Local noise yielding full control
Christian Arenz and Daniel BurgarthDepartment of Mathematics, Aberystwyth University, Wales
QuTiP
C. Arenz QCC Nottingham 23rd Jan 2015
Quantum control on one slide• Given a set of control Hamiltonians {H(1), ..., H(n)}
C. Arenz QCC Nottingham 23rd Jan 2015
Quantum control on one slide• Given a set of control Hamiltonians {H(1), ..., H(n)}
Which unitary operations can we implement?
C. Arenz QCC Nottingham 23rd Jan 2015
Quantum control on one slide• Given a set of control Hamiltonians {H(1), ..., H(n)}
Which unitary operations can we implement?
How can we implement them?
C. Arenz QCC Nottingham 23rd Jan 2015
Quantum control on one slide• Given a set of control Hamiltonians {H(1), ..., H(n)}
Which unitary operations can we implement?
How can we implement them?
Lie algebraic methods,…
C. Arenz QCC Nottingham 23rd Jan 2015
Quantum control on one slide• Given a set of control Hamiltonians {H(1), ..., H(n)}
Which unitary operations can we implement?
How can we implement them?
Lie algebraic methods,…
Numerical optimization of the control pulses for a given target,…QuTiP qutip.org
C. Arenz QCC Nottingham 23rd Jan 2015
Quantum control on one slide• Given a set of control Hamiltonians {H(1), ..., H(n)}
Which unitary operations can we implement?
How can we implement them?
Control pulses
1U
Unitary control
Numerical optimization of the control pulses for a given target,…QuTiP qutip.org
Lie algebraic methods,…
C. Arenz QCC Nottingham 23rd Jan 2015
Quantum control on one slide• Given a set of control Hamiltonians {H(1), ..., H(n)}
Which unitary operations can we implement?
How can we implement them?
• The dynamical Lie algebra:
Every U = eA with A 2 L
can be implemented.
• Full controllability: L = su(d)
Control pulses
1U
Unitary control
Numerical optimization of the control pulses for a given target,…QuTiP qutip.org
Lie algebraic methods,…
(finite dimensional systems)
L = Lie(iH(1), ..., iH(n))
C. Arenz QCC Nottingham 23rd Jan 2015
Control and measurement
• The idea: [H(1), H(2)] = 0 ; [PH(1)P, PH(2)P ] = 0
with P = P 2 a projection.
D. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity
C. Arenz QCC Nottingham 23rd Jan 2015
Control and measurement
• The idea: [H(1), H(2)] = 0 ; [PH(1)P, PH(2)P ] = 0
with P = P 2 a projection.
P
D. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity
• Frequently repeated projective measurements
C. Arenz QCC Nottingham 23rd Jan 2015
• The idea: [H(1), H(2)] = 0 ; [PH(1)P, PH(2)P ] = 0
with P = P 2 a projection.
P
• Frequently repeated projective measurements
Control and measurementD. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity
limN!1
⇣Pe�iH(j)t/NP
⌘N
= e�iPH(j)PtP
C. Arenz QCC Nottingham 23rd Jan 2015
• The idea: [H(1), H(2)] = 0 ; [PH(1)P, PH(2)P ] = 0
with P = P 2 a projection.
limN!1
⇣Pe�iH(j)t/NP
⌘N
= e�iPH(j)PtP
P
Zeno dynamics
Control and measurementD. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity
• Frequently repeated projective measurements
C. Arenz QCC Nottingham 23rd Jan 2015
• The idea: [H(1), H(2)] = 0 ; [PH(1)P, PH(2)P ] = 0
with P = P 2 a projection.
limN!1
⇣Pe�iH(j)t/NP
⌘N
= e�iPH(j)PtP
P
Zeno dynamics
LZeno
= Lie(iPH(1)P, ..., iPH(n)P )
Control and measurementD. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity
• Frequently repeated projective measurements
C. Arenz QCC Nottingham 23rd Jan 2015
• The idea: [H(1), H(2)] = 0 ; [PH(1)P, PH(2)P ] = 0
with P = P 2 a projection.
limN!1
⇣Pe�iH(j)t/NP
⌘N
= e�iPH(j)PtP
P
Zeno dynamics
LZeno
= Lie(iPH(1)P, ..., iPH(n)P )
Control in a projected subspace
Control and measurementD. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity
• Frequently repeated projective measurements
C. Arenz QCC Nottingham 23rd Jan 2015
• The idea: [H(1), H(2)] = 0 ; [PH(1)P, PH(2)P ] = 0
with P = P 2 a projection.
limN!1
⇣Pe�iH(j)t/NP
⌘N
= e�iPH(j)PtP
P
Zeno dynamics
LZeno
= Lie(iPH(1)P, ..., iPH(n)P )
Control in a projected subspace
• Example: H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
P = 1⌦ |0i h0|z
Control and measurementD. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity
• Frequently repeated projective measurements
C. Arenz QCC Nottingham 23rd Jan 2015
• The idea: [H(1), H(2)] = 0 ; [PH(1)P, PH(2)P ] = 0
with P = P 2 a projection.
limN!1
⇣Pe�iH(j)t/NP
⌘N
= e�iPH(j)PtP Zeno dynamics
LZeno
= Lie(iPH(1)P, ..., iPH(n)P )
LZeno
= su(2)⌦ |0i h0|zQubit 1 is fully controllable
Control and measurementD. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity
• Example: H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
P = 1⌦ |0i h0|z
P
Control in a projected subspace
• Frequently repeated projective measurements
C. Arenz QCC Nottingham 23rd Jan 2015
The environment is watching
C. Arenz QCC Nottingham 23rd Jan 2015
The environment is watching• Markovian Lindblad generator: L0
C. Arenz QCC Nottingham 23rd Jan 2015
The environment is watching• Markovian Lindblad generator: L0
Time evolution: ⇤t = e�tL0
C. Arenz QCC Nottingham 23rd Jan 2015
The environment is watching• Markovian Lindblad generator:
• Attractive steady state manifold:
with P = P2 a (super) projector.
L0
(strong damping limit)
lim�!1
e�tL0 = P
Time evolution: ⇤t = e�tL0
C. Arenz QCC Nottingham 23rd Jan 2015
The environment is watching• Markovian Lindblad generator:
• Attractive steady state manifold:
• Additional Hamiltonian part:
with P = P2 a (super) projector.
L = �L0 + gK with the unitary generator K = �i[H, ·]
L0
(strong damping limit)
lim�!1
e�tL0 = P
Time evolution: ⇤t = e�tL0
C. Arenz QCC Nottingham 23rd Jan 2015
The environment is watching• Markovian Lindblad generator:
• Attractive steady state manifold:
• Additional Hamiltonian part:
with P = P2 a (super) projector.
L = �L0 + gK with the unitary generator K = �i[H, ·]
• Projection theorem: || �etL � eKeff�P|| O(g/�)
Ke↵ = tgPKPEffective unitary generator:
P. Zanardi and C.V. LorenzoPRL 113, 240406 (2014).
L0
(strong damping limit)
lim�!1
e�tL0 = P
Time evolution: ⇤t = e�tL0
C. Arenz QCC Nottingham 23rd Jan 2015
Two qubits
LAD �•Second qubit is subject to an amplitude damping channel described by with rate
C. Arenz QCC Nottingham 23rd Jan 2015
Two qubits
LAD �•Second qubit is subject to an amplitude damping channel described by with rate
•Strong damping limit: , �t � 1 dynamics is governed by:
M0 = 1⌦ |0i h0|zet�LAD ! P(·) = M0(·)M†0 +M1(·)M†
1
M1 = 1⌦ |0i h1|z
C. Arenz QCC Nottingham 23rd Jan 2015
Two qubits
LAD �•Second qubit is subject to an amplitude damping channel described by with rate
•Strong damping limit: , �t � 1 dynamics is governed by:
et�LAD ! P(·) = M0(·)M†0 +M1(·)M†
1
such that
M0
}
LZeno
= Lie(M0
H(1)M0
, ...,M0
H(n)M0
)
•Using the projection theorem one finds for a given Hamiltonian an effective evolution:
H(j)
⇤(j)t = e�itg[M0H
(j)M0,·]P
⇤(j)t
M0 = 1⌦ |0i h0|z
M1 = 1⌦ |0i h1|z
C. Arenz QCC Nottingham 23rd Jan 2015
H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
• Control Hamiltonians:
• Target operation:
Hadamard- and T- gate on qubit 1
Numerical optimization QuTiP
C. Arenz QCC Nottingham 23rd Jan 2015
H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
• Control Hamiltonians:
• Target operation:
Hadamard- and T- gate on qubit 1
Numerical optimization QuTiP
C. Arenz QCC Nottingham 23rd Jan 2015
H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
• Control Hamiltonians:
• Target operation:
Hadamard- and T- gate on qubit 1
Control pulsesNumerical optimization QuTiP
C. Arenz QCC Nottingham 23rd Jan 2015
H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
• Control Hamiltonians:
• Target operation:
Hadamard- and T- gate on qubit 1
Control pulsesNumerical optimization QuTiP
C. Arenz QCC Nottingham 23rd Jan 2015
H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
• Control Hamiltonians:
• Target operation:
Hadamard- and T- gate on qubit 1
Control pulsesNumerical optimization QuTiP
H(2)• Now as a drift with strength g
C. Arenz QCC Nottingham 23rd Jan 2015
H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
• Control Hamiltonians:
• Target operation:
Hadamard- and T- gate on qubit 1
Control pulsesNumerical optimization QuTiP
H(2)• Now as a drift with strength g
C. Arenz QCC Nottingham 23rd Jan 2015
H(1) = �x
⌦ (�x
+ �z
),
H(2) = �y
⌦ (�x
� �z
)
• Control Hamiltonians:
• Target operation:
Hadamard- and T- gate on qubit 1
Control pulsesNumerical optimization QuTiP
H(2)• Now as a drift with strength g
C. Arenz QCC Nottingham 23rd Jan 2015
N-level atom•A metastable level decays
with rates to the lower lying levels described by
|ei�1, ..., �N|1i , ..., |Ni
LN
|1i|2i
|3i
|Ni
|ei
�N�3�2�1
C. Arenz QCC Nottingham 23rd Jan 2015
N-level atom•A metastable level decays
with rates to the lower lying levels described by
|ei�1, ..., �N|1i , ..., |Ni
LN
|1i|2i
|3i
|Ni
|ei
�N�3�2�1
•Strong damping limit: �t � 1
P(·) = M0(·)M†0 +
1
�
NX
j=1
�jMj(·)M†j M0 = 1� |ei he| , Mj = |ji he|
C. Arenz QCC Nottingham 23rd Jan 2015
N-level atom•A metastable level decays
with rates to the lower lying levels described by
|ei�1, ..., �N|1i , ..., |Ni
LN
|1i|2i
|3i
|Ni
|ei
�N�3�2�1
•Strong damping limit: �t � 1
P(·) = M0(·)M†0 +
1
�
NX
j=1
�jMj(·)M†j M0 = 1� |ei he| , Mj = |ji he|
•Drift term:
with
L = LN � ig[H0, ·]
H0 =p2 |ei h2|+
NX
j=1
|ji hj + 1|+ h.c.
C. Arenz QCC Nottingham 23rd Jan 2015
N-level atom•A metastable level decays
with rates to the lower lying levels described by
|ei�1, ..., �N|1i , ..., |Ni
LN
|1i|2i
|3i
|Ni
|ei
�N�3�2�1
•Strong damping limit: �t � 1
P(·) = M0(·)M†0 +
1
�
NX
j=1
�jMj(·)M†j M0 = 1� |ei he| , Mj = |ji he|
•Drift term:
with
•Control Hamiltonian:
L = LN � ig[H0, ·]
H0 =p2 |ei h2|+
NX
j=1
|ji hj + 1|+ h.c.
H(1) =1
2|ei he|+ |1i h1|� 1p
2(|ei h1|+ |1i he|)
C. Arenz QCC Nottingham 23rd Jan 2015
N-level atom•A metastable level decays
with rates to the lower lying levels described by
|ei�1, ..., �N|1i , ..., |Ni
LN
|1i|2i
|3i
|Ni
|ei
�N�3�2�1
•Strong damping limit: �t � 1
P(·) = M0(·)M†0 +
1
�
NX
j=1
�jMj(·)M†j M0 = 1� |ei he| , Mj = |ji he|
•Drift term:
with
•Control Hamiltonian:
L = LN � ig[H0, ·]
H0 =p2 |ei h2|+
NX
j=1
|ji hj + 1|+ h.c.
H(1) =1
2|ei he|+ |1i h1|� 1p
2(|ei h1|+ |1i he|)
Control
Projected subspace
C. Arenz QCC Nottingham 23rd Jan 2015
N-level atom•A metastable level decays
with rates to the lower lying levels described by
|ei�1, ..., �N|1i , ..., |Ni
LN
|1i|2i
|3i
|Ni
|ei
�N�3�2�1
•Strong damping limit: �t � 1
P(·) = M0(·)M†0 +
1
�
NX
j=1
�jMj(·)M†j M0 = 1� |ei he| , Mj = |ji he|
•Drift term:
with
•Control Hamiltonian:
L = LN � ig[H0, ·]
LZeno
= u(N)M0
Appropriate choice of and use of the projection theorem leads to:
Full control within the subspace of the lower lying levels.
H0 =p2 |ei h2|+
NX
j=1
|ji hj + 1|+ h.c.
g
H(1) =1
2|ei he|+ |1i h1|� 1p
2(|ei h1|+ |1i he|)
Control
Projected subspace
C. Arenz QCC Nottingham 23rd Jan 2015
Numerical optimization QuTiP
•Target gate:pSWAP between |1i and |Ni
C. Arenz QCC Nottingham 23rd Jan 2015
Numerical optimization QuTiP
•Target gate:
•Optimization for different N and total times (searching for the minimum gate time), decay to all lower lying levels with the same rate � = 50, g/� = 2 · 10�3
pSWAP between |1i and |Ni
C. Arenz QCC Nottingham 23rd Jan 2015
Numerical optimization QuTiP
•Target gate:
•Optimization for different N and total times (searching for the minimum gate time), decay to all lower lying levels with the same rate � = 50, g/� = 2 · 10�3
pSWAP between |1i and |Ni
C. Arenz QCC Nottingham 23rd Jan 2015
Numerical optimization QuTiP
•Target gate:
•Optimization for different N and total times (searching for the minimum gate time), decay to all lower lying levels with the same rate � = 50, g/� = 2 · 10�3
Upper level removed, pure unitary dynamics
pSWAP between |1i and |Ni
C. Arenz QCC Nottingham 23rd Jan 2015
Numerical optimization QuTiP
•Target gate:
•Optimization for different N and total times (searching for the minimum gate time), decay to all lower lying levels with the same rate � = 50, g/� = 2 · 10�3
Upper level removed, pure unitary dynamics
Control pulses forN = 5, T = 242
pSWAP between |1i and |Ni
C. Arenz QCC Nottingham 23rd Jan 2015
Conclusions
1U
Unitary control
C. Arenz QCC Nottingham 23rd Jan 2015
Conclusions
•Frequent measurements of a dynamical system can induce an enhancement in the complexity of the system dynamics
1U
•A strong local noise process can have the same effect.
Frequent measurements/Strong local noise
Unitary control
Control in a projected subspace
C. Arenz QCC Nottingham 23rd Jan 2015
Conclusions
•Frequent measurements of a dynamical system can induce an enhancement in the complexity of the system dynamics
1U
A small number of quantum gates can be transformed into a universal set capable of performing arbitrary quantum computational tasks
•A strong local noise process can have the same effect.
Full control within a subspace can be achieved
Frequent measurements/Strong local noise
QuTiP
Unitary control
Control in a projected subspace