The document discusses quantum computation and entanglement dynamics. It introduces quantum bits and entanglement, describing how quantum gates can generate entanglement. It then discusses implementations of quantum computers using superconducting qubits and Josephson junctions. Noise processes that affect superconducting qubits are also examined. The document analyzes the entanglement dynamics between two superconducting qubits subject to random noise.
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Presentazone15
1. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Entanglement Dynamics of Two Superconducting Qubits
Subject to Random Telegraph Noise
Marta Agati
Università degl Studi di Catania
Dipartimento di Fisica e Astronomia
Corso di Laurea in Fisica
Matis CNR-IMM UOS Catania
Centro Siciliano Fisica Nucleare e
Struttura della Materia (CSFNSM)
QUINN QUantum INformation and
Nanonsystems group
Relatore
Prof.ssa Elisabetta Paladino
Correlatore
Prof. Giuseppe Falci
Dott. Antonio D’Arrigo
July 16, 2013
Marta Agati Entanglement Dynamics
2. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Contents
1 Quantum Computation
Quantum Computing and Quantum Mechanics
2 Superconducting Qubits
Charge Qubit
3 Noise in Josephson Qubits
Methods
4 Entanglement Dynamics
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
3. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Introduction to Quantum Computation
Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010
Marta Agati Entanglement Dynamics
4. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Contents
1 Quantum Computation
Quantum Computing and Quantum Mechanics
2 Superconducting Qubits
Charge Qubit
3 Noise in Josephson Qubits
Methods
4 Entanglement Dynamics
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
5. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Unit of Quantum Information
Quantum bit or Qubit
Quantum bit or Qubit ψ = α0|0 + α1|1 Superposition Principle
Multiple-Qubit state
Two qubits ψ = α00|00 + α01|01 + α10|10 + α11|11
Product State
ψS = |01 +|11
√
2
= |0 +|1
√
2
⊗ |1
Entangled State
(Bell State)
ψE = |00 +|11
√
2
Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010
Marta Agati Entanglement Dynamics
6. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Unit of Quantum Information
Quantum bit or Qubit
Quantum bit or Qubit ψ = α0|0 + α1|1 Superposition Principle
Multiple-Qubit state
Two qubits ψ = α00|00 + α01|01 + α10|10 + α11|11
Product State
ψS = |01 +|11
√
2
= |0 +|1
√
2
⊗ |1
Entangled State
(Bell State)
ψE = |00 +|11
√
2
Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010
Marta Agati Entanglement Dynamics
7. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Unit of Quantum Information
Quantum bit or Qubit
Quantum bit or Qubit ψ = α0|0 + α1|1 Superposition Principle
Multiple-Qubit state
Two qubits ψ = α00|00 + α01|01 + α10|10 + α11|11
Product State
ψS = |01 +|11
√
2
= |0 +|1
√
2
⊗ |1
Entangled State
(Bell State)
ψE = |00 +|11
√
2
Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010
Marta Agati Entanglement Dynamics
8. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Entanglement Quantifiers
ρ ≡ Two-Qubit Density Matrix
=⇒ ˜ρ = (σy ⊗ σy ) ρ (σy ⊗ σy )
Wootters Concurrence
C(t) = 2Max 0,
√
λ1 −
√
λ2 −
√
λ3 −
√
λ4
λi , i = {1, . . . , 4}, eigenvalues of the matrix ρ˜ρ arranged in decreasing order.
Maximally Entangled States C=1
Product States C=0
Invariance for Local Unitary Transformations.
W. K. Wotters, Entanglement of Formation of an Arbitrary State of two Qubits, Phys. Rev. Lett., 80, 10,( 1998)
Marta Agati Entanglement Dynamics
9. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Entanglement Quantifiers
ρ ≡ Two-Qubit Density Matrix
=⇒ ˜ρ = (σy ⊗ σy ) ρ (σy ⊗ σy )
Wootters Concurrence
C(t) = 2Max 0,
√
λ1 −
√
λ2 −
√
λ3 −
√
λ4
λi , i = {1, . . . , 4}, eigenvalues of the matrix ρ˜ρ arranged in decreasing order.
Maximally Entangled States C=1
Product States C=0
Invariance for Local Unitary Transformations.
W. K. Wotters, Entanglement of Formation of an Arbitrary State of two Qubits, Phys. Rev. Lett., 80, 10,( 1998)
Marta Agati Entanglement Dynamics
10. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Quantum Gates
Universary set of Quantum Gates
Any multiple qubits logic gate may be composed of single qubit gates and at
least one entanglement-generating two-qubit gate.
CNot Gate
(|0 + |1 ) |0
√
2
⇒
|00 + |11
√
2
Motivation for our study on the
sensitivity of the entanglement
to external influences (environment)
Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010
Marta Agati Entanglement Dynamics
11. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Quantum Gates
Universary set of Quantum Gates
Any multiple qubits logic gate may be composed of single qubit gates and at
least one entanglement-generating two-qubit gate.
CNot Gate
(|0 + |1 ) |0
√
2
⇒
|00 + |11
√
2
Motivation for our study on the
sensitivity of the entanglement
to external influences (environment)
Michael A. Nielsen, Isaac L. Chuang; Quantum Computation and Quantum Information, Cambridge University Press, 2010
Marta Agati Entanglement Dynamics
12. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Quantum Computing and Quantum Mechanics
Quantum Computers Implementations
G. Chen, D. A. Church, B.G. Englert, C. Henkel, B. Ronwedder, M. O. Scully, M. Zubairy, Quantum Computing Devices: principles,
Designs and Analysis, Chapman et Hall/CRC, 2007
Marta Agati Entanglement Dynamics
13. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Superconducting materials and Josephson junctions
Characteristics of Superconducting Materials
Hallmarks:
Perfect Conductivity
Perfect Diamagnetism (Meissner Effect)
Cooper pairs
Josephson Effect
Josephson Equations
I = IC sin φ
Stationary Josephson Effect:
a current flows at 0 Voltage.
V(t) = 2e
∂
∂t
φ
A.C. Josephson Effect
Michael Tinkham, Introduction to Superconductivity, McGRAW-HILL EDITIONS, 1996
Marta Agati Entanglement Dynamics
14. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Superconducting materials and Josephson junctions
Characteristics of Superconducting Materials
Hallmarks:
Perfect Conductivity
Perfect Diamagnetism (Meissner Effect)
Cooper pairs
Josephson Effect
Josephson Equations
I = IC sin φ
Stationary Josephson Effect:
a current flows at 0 Voltage.
V(t) = 2e
∂
∂t
φ
A.C. Josephson Effect
Michael Tinkham, Introduction to Superconductivity, McGRAW-HILL EDITIONS, 1996
Marta Agati Entanglement Dynamics
15. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Superconducting Qubits
Charge Qubit Phase Qubit
Other qubits based on Cooper Pair
Box: Quantronium and Trasmon
Flux Qubit
Appunti del Corso di fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
16. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Superconducting Qubits
Charge Qubit Phase Qubit
Other qubits based on Cooper Pair
Box: Quantronium and Trasmon
Flux Qubit
Appunti del Corso di fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
17. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Contents
1 Quantum Computation
Quantum Computing and Quantum Mechanics
2 Superconducting Qubits
Charge Qubit
3 Noise in Josephson Qubits
Methods
4 Entanglement Dynamics
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
18. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Network Equations for Josephson Circuits (Lagrangian form)
Electrostatic Energy
K = CΣ
2 2e
˙φ +
Cg
CΣ
Vg
2
CΣ ≡ (C + Cg)
Magnetic Energy
UJ (φ) =
t
0
dt I(t ) ˙Φ(t ) = EJ (1 − cosφ)
EJ ≡ 2e
Ic
Lagrangian
L(2e
φ, 2e
˙φ) = K( ˙φ) − U(φ)
Classical Hamiltonian
H(Q,
2e
φ) =
1
2CΣ
2e
(Q − CgVg)2
+ EJ (1 − cos φ)
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
19. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Network Equations for Josephson Circuits (Lagrangian form)
Electrostatic Energy
K = CΣ
2 2e
˙φ +
Cg
CΣ
Vg
2
CΣ ≡ (C + Cg)
Magnetic Energy
UJ (φ) =
t
0
dt I(t ) ˙Φ(t ) = EJ (1 − cosφ)
EJ ≡ 2e
Ic
Lagrangian
L(2e
φ, 2e
˙φ) = K( ˙φ) − U(φ)
Classical Hamiltonian
H(Q,
2e
φ) =
1
2CΣ
2e
(Q − CgVg)2
+ EJ (1 − cos φ)
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
20. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Network Equations for Josephson Circuits (Lagrangian form)
Electrostatic Energy
K = CΣ
2 2e
˙φ +
Cg
CΣ
Vg
2
CΣ ≡ (C + Cg)
Magnetic Energy
UJ (φ) =
t
0
dt I(t ) ˙Φ(t ) = EJ (1 − cosφ)
EJ ≡ 2e
Ic
Lagrangian
L(2e
φ, 2e
˙φ) = K( ˙φ) − U(φ)
Classical Hamiltonian
H(Q,
2e
φ) =
1
2CΣ
2e
(Q − CgVg)2
+ EJ (1 − cos φ)
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
21. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Charge Qubit Hamiltonian
|n , |n + 1 ≡ Eigenstates of the charge in the island.
Quantum Hamiltonian (in the charge basis)
ˆH = EC
n
(n − qg)2
|n n| −
EJ
2 n
|n n + 1| + |n + 1 n|
Projection on to the lowest energy bidimensional subspace
Charge Qubit Hamiltonian
Hq = −1
2
σz − 1
2
∆σx
≡ 4EC(1 − 2qx )
∆ ≡ EJ
σi ≡ Pauli Matrices
Phenomenological Quantization of
the Phase φ
2e
, Q = i
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
22. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Charge Qubit
Charge Qubit Hamiltonian
|n , |n + 1 ≡ Eigenstates of the charge in the island.
Quantum Hamiltonian (in the charge basis)
ˆH = EC
n
(n − qg)2
|n n| −
EJ
2 n
|n n + 1| + |n + 1 n|
Projection on to the lowest energy bidimensional subspace
Charge Qubit Hamiltonian
Hq = −1
2
σz − 1
2
∆σx
≡ 4EC(1 − 2qx )
∆ ≡ EJ
σi ≡ Pauli Matrices
Phenomenological Quantization of
the Phase φ
2e
, Q = i
Appunti del Corso di Fisica dei Nanosistemi, Giuseppe Falci, AA 2012-2013
Marta Agati Entanglement Dynamics
23. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Noise Sources
Quantum Coherence
|ψ, t = q1,··· ,qN
cq1,··· ,qN (t)|q1, · · · , qN =⇒ it exists a well defined
deterministic relation between the complex amplitudes cqi (t) provided by the
Schrödinger equation.
Open Quantum System
Decoherence
Noise
Classical Stochastic Process
Htot = −1
2
σz − 1
2
∆σx − 1
2
ξ(t)v · −→σ
Particular coupling conditions
Longitudinal Coupling v H
Transvers Coupling v ⊥ H
−→ Density Matrix Formalism
G. Falci, E. Paladino, R. Fazio, Decoherence in Josephson Qubits, Varenna Review 2003
Marta Agati Entanglement Dynamics
24. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Noise Sources
Quantum Coherence
|ψ, t = q1,··· ,qN
cq1,··· ,qN (t)|q1, · · · , qN =⇒ it exists a well defined
deterministic relation between the complex amplitudes cqi (t) provided by the
Schrödinger equation.
Open Quantum System
Decoherence
Noise
Classical Stochastic Process
Htot = −1
2
σz − 1
2
∆σx − 1
2
ξ(t)v · −→σ
Particular coupling conditions
Longitudinal Coupling v H
Transvers Coupling v ⊥ H
−→ Density Matrix Formalism
G. Falci, E. Paladino, R. Fazio, Decoherence in Josephson Qubits, Varenna Review 2003
Marta Agati Entanglement Dynamics
25. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Noise Sources
Quantum Coherence
|ψ, t = q1,··· ,qN
cq1,··· ,qN (t)|q1, · · · , qN =⇒ it exists a well defined
deterministic relation between the complex amplitudes cqi (t) provided by the
Schrödinger equation.
Open Quantum System
Decoherence
Noise
Classical Stochastic Process
Htot = −1
2
σz − 1
2
∆σx − 1
2
ξ(t)v · −→σ
Particular coupling conditions
Longitudinal Coupling v H
Transvers Coupling v ⊥ H
−→ Density Matrix Formalism
G. Falci, E. Paladino, R. Fazio, Decoherence in Josephson Qubits, Varenna Review 2003
Marta Agati Entanglement Dynamics
26. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Noise in Josephson Qubits
Internal sources: Exitation of Quasi-particles
External environment:
Circuit
Preparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Background
fluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer of
Charge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2
γ
γ2+ω2
E. Paladino, Y. M. Galperin, G. Falci, B. L. Altshuler, 1/f noise: implications for solid-state quantum information,a rXiv:1304.7925,
submitted to RMP
Marta Agati Entanglement Dynamics
27. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Noise in Josephson Qubits
Internal sources: Exitation of Quasi-particles
External environment:
Circuit
Preparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Background
fluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer of
Charge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2
γ
γ2+ω2
E. Paladino, Y. M. Galperin, G. Falci, B. L. Altshuler, 1/f noise: implications for solid-state quantum information,a rXiv:1304.7925,
submitted to RMP
Marta Agati Entanglement Dynamics
28. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Noise in Josephson Qubits
Internal sources: Exitation of Quasi-particles
External environment:
Circuit
Preparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Background
fluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer of
Charge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2
γ
γ2+ω2
E. Paladino, Y. M. Galperin, G. Falci, B. L. Altshuler, 1/f noise: implications for solid-state quantum information,a rXiv:1304.7925,
submitted to RMP
Marta Agati Entanglement Dynamics
29. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Noise in Josephson Qubits
Internal sources: Exitation of Quasi-particles
External environment:
Circuit
Preparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Background
fluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer of
Charge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2
γ
γ2+ω2
E. Paladino, Y. M. Galperin, G. Falci, B. L. Altshuler, 1/f noise: implications for solid-state quantum information,a rXiv:1304.7925,
submitted to RMP
Marta Agati Entanglement Dynamics
30. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Noise in Josephson Qubits
Internal sources: Exitation of Quasi-particles
External environment:
Circuit
Preparation, Control and measurement apparata
Dynamic defects fluctuating between two localized states (Background
fluctuators) produce random telegraph noise (RTN)
Example
Background charged impurities trapped close to the insulating layer of
Charge Qubits or in the substrate.
Power Spectrum RTN
S(ω) = v2
2
γ
γ2+ω2
E. Paladino, Y. M. Galperin, G. Falci, B. L. Altshuler, 1/f noise: implications for solid-state quantum information,a rXiv:1304.7925,
submitted to RMP
Marta Agati Entanglement Dynamics
31. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Contents
1 Quantum Computation
Quantum Computing and Quantum Mechanics
2 Superconducting Qubits
Charge Qubit
3 Noise in Josephson Qubits
Methods
4 Entanglement Dynamics
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
32. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Master Equation
Weak coupling and fast fluctuator: v Ω and v γ
ΓR = 1
2
sin2
θS(Ω) Relaxation Rate
(Decay of z-component of the
qubit Bloch vector)
Γφ = Γ0
φ + 1
2
ΓR = 1
2
cos2
θS(0) + 1
2
ΓR
Dephasing Rate (Decay of x- and
y-components of the
qubit Bloch vector)
Microscopic Model of Background Charges
ˆH = −1
2
σz − 1
2
∆σx + b+
b + k [Tk c+
k b + h.c.] + k k c+
k ck + (v/2)σz b+
b
ξ(t) = 0, +1 Asymmetric fluctuator
ξ(t) = −1, +1 Symmetric fluctuator
E. Paladino, L. Faoro, G. Falci, Decoherence Due to Discrete Noise in Josephson Qubits, Adv. in Sol. St. Phys., 43, (2003)
Marta Agati Entanglement Dynamics
33. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Master Equation
Weak coupling and fast fluctuator: v Ω and v γ
ΓR = 1
2
sin2
θS(Ω) Relaxation Rate
(Decay of z-component of the
qubit Bloch vector)
Γφ = Γ0
φ + 1
2
ΓR = 1
2
cos2
θS(0) + 1
2
ΓR
Dephasing Rate (Decay of x- and
y-components of the
qubit Bloch vector)
Microscopic Model of Background Charges
ˆH = −1
2
σz − 1
2
∆σx + b+
b + k [Tk c+
k b + h.c.] + k k c+
k ck + (v/2)σz b+
b
ξ(t) = 0, +1 Asymmetric fluctuator
ξ(t) = −1, +1 Symmetric fluctuator
E. Paladino, L. Faoro, G. Falci, Decoherence Due to Discrete Noise in Josephson Qubits, Adv. in Sol. St. Phys., 43, (2003)
Marta Agati Entanglement Dynamics
34. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Master Equation
Weak coupling and fast fluctuator: v Ω and v γ
ΓR = 1
2
sin2
θS(Ω) Relaxation Rate
(Decay of z-component of the
qubit Bloch vector)
Γφ = Γ0
φ + 1
2
ΓR = 1
2
cos2
θS(0) + 1
2
ΓR
Dephasing Rate (Decay of x- and
y-components of the
qubit Bloch vector)
Microscopic Model of Background Charges
ˆH = −1
2
σz − 1
2
∆σx + b+
b + k [Tk c+
k b + h.c.] + k k c+
k ck + (v/2)σz b+
b
ξ(t) = 0, +1 Asymmetric fluctuator
ξ(t) = −1, +1 Symmetric fluctuator
E. Paladino, L. Faoro, G. Falci, Decoherence Due to Discrete Noise in Josephson Qubits, Adv. in Sol. St. Phys., 43, (2003)
Marta Agati Entanglement Dynamics
35. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Methods
Quasi-Hamiltonian Method
Transition Probability Matrix
(RTN)
W =
1 − p p
p 1 − p
Element of Qubit Transfer Matrix T(without
noise)
Tijξi
(∆t) = 1
2
Tr[σi Uξi
(∆t)σj U+
ξi
(∆t)]
Average Tranfer Matrix
T(t) ≡ xf |ΓN
|if
Γ ≡ W ⊗ T
Quasi-Hamiltonian HqH
ΓN
(t) ≡ (Γ(∆t))N
∼ (I − iHqH ∆t)N
∼ exp(−iHqH t)
First order expansion
Bloch vector evolution under noise
n(t) = xf | ψ |ψ eiωψt
ψ| |if n(0)
B. Cheng, Q.-H. Wang and R. Joynt, Transfer matrix solution of a model of qubit dechoerence due to telegraph noise, Physical Review A,
78, (2008)
Marta Agati Entanglement Dynamics
36. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Two-Qubit System
Two-Qubit Density Matrix
Two uncorrelated systems, each composed of a single
qubit and a background charge.
The two-qubit density matrix depends on the initial
conditions ρ(0) and on the time-evolution of each qubit,
namely qubit A and qubit B under their own source of
noise.
The time-evolution is obtained the average transfer
matrices relative to qubit A and B: TA(t), TB(t).
ρ(t) = f(TA(t) ⊗ TB(t), ρ(0))
Marta Agati Entanglement Dynamics
37. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Two-Qubit System
Two-Qubit Density Matrix
Two uncorrelated systems, each composed of a single
qubit and a background charge.
The two-qubit density matrix depends on the initial
conditions ρ(0) and on the time-evolution of each qubit,
namely qubit A and qubit B under their own source of
noise.
The time-evolution is obtained the average transfer
matrices relative to qubit A and B: TA(t), TB(t).
ρ(t) = f(TA(t) ⊗ TB(t), ρ(0))
Marta Agati Entanglement Dynamics
38. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Entanglement time-evolution
Entanglement Sudden Death
Markovian noise, weak coupling
Entanglement Revivals
Markovian noise, strong coupling
Non-Markovian noise
Initial Conditions: Extended Werner Like (EWL) States
ˆρΦ = r|Φ Φ| + 1−r
4
I ˆρΨ = r|Ψ Ψ| + 1−r
4
I
r quantifies the mixedness;
|Φ = a|00 ± b|11 |Ψ = a|01 ± b|10
where a represents the initial degree of entanglement
of the pure part and |a|2
+ |b|2
= 1.
T. Yu and J. H. Eberly, Sudden Death of Entanglement,Science, 323, (2009)
Marta Agati Entanglement Dynamics
39. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Entanglement time-evolution
Entanglement Sudden Death
Markovian noise, weak coupling
Entanglement Revivals
Markovian noise, strong coupling
Non-Markovian noise
Initial Conditions: Extended Werner Like (EWL) States
ˆρΦ = r|Φ Φ| + 1−r
4
I ˆρΨ = r|Ψ Ψ| + 1−r
4
I
r quantifies the mixedness;
|Φ = a|00 ± b|11 |Ψ = a|01 ± b|10
where a represents the initial degree of entanglement
of the pure part and |a|2
+ |b|2
= 1.
T. Yu and J. H. Eberly, Sudden Death of Entanglement,Science, 323, (2009)
Marta Agati Entanglement Dynamics
40. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Contents
1 Quantum Computation
Quantum Computing and Quantum Mechanics
2 Superconducting Qubits
Charge Qubit
3 Noise in Josephson Qubits
Methods
4 Entanglement Dynamics
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
41. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Noise on one qubit: Concurrence Decay and Revivals
r=1
Weak Coupling −→
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nΤ
a
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
Τ
Λ1Τ,Λ2Τ,Λ3Τ,Λ4Τ
b
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.0
Τ
CΤ
c
Transition Region −→
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nΤ
a
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
Τ
Λ1Τ,Λ2Τ,Λ3Τ,Λ4Τ
b
0.00.51.01.52.02.5
0.0
0.2
0.4
0.6
0.8
1.0
Τ
CΤ
c
Strong Coupling
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nΤ
a
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
Τ
Λ1Τ,Λ2Τ,Λ3Τ,Λ4Τ
b
0.00.51.01.52.02.5
0.0
0.2
0.4
0.6
0.8
1.0
Τ
CΤ
c
Marta Agati Entanglement Dynamics
42. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Noise on both qubits
Equal weakly coupled noise
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
Τ
nyΤ
a
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nzΤ
b
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nΤ
a
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
CΤ
d
Wekly coupled noise on one qubit
and Strong coupled noise on the other
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
Τ
nyΤ
a
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
Τ
nyΤ
a
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nzΤ
b
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nzΤ
b
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nΤ a
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nΤ
a
0 1 2 3 4
0.0
0.2
0.4
0.6
0.8
1.0
Τ
CΤ
d
Marta Agati Entanglement Dynamics
43. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Contents
1 Quantum Computation
Quantum Computing and Quantum Mechanics
2 Superconducting Qubits
Charge Qubit
3 Noise in Josephson Qubits
Methods
4 Entanglement Dynamics
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
44. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Noise on one qubit
r=1
Asymmetric versus Symmetric
Weak coupling
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
Τ
nyΤ
Γ
40,
v
Γ
2
"Strong" coupling
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
Τ
nyΤ
Γ
40,
v
Γ
18
Transition Region
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
Τ
nyΤ
Γ
40,
v
Γ
9
"Strong" Symmetric
Fluctuator
0 1 2 3 4
1.0
0.5
0.0
0.5
1.0
Τ
nyΤ
a
v Γ 14
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Τ
nzΤ
b
v Γ 14
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
Τ
nΤ
c
v Γ 14
0 20 40 60 80 100
0.0
0.2
0.4
0.6
0.8
1.0
Τ
CΤ
v Γ 14
Marta Agati Entanglement Dynamics
45. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Contents
1 Quantum Computation
Quantum Computing and Quantum Mechanics
2 Superconducting Qubits
Charge Qubit
3 Noise in Josephson Qubits
Methods
4 Entanglement Dynamics
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
5 Conclusions
Marta Agati Entanglement Dynamics
46. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Transvers Coupling, Asymmetric Fluctuator
Transvers Coupling, Comparison with Symmetric Fluctuator
Comparison with Longitudinal Coupling
Noise on one qubit
r=0.91
Longitudinal Coupling
v/γ = 0.5 Weak Coupling
v/γ = 5 Strong Coupling
Transvers Coupling (Crossover)
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
Τ
CΤ
v Γ 2
v Γ 5
v Γ 9
v Γ 14
v Γ 18
R. Lo Franco, A. D’Arrigo, G. Falci, C. Compagno, E. Paladino, Entanglement dynamics in superconducting qubits affected by local
bistable impurities, Phys.Scr., 9, (2012)
Marta Agati Entanglement Dynamics
47. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Epilogue
Two superconducting qubits, each subject indipendently to Random
Telegraph Noise.
Example: Random Telegraph Noise by charged impurities trapped close to
a charge Josephson qubit.
Microscopic model of the RTN generation.
Application of the Quasi-Hamiltonian method.
Evaluation of the two-qubit density matrix.
Evaluation of the concurrence
Marta Agati Entanglement Dynamics
48. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
Results
−→ Crossover between weak coupling and strong coupling.
−→ Asymmetric and symmetric fluctuator and comparison.
−→ Initial conditions as pure state and Extended Werner Like (EWL)
state.
−→ Analogous behaviour for the states ρΦ and ρΨ.
−→ For an asymmetric fluctuator model in weak coupling conditions the
entanglement displays ESD, while in strong coupling conditions the
entanglement displays dark peridos and revivals.
−→ For a symmetric fluctuator model the entanglement decays both in
weak coupling conditions and in strong coupling conditions. The
entanglement can also definitively vanish starting with a pure state or
an EWL state.
Marta Agati Entanglement Dynamics
49. Entanglement
Dynamics
Marta Agati
Quantum
Computa-
tion
Quantum
Computing
and Quantum
Mechanics
Superconducting
Qubits
Charge Qubit
Noise in
Josephson
Qubits
Methods
Entanglement
Dynamics
Transvers
Coupling,
Asymmetric
Fluctuator
Transvers
Coupling,
Comparison
with
Symmetric
Fluctuator
Comparison
with
Longitudinal
Coupling
Conclusions
Quantum Computation
Superconducting Qubits
Noise in Josephson Qubits
Entanglement Dynamics
Conclusions
So...
THANK YOU FOR THE KIND ATTENTION
Marta Agati Entanglement Dynamics