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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 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
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Page 1: Presentazone15

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

Page 2: Presentazone15

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

Page 3: Presentazone15

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

Page 4: Presentazone15

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

Page 5: Presentazone15

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

Page 6: Presentazone15

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

Page 7: Presentazone15

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

Page 8: Presentazone15

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

Page 9: Presentazone15

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

Page 10: Presentazone15

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

Page 11: Presentazone15

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

Page 12: Presentazone15

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

Page 13: Presentazone15

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

Page 14: Presentazone15

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

Page 15: Presentazone15

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

Page 16: Presentazone15

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

Page 17: Presentazone15

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

Page 18: Presentazone15

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

Page 19: Presentazone15

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

Page 20: Presentazone15

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

Page 21: Presentazone15

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

Page 22: Presentazone15

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

Page 23: Presentazone15

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

Page 24: Presentazone15

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

Page 25: Presentazone15

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

Page 26: Presentazone15

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

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

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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

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

Page 28: Presentazone15

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

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

Page 29: Presentazone15

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

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

Page 30: Presentazone15

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

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

Page 31: Presentazone15

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

Page 32: Presentazone15

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

Page 33: Presentazone15

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

Page 34: Presentazone15

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

Page 35: Presentazone15

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

Page 36: Presentazone15

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

Page 37: Presentazone15

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

Page 38: Presentazone15

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

Page 39: Presentazone15

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

Page 40: Presentazone15

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

Page 41: Presentazone15

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

Page 42: Presentazone15

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

Page 43: Presentazone15

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

Page 44: Presentazone15

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

Page 45: Presentazone15

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

Page 46: Presentazone15

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

Page 47: Presentazone15

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

Page 48: Presentazone15

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

Page 49: Presentazone15

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