Hamiltonian (in the charge basis) ˆH = EC n (n − qg)2 |n n| − EJ 2 n |n n + 1| + |n + 1 n| Projection on to the lowest energy bidimensional subspace Charge Qubit Hamiltonian Hq = −1 2 σz − 1 2 ∆σx ≡ 4EC(1 − 2qx ) ∆ ≡ EJ σi ≡ Pauli Matrices Phenomenological Quantization of the Phase φ 2e , Q = i Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013 Marta Agati Entanglement Dynamics

22. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Charge Qubit Charge Qubit Hamiltonian |n , |n + 1 ≡ Eigenstates of the charge in the island. Quantum Hamiltonian (in the charge basis) ˆH = EC n (n − qg)2 |n n| − EJ 2 n |n n + 1| + |n + 1 n| Projection on to the lowest energy bidimensional subspace Charge Qubit Hamiltonian Hq = −1 2 σz − 1 2 ∆σx ≡ 4EC(1 − 2qx ) ∆ ≡ EJ σi ≡ Pauli Matrices Phenomenological Quantization of the Phase φ 2e , Q = i Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013 Marta Agati Entanglement Dynamics

23. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Methods Noise Sources Quantum Coherence |ψ, t = q1,··· ,qN cq1,··· ,qN (t)|q1, · · · , qN =⇒ it exists a well deﬁned deterministic relation between the complex amplitudes cqi (t) provided by the Schrödinger equation. Open Quantum System Decoherence Noise Classical Stochastic Process Htot = −1 2 σz − 1 2 ∆σx − 1 2 ξ(t)v · −→σ Particular coupling conditions Longitudinal Coupling v H Transvers Coupling v ⊥ H −→ Density Matrix Formalism G. Falci, E. Paladino, R. Fazio, Decoherence in Josephson Qubits, Varenna Review 2003 Marta Agati Entanglement Dynamics

26. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Methods Noise in Josephson Qubits Internal sources: Exitation of Quasi-particles External environment: Circuit Preparation, Control and measurement apparata Dynamic defects ﬂuctuating between two localized states (Background ﬂuctuators) produce random telegraph noise (RTN) Example Background charged impurities trapped close to the insulating layer of Charge Qubits or in the substrate. Power Spectrum RTN S(ω) = v2 2 γ γ2+ω2 E. Paladino, Y. M. Galperin, G. Falci, B. L. Altshuler, 1/f noise: implications for solid-state quantum information,a rXiv:1304.7925, submitted to RMP Marta Agati Entanglement Dynamics

31. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Methods Contents 1 Quantum Computation Quantum Computing and Quantum Mechanics 2 Superconducting Qubits Charge Qubit 3 Noise in Josephson Qubits Methods 4 Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling 5 Conclusions Marta Agati Entanglement Dynamics

32. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Methods Master Equation Weak coupling and fast fluctuator: v Ω and v γ ΓR = 1 2 sin2 θS(Ω) Relaxation Rate (Decay of z-component of the qubit Bloch vector) Γφ = Γ0 φ + 1 2 ΓR = 1 2 cos2 θS(0) + 1 2 ΓR Dephasing Rate (Decay of x- and y-components of the qubit Bloch vector) Microscopic Model of Background Charges ˆH = −1 2 σz − 1 2 ∆σx + b+ b + k [Tk c+ k b + h.c.] + k k c+ k ck + (v/2)σz b+ b ξ(t) = 0, +1 Asymmetric fluctuator ξ(t) = −1, +1 Symmetric fluctuator E. Paladino, L. Faoro, G. Falci, Decoherence Due to Discrete Noise in Josephson Qubits, Adv. in Sol. St. Phys., 43, (2003) Marta Agati Entanglement Dynamics

35. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Methods Quasi-Hamiltonian Method Transition Probability Matrix (RTN) W = 1 − p p p 1 − p Element of Qubit Transfer Matrix T(without noise) Tijξi (∆t) = 1 2 Tr[σi Uξi (∆t)σj U+ ξi (∆t)] Average Tranfer Matrix T(t) ≡ xf |ΓN |if Γ ≡ W ⊗ T Quasi-Hamiltonian HqH ΓN (t) ≡ (Γ(∆t))N ∼ (I − iHqH ∆t)N ∼ exp(−iHqH t) First order expansion Bloch vector evolution under noise n(t) = xf | ψ |ψ eiωψt ψ| |if n(0) B. Cheng, Q.-H. Wang and R. Joynt, Transfer matrix solution of a model of qubit dechoerence due to telegraph noise, Physical Review A, 78, (2008) Marta Agati Entanglement Dynamics

36. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Two-Qubit System Two-Qubit Density Matrix Two uncorrelated systems, each composed of a single qubit and a background charge. The two-qubit density matrix depends on the initial conditions ρ(0) and on the time-evolution of each qubit, namely qubit A and qubit B under their own source of noise. The time-evolution is obtained the average transfer matrices relative to qubit A and B: TA(t), TB(t). ρ(t) = f(TA(t) ⊗ TB(t), ρ(0)) Marta Agati Entanglement Dynamics

37. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Two-Qubit System Two-Qubit Density Matrix Two uncorrelated systems, each composed of a single qubit and a background charge. The two-qubit density matrix depends on the initial conditions ρ(0) and on the time-evolution of each qubit, namely qubit A and qubit B under their own source of noise. The time-evolution is obtained the average transfer matrices relative to qubit A and B: TA(t), TB(t). ρ(t) = f(TA(t) ⊗ TB(t), ρ(0)) Marta Agati Entanglement Dynamics

38. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Entanglement time-evolution Entanglement Sudden Death Markovian noise, weak coupling Entanglement Revivals Markovian noise, strong coupling Non-Markovian noise Initial Conditions: Extended Werner Like (EWL) States ˆρΦ = r|Φ Φ| + 1−r 4 I ˆρΨ = r|Ψ Ψ| + 1−r 4 I r quantiﬁes the mixedness; |Φ = a|00 ± b|11 |Ψ = a|01 ± b|10 where a represents the initial degree of entanglement of the pure part and |a|2 + |b|2 = 1. T. Yu and J. H. Eberly, Sudden Death of Entanglement,Science, 323, (2009) Marta Agati Entanglement Dynamics

39. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Entanglement time-evolution Entanglement Sudden Death Markovian noise, weak coupling Entanglement Revivals Markovian noise, strong coupling Non-Markovian noise Initial Conditions: Extended Werner Like (EWL) States ˆρΦ = r|Φ Φ| + 1−r 4 I ˆρΨ = r|Ψ Ψ| + 1−r 4 I r quantiﬁes the mixedness; |Φ = a|00 ± b|11 |Ψ = a|01 ± b|10 where a represents the initial degree of entanglement of the pure part and |a|2 + |b|2 = 1. T. Yu and J. H. Eberly, Sudden Death of Entanglement,Science, 323, (2009) Marta Agati Entanglement Dynamics

40. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Contents 1 Quantum Computation Quantum Computing and Quantum Mechanics 2 Superconducting Qubits Charge Qubit 3 Noise in Josephson Qubits Methods 4 Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling 5 Conclusions Marta Agati Entanglement Dynamics

41. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Noise on one qubit: Concurrence Decay and Revivals r=1 Weak Coupling −→ 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nΤ a 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0 Τ Λ1Τ,Λ2Τ,Λ3Τ,Λ4Τ b 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.2 0.4 0.6 0.8 1.0 Τ CΤ c Transition Region −→ 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nΤ a 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0 Τ Λ1Τ,Λ2Τ,Λ3Τ,Λ4Τ b 0.00.51.01.52.02.5 0.0 0.2 0.4 0.6 0.8 1.0 Τ CΤ c Strong Coupling 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nΤ a 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 1.0 Τ Λ1Τ,Λ2Τ,Λ3Τ,Λ4Τ b 0.00.51.01.52.02.5 0.0 0.2 0.4 0.6 0.8 1.0 Τ CΤ c Marta Agati Entanglement Dynamics

42. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Noise on both qubits Equal weakly coupled noise 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 Τ nyΤ a 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nzΤ b 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nΤ a 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ CΤ d Wekly coupled noise on one qubit and Strong coupled noise on the other 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 Τ nyΤ a 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 Τ nyΤ a 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nzΤ b 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nzΤ b 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nΤ a 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ nΤ a 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Τ CΤ d Marta Agati Entanglement Dynamics

44. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Noise on one qubit r=1 Asymmetric versus Symmetric Weak coupling 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 Τ nyΤ Γ 40, v Γ 2 "Strong" coupling 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 Τ nyΤ Γ 40, v Γ 18 Transition Region 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 Τ nyΤ Γ 40, v Γ 9 "Strong" Symmetric Fluctuator 0 1 2 3 4 1.0 0.5 0.0 0.5 1.0 Τ nyΤ a v Γ 14 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Τ nzΤ b v Γ 14 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Τ nΤ c v Γ 14 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 1.0 Τ CΤ v Γ 14 Marta Agati Entanglement Dynamics

46. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Noise on one qubit r=0.91 Longitudinal Coupling v/γ = 0.5 Weak Coupling v/γ = 5 Strong Coupling Transvers Coupling (Crossover) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.2 0.4 0.6 0.8 Τ CΤ v Γ 2 v Γ 5 v Γ 9 v Γ 14 v Γ 18 R. Lo Franco, A. D’Arrigo, G. Falci, C. Compagno, E. Paladino, Entanglement dynamics in superconducting qubits affected by local bistable impurities, Phys.Scr., 9, (2012) Marta Agati Entanglement Dynamics

47. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Epilogue Two superconducting qubits, each subject indipendently to Random Telegraph Noise. Example: Random Telegraph Noise by charged impurities trapped close to a charge Josephson qubit. Microscopic model of the RTN generation. Application of the Quasi-Hamiltonian method. Evaluation of the two-qubit density matrix. Evaluation of the concurrence Marta Agati Entanglement Dynamics

48. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions Results −→ Crossover between weak coupling and strong coupling. −→ Asymmetric and symmetric fluctuator and comparison. −→ Initial conditions as pure state and Extended Werner Like (EWL) state. −→ Analogous behaviour for the states ρΦ and ρΨ. −→ For an asymmetric fluctuator model in weak coupling conditions the entanglement displays ESD, while in strong coupling conditions the entanglement displays dark peridos and revivals. −→ For a symmetric fluctuator model the entanglement decays both in weak coupling conditions and in strong coupling conditions. The entanglement can also definitively vanish starting with a pure state or an EWL state. Marta Agati Entanglement Dynamics

49. Entanglement Dynamics Marta Agati Quantum Computa- tion Quantum Computing and Quantum Mechanics Superconducting Qubits Charge Qubit Noise in Josephson Qubits Methods Entanglement Dynamics Transvers Coupling, Asymmetric Fluctuator Transvers Coupling, Comparison with Symmetric Fluctuator Comparison with Longitudinal Coupling Conclusions Quantum Computation Superconducting Qubits Noise in Josephson Qubits Entanglement Dynamics Conclusions So... THANK YOU FOR THE KIND ATTENTION Marta Agati Entanglement Dynamics

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