local noise yielding full control · c. arenz qcc nottingham 23rd jan 2015 n-level atom •a...

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)}



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

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Control pulses

1U

Unitary control



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• The dynamical Lie algebra:

Every U = eA with A 2 L

can be implemented.

• Full controllability: L = su(d)

Control pulses

1U

Unitary control



(finite dimensional systems)

L = Lie(iH(1), ..., iH(n))

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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


Control and measurement



P

D. Burgarth et al. , Nature Communications 5 (2014)A possible rise of complexity

• Frequently repeated projective measurements




P


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




limN!1

⇣Pe�iH(j)t/NP

⌘N

= e�iPH(j)PtP

P

Zeno dynamics






limN!1

⇣Pe�iH(j)t/NP

⌘N

= e�iPH(j)PtP

P

Zeno dynamics

LZeno

= Lie(iPH(1)P, ..., iPH(n)P )






limN!1

⇣Pe�iH(j)t/NP

⌘N

= e�iPH(j)PtP

P

Zeno dynamics

LZeno


Control in a projected subspace






limN!1

⇣Pe�iH(j)t/NP

⌘N

= e�iPH(j)PtP

P

Zeno dynamics

LZeno



• Example: H(1) = �x

⌦ (�x

+ �z

),

H(2) = �y

⌦ (�x

� �z

)

P = 1⌦ |0i h0|z






limN!1

⇣Pe�iH(j)t/NP

⌘N

= e�iPH(j)PtP Zeno dynamics

LZeno


LZeno

= su(2)⌦ |0i h0|zQubit 1 is fully controllable


• Example: H(1) = �x

⌦ (�x

+ �z

),

H(2) = �y

⌦ (�x

� �z

)

P = 1⌦ |0i h0|z

P




The environment is watching


The environment is watching• Markovian Lindblad generator: L0


The environment is watching• Markovian Lindblad generator: L0

Time evolution: ⇤t = e�tL0


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





• Additional Hamiltonian part:


L = �L0 + gK with the unitary generator K = �i[H, ·]

L0


lim�!1

e�tL0 = P





• Additional Hamiltonian part:


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


lim�!1

e�tL0 = P



Two qubits

LAD �•Second qubit is subject to an amplitude damping channel described by with rate


Two qubits


•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


Two qubits


•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


H(1) = �x

⌦ (�x

+ �z

),

H(2) = �y

⌦ (�x

� �z

)

• Control Hamiltonians:

• Target operation:

Hadamard- and T- gate on qubit 1

Numerical optimization QuTiP


H(1) = �x

⌦ (�x

+ �z

),

H(2) = �y

⌦ (�x

� �z

)




Control pulsesNumerical optimization QuTiP


H(1) = �x

⌦ (�x

+ �z

),

H(2) = �y

⌦ (�x

� �z

)




Control pulsesNumerical optimization QuTiP

H(2)• Now as a drift with strength g


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




|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|




|ei�1, ..., �N|1i , ..., |Ni

LN

|1i|2i

|3i

|Ni

|ei

�N�3�2�1


P(·) = M0(·)M†0 +

1

�

NX

j=1


•Drift term:

with

L = LN � ig[H0, ·]

H0 =p2 |ei h2|+

NX

j=1

|ji hj + 1|+ h.c.




|ei�1, ..., �N|1i , ..., |Ni

LN

|1i|2i

|3i

|Ni

|ei

�N�3�2�1


P(·) = M0(·)M†0 +

1

�

NX

j=1


•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|)




|ei�1, ..., �N|1i , ..., |Ni

LN

|1i|2i

|3i

|Ni

|ei

�N�3�2�1


P(·) = M0(·)M†0 +

1

�

NX

j=1


•Drift term:

with


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




|ei�1, ..., �N|1i , ..., |Ni

LN

|1i|2i

|3i

|Ni

|ei

�N�3�2�1


P(·) = M0(·)M†0 +

1

�

NX

j=1


•Drift term:

with


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



•Target gate:pSWAP between |1i and |Ni



•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



•Target gate:


Upper level removed, pure unitary dynamics




•Target gate:


Upper level removed, pure unitary dynamics

Control pulses forN = 5, T = 242



Conclusions

1U

Unitary control


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



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


local noise yielding full control · c. arenz qcc nottingham 23rd jan 2015 n-level atom •a...

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