EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Entanglement Dynamics of Two Superconducting QubitsSubject to Random Telegraph Noise
Marta Agati
Università degl Studi di CataniaDipartimento di Fisica e AstronomiaCorso di Laurea in Fisica
Matis CNR-IMM UOS CataniaCentro Siciliano Fisica Nucleare eStruttura della Materia (CSFNSM)QUINN QUantum INformation andNanonsystems group
RelatoreProf.ssa Elisabetta Paladino
CorrelatoreProf. Giuseppe FalciDott. Antonio D’Arrigo
July 16, 2013
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Contents
1 Quantum ComputationQuantum Computing and Quantum Mechanics
2 Superconducting QubitsCharge Qubit
3 Noise in Josephson QubitsMethods
4 Entanglement DynamicsTransvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Contents
1 Quantum ComputationQuantum Computing and Quantum Mechanics
2 Superconducting QubitsCharge Qubit
3 Noise in Josephson QubitsMethods
4 Entanglement DynamicsTransvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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 ρρ arranged in decreasing order.
Maximally Entangled States C=1Product States C=0Invariance 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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 ρρ arranged in decreasing order.
Maximally Entangled States C=1Product States C=0Invariance 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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 atleast one entanglement-generating two-qubit gate.
CNot Gate
(|0〉+ |1〉) |0〉√2
⇒ |00〉+ |11〉√2
Motivation for our study on thesensitivity of the entanglementto external influences (environment)
Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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 atleast one entanglement-generating two-qubit gate.
CNot Gate
(|0〉+ |1〉) |0〉√2
⇒ |00〉+ |11〉√2
Motivation for our study on thesensitivity of the entanglementto external influences (environment)
Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Superconducting materials and Josephson junctions
Characteristics of Superconducting Materials
Hallmarks:Perfect ConductivityPerfect Diamagnetism (Meissner Effect)
Cooper pairsJosephson Effect
Josephson Equations
I = IC sinφStationary 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Superconducting materials and Josephson junctions
Characteristics of Superconducting Materials
Hallmarks:Perfect ConductivityPerfect Diamagnetism (Meissner Effect)
Cooper pairsJosephson Effect
Josephson Equations
I = IC sinφStationary 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Superconducting Qubits
Charge Qubit Phase Qubit
Other qubits based on Cooper PairBox: Quantronium and Trasmon
Flux Qubit
Appunti del Corso di fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Superconducting Qubits
Charge Qubit Phase Qubit
Other qubits based on Cooper PairBox: Quantronium and Trasmon
Flux Qubit
Appunti del Corso di fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Contents
1 Quantum ComputationQuantum Computing and Quantum Mechanics
2 Superconducting QubitsCharge Qubit
3 Noise in Josephson QubitsMethods
4 Entanglement DynamicsTransvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Network Equations for Josephson Circuits (Lagrangian form)
Electrostatic Energy
K = CΣ2
(~2e φ+
CgCΣ
Vg
)2
CΣ ≡ (C + Cg)
Magnetic Energy
UJ (φ) =∫ t
0 dt ′I(t ′)Φ(t ′) = EJ (1− cosφ)
EJ ≡ ~2e Ic
Lagrangian
L( ~2eφ,
~2e φ) = K (φ)− U(φ)
Classical Hamiltonian
H(Q,~
2eφ) =
12CΣ
(2e~
)(Q − CgVg)2 + EJ (1− cosφ)
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Network Equations for Josephson Circuits (Lagrangian form)
Electrostatic Energy
K = CΣ2
(~2e φ+
CgCΣ
Vg
)2
CΣ ≡ (C + Cg)
Magnetic Energy
UJ (φ) =∫ t
0 dt ′I(t ′)Φ(t ′) = EJ (1− cosφ)
EJ ≡ ~2e Ic
Lagrangian
L( ~2eφ,
~2e φ) = K (φ)− U(φ)
Classical Hamiltonian
H(Q,~
2eφ) =
12CΣ
(2e~
)(Q − CgVg)2 + EJ (1− cosφ)
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Network Equations for Josephson Circuits (Lagrangian form)
Electrostatic Energy
K = CΣ2
(~2e φ+
CgCΣ
Vg
)2
CΣ ≡ (C + Cg)
Magnetic Energy
UJ (φ) =∫ t
0 dt ′I(t ′)Φ(t ′) = EJ (1− cosφ)
EJ ≡ ~2e Ic
Lagrangian
L( ~2eφ,
~2e φ) = K (φ)− U(φ)
Classical Hamiltonian
H(Q,~
2eφ) =
12CΣ
(2e~
)(Q − CgVg)2 + EJ (1− cosφ)
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Charge Qubit Hamiltonian
|n〉, |n + 1〉 ≡ Eigenstates of the charge in the island.
Quantum Hamiltonian (in the charge basis)
H = 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 = − 12 εσz − 1
2 ∆σx
ε ≡ 4EC(1− 2qx )∆ ≡ EJ
σi ≡ Pauli Matrices
Phenomenological Quantization ofthe Phase
[ ~φ2e ,Q
]= i~
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Charge Qubit
Charge Qubit Hamiltonian
|n〉, |n + 1〉 ≡ Eigenstates of the charge in the island.
Quantum Hamiltonian (in the charge basis)
H = 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 = − 12 εσz − 1
2 ∆σx
ε ≡ 4EC(1− 2qx )∆ ≡ EJ
σi ≡ Pauli Matrices
Phenomenological Quantization ofthe Phase
[ ~φ2e ,Q
]= i~
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Noise Sources
Quantum Coherence
|ψ, t〉 =∑
q1,··· ,qNcq1,··· ,qN (t)|q1, · · · , qN〉 =⇒ it exists a well defined
deterministic relation between the complex amplitudes cqi (t) provided by theSchrödinger equation.
Open Quantum SystemDecoherenceNoise
Classical Stochastic ProcessHtot = − 1
2 εσz − 12 ∆σx − 1
2ξ(t)v · −→σ
Particular coupling conditions
Longitudinal Coupling v ‖ HTransvers Coupling v ⊥ H
−→ Density Matrix Formalism
G. Falci, E. Paladino, R. Fazio, Decoherence in Josephson Qubits, Varenna Review 2003
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Noise Sources
Quantum Coherence
|ψ, t〉 =∑
q1,··· ,qNcq1,··· ,qN (t)|q1, · · · , qN〉 =⇒ it exists a well defined
deterministic relation between the complex amplitudes cqi (t) provided by theSchrödinger equation.
Open Quantum SystemDecoherenceNoise
Classical Stochastic ProcessHtot = − 1
2 εσz − 12 ∆σx − 1
2ξ(t)v · −→σ
Particular coupling conditions
Longitudinal Coupling v ‖ HTransvers Coupling v ⊥ H
−→ Density Matrix Formalism
G. Falci, E. Paladino, R. Fazio, Decoherence in Josephson Qubits, Varenna Review 2003
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Noise Sources
Quantum Coherence
|ψ, t〉 =∑
q1,··· ,qNcq1,··· ,qN (t)|q1, · · · , qN〉 =⇒ it exists a well defined
deterministic relation between the complex amplitudes cqi (t) provided by theSchrödinger equation.
Open Quantum SystemDecoherenceNoise
Classical Stochastic ProcessHtot = − 1
2 εσz − 12 ∆σx − 1
2ξ(t)v · −→σ
Particular coupling conditions
Longitudinal Coupling v ‖ HTransvers Coupling v ⊥ H
−→ Density Matrix Formalism
G. Falci, E. Paladino, R. Fazio, Decoherence in Josephson Qubits, Varenna Review 2003
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Noise in Josephson QubitsInternal sources: Exitation of Quasi-particlesExternal environment:
CircuitPreparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Backgroundfluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer ofCharge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2γ
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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Noise in Josephson QubitsInternal sources: Exitation of Quasi-particlesExternal environment:
CircuitPreparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Backgroundfluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer ofCharge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2γ
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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Noise in Josephson QubitsInternal sources: Exitation of Quasi-particlesExternal environment:
CircuitPreparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Backgroundfluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer ofCharge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2γ
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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Noise in Josephson QubitsInternal sources: Exitation of Quasi-particlesExternal environment:
CircuitPreparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Backgroundfluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer ofCharge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2γ
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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Noise in Josephson QubitsInternal sources: Exitation of Quasi-particlesExternal environment:
CircuitPreparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Backgroundfluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer ofCharge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2γ
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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Contents
1 Quantum ComputationQuantum Computing and Quantum Mechanics
2 Superconducting QubitsCharge Qubit
3 Noise in Josephson QubitsMethods
4 Entanglement DynamicsTransvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Master Equation
Weak coupling and fast fluctuator: v Ω and v γ
ΓR = 12 sin2 θS(Ω) Relaxation Rate
(Decay of z-component of thequbit Bloch vector)
Γφ = Γ0φ + 1
2 ΓR = 12 cos2θS(0) + 1
2 ΓR
Dephasing Rate (Decay of x- andy -components of thequbit Bloch vector)
Microscopic Model of Background Charges
H = − 12 εσz − 1
2 ∆σx + εb+b +∑
k [Tk c+k b + h.c.] +
∑k εk c+
k ck + (v/2)σzb+b
ξ(t) = 0,+1 Asymmetric fluctuatorξ(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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Master Equation
Weak coupling and fast fluctuator: v Ω and v γ
ΓR = 12 sin2 θS(Ω) Relaxation Rate
(Decay of z-component of thequbit Bloch vector)
Γφ = Γ0φ + 1
2 ΓR = 12 cos2θS(0) + 1
2 ΓR
Dephasing Rate (Decay of x- andy -components of thequbit Bloch vector)
Microscopic Model of Background Charges
H = − 12 εσz − 1
2 ∆σx + εb+b +∑
k [Tk c+k b + h.c.] +
∑k εk c+
k ck + (v/2)σzb+b
ξ(t) = 0,+1 Asymmetric fluctuatorξ(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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Master Equation
Weak coupling and fast fluctuator: v Ω and v γ
ΓR = 12 sin2 θS(Ω) Relaxation Rate
(Decay of z-component of thequbit Bloch vector)
Γφ = Γ0φ + 1
2 ΓR = 12 cos2θS(0) + 1
2 ΓR
Dephasing Rate (Decay of x- andy -components of thequbit Bloch vector)
Microscopic Model of Background Charges
H = − 12 εσz − 1
2 ∆σx + εb+b +∑
k [Tk c+k b + h.c.] +
∑k εk c+
k ck + (v/2)σzb+b
ξ(t) = 0,+1 Asymmetric fluctuatorξ(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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Methods
Quasi-Hamiltonian Method
Transition Probability Matrix(RTN)
W =
[1− p p
p 1− p
] Element of Qubit Transfer Matrix T (withoutnoise)
Tijξi(∆t) = 1
2 Tr [σiUξi (∆t)σjU+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〉eiωψ t〈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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Two-Qubit System
Two-Qubit Density Matrix
Two uncorrelated systems, each composed of a singlequbit and a background charge.The two-qubit density matrix depends on the initialconditions ρ(0) and on the time-evolution of each qubit,namely qubit A and qubit B under their own source ofnoise.The time-evolution is obtained the average transfermatrices relative to qubit A and B: T A(t), T B(t).
ρ(t) = f (T A(t)⊗ T B(t), ρ(0))
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Two-Qubit System
Two-Qubit Density Matrix
Two uncorrelated systems, each composed of a singlequbit and a background charge.The two-qubit density matrix depends on the initialconditions ρ(0) and on the time-evolution of each qubit,namely qubit A and qubit B under their own source ofnoise.The time-evolution is obtained the average transfermatrices relative to qubit A and B: T A(t), T B(t).
ρ(t) = f (T A(t)⊗ T B(t), ρ(0))
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Entanglement time-evolution
Entanglement Sudden DeathMarkovian noise, weak coupling
Entanglement RevivalsMarkovian noise, strong couplingNon-Markovian noise
Initial Conditions: Extended Werner Like (EWL) States
ρΦ = r |Φ〉〈Φ|+ 1−r4 I ρΨ = 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 entanglementof 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Entanglement time-evolution
Entanglement Sudden DeathMarkovian noise, weak coupling
Entanglement RevivalsMarkovian noise, strong couplingNon-Markovian noise
Initial Conditions: Extended Werner Like (EWL) States
ρΦ = r |Φ〉〈Φ|+ 1−r4 I ρΨ = 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 entanglementof 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Contents
1 Quantum ComputationQuantum Computing and Quantum Mechanics
2 Superconducting QubitsCharge Qubit
3 Noise in Josephson QubitsMethods
4 Entanglement DynamicsTransvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Noise on one qubit: Concurrence Decay and Revivalsr=1
Weak Coupling −→
0 1 2 3 40.00.20.40.60.81.0
Τ
ÈnHΤLÈ
a
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0
0.2
0.4
0.6
0.8
1.0
Τ
Λ1
HΤL,Λ
2HΤL,
Λ3
HΤL,Λ
4HΤL
b
0.0 0.5 1.0 1.5 2.0 2.50.00.20.40.60.81.0
Τ
CHΤL
c
Transition Region −→
0 1 2 3 40.00.20.40.60.81.0
Τ
ÈnHΤLÈ
a
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0
0.2
0.4
0.6
0.8
1.0
Τ
Λ1
HΤL,Λ
2HΤL,
Λ3
HΤL,Λ
4HΤL b
0.00.51.01.52.02.50.00.20.40.60.81.0
Τ
CHΤL
c
Strong Coupling
0 1 2 3 40.00.20.40.60.81.0
Τ
ÈnHΤLÈ
a
0.5 1.0 1.5 2.0 2.5 3.0 3.50.0
0.2
0.4
0.6
0.8
1.0
Τ
Λ1
HΤL,Λ
2HΤL,
Λ3
HΤL,Λ
4HΤL b
0.00.51.01.52.02.50.00.20.40.60.81.0
Τ
CHΤL
c
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Noise on both qubits
Equal weakly coupled noise
0 1 2 3 4-1.0-0.5
0.00.51.0
Τ
nyHΤL
a
0 1 2 3 40.00.20.40.60.81.0
Τ
n zHΤL
b
0 1 2 3 40.00.20.40.60.81.0
Τ
ÈnHΤLÈ
a
0 1 2 3 40.00.20.40.60.81.0
Τ
CHΤL
d
Wekly coupled noise on one qubitand Strong coupled noise on the other
0 1 2 3 4-1.0-0.5
0.00.51.0
Τ
nyHΤL
a
0 1 2 3 4-1.0-0.5
0.00.51.0
Τ
nyHΤL
a
0 1 2 3 40.00.20.40.60.81.0
Τ
n zHΤL
b
0 1 2 3 40.00.20.40.60.81.0
Τ
n zHΤL
b
0 1 2 3 40.00.20.40.60.81.0
Τ
ÈnHΤLÈ
a
0 1 2 3 40.00.20.40.60.81.0
Τ
ÈnHΤLÈ
a
0 1 2 3 40.00.20.40.60.81.0
Τ
CHΤL
d
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Contents
1 Quantum ComputationQuantum Computing and Quantum Mechanics
2 Superconducting QubitsCharge Qubit
3 Noise in Josephson QubitsMethods
4 Entanglement DynamicsTransvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Noise on one qubitr=1
Asymmetric versus SymmetricWeak coupling
0 1 2 3 4-1.0
-0.5
0.0
0.5
1.0
Τ
nyHΤ
L
W
Γ
=40,v
Γ
=2
"Strong" coupling
0 1 2 3 4-1.0
-0.5
0.0
0.5
1.0
Τ
nyHΤ
L
W
Γ
=40,v
Γ
=18
Transition Region
0 1 2 3 4-1.0
-0.5
0.0
0.5
1.0
Τ
nyHΤ
L
W
Γ
=40,v
Γ
=9
"Strong" SymmetricFluctuator
0 1 2 3 4
-1.0-0.5
0.00.51.0
Τ
nyHΤL
avΓ=14
0 20 40 60 80 1000.00.20.40.60.81.01.2
Τ
n zHΤL
bvΓ=14
0 20 40 60 80 1000.00.20.40.60.81.0
Τ
nHΤLÈ
cvΓ=14
0 20 40 60 80 1000.00.20.40.60.81.0
Τ
CHΤL
vΓ=14
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Contents
1 Quantum ComputationQuantum Computing and Quantum Mechanics
2 Superconducting QubitsCharge Qubit
3 Noise in Josephson QubitsMethods
4 Entanglement DynamicsTransvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric FluctuatorTransvers Coupling, Comparison with Symmetric FluctuatorComparison with Longitudinal Coupling
Noise on one qubitr=0.91
Longitudinal Couplingv/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 3.50.0
0.2
0.4
0.6
0.8
Τ
CHΤ
L
vΓ=2
vΓ=5
vΓ=9vΓ=14
vΓ=18
R. Lo Franco, A. D’Arrigo, G. Falci, C. Compagno, E. Paladino, Entanglement dynamics in superconducting qubits affected by localbistable impurities, Phys.Scr., 9, (2012)
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
Epilogue
Two superconducting qubits, each subject indipendently to RandomTelegraph Noise.
Example: Random Telegraph Noise by charged impurities trapped close toa 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
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement 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 theentanglement displays ESD, while in strong coupling conditions theentanglement displays dark peridos and revivals.
−→ For a symmetric fluctuator model the entanglement decays both inweak coupling conditions and in strong coupling conditions. Theentanglement can also definitively vanish starting with a pure state oran EWL state.
Marta Agati Entanglement Dynamics
EntanglementDynamics
Marta Agati
QuantumComputa-tionQuantumComputingand QuantumMechanics
SuperconductingQubitsCharge Qubit
Noise inJosephsonQubitsMethods
EntanglementDynamicsTransversCoupling,AsymmetricFluctuator
TransversCoupling,ComparisonwithSymmetricFluctuator
ComparisonwithLongitudinalCoupling
Conclusions
Quantum ComputationSuperconducting Qubits
Noise in Josephson QubitsEntanglement Dynamics
Conclusions
So...
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Marta Agati Entanglement Dynamics
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