Geometric phase and Geometric Quantum Computation'

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Geometric phase andGeometric Quantum
Computation.

• Angelo Carollo
• a.carollo_at_imperial.ac.uk

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Why studying geometric phase?

• It is an interesting phenomenon of Quantum
  mechanics, which can be observed in many physical
  systems...
• It has interesting properties that can be
  exploited to increase the robustness of Quantum
  Computation
• Geometric Quantum Computation

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Quantum Computation
Geometric
U1
Quantum register
U2
UN
Ui
gates
Measurement
Dynamical evolution
Geometric gate
Geometric gates can be more robust against
different sources of noise
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General Definitions

• What is meant by geometrical in the geometric
  phase?

A physically observable feature of a quantum
system that depends only on the path described
during its evolution.
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..Lets be more precise..

• Let H be a Hilbert space, with elements denoted
  by y, y, f, f
• We are generally interested in a particular
  subspace of it

• This space is invariant under the group U(1), of
  phase transformation

• This group leads to a corresponding equivalence
  relation
• Two states of L are equivalent if one goes
  into the other via a U(1) transformation (..i.e.
  if they are the same up to a phase change).
• We can thus define the space of equivalent
  classes, formally denoted by

Projective Hilbert Space
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• Physically, Each element of the space P
  represents, a pure state, up to unessential
  normalisation condition and overall phase
  factors.

• A quantum (pure) state at a given instant of time
  is represented by a point in P and its evolution
  is given by a curve C in P.

• Geometric Property of evolution (more precisely)
• Any quantity that depends only on the curve C
  traversed by the system in the Projective Hilbert
  space.

• I.e.
• That does not depend on the phase of the wave
  function
• And is independent of the details of the time
  evolution

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...search for a geometric property of quantum
states..

• (..i.e. an invariant under U(1) transformations..)

• Given two states
• with two independent transformation U(1)

Just
(Trivial!!!)

• What about three states?

Apart from
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Geometric phase and distant pararellelism
f12
f13
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• Additive property of g

This quantity can be generalized in the case of
four states
This additive property is due to its nature of
integral on an area in the space P
This property is at the origin of the intrinsic
robustness of geometric phase...
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Continuous case
Given a curve Y(s) in the space L
Define the subdivision s s1lt s2 lt sN-1 ltsN
with si1si1D
Calculate
In the continuous limit Dltlt1
Y(sN-1)
Y(sN)
Y(s1)
Y(s7)
Y(s6)
Y(s2)
Y(s4)
Y(s3)
Y(s5)
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Geometric(?) phase

• Gauge invariance

Consider the generalization of the U(1) in the
continuous case, which amounts in
(Gauge transformation)
The corresponding curve C(s) in the projective
Hilbert space P is the same for both Y(s) and
Y(s). (..as they differ for a phase
transformation..) (Y(s) and Y(s) are called
lift of the curve C(s) )
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It can be easily seen that g is invariant under
gauge transformation i.e.
Which means that given any lift Y(s) of the curve
C(s) g is always the same
Parameterisation invariance
Suppose that we replace the parameter s with
another parameter s which is a smooth monotonic
function of s, the curve Y(s) is formally
replaced by a curve Y(s) with the same support,
but with a different rate of traversal.
Due to its form of a path integral g is invariant
under this transformation.
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• Combining
• Gauge invariance
• and
• Parameterisation invariance,
• The functional

Depends only on the support of the curve C(t) in
the Projective Hilbert space P.
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Physical meaning?

• Geometric phase and Schrödinger evolution
  (Berrys original idea)

• Suppose that an Hamiltonian H(r) depends on a set
  of parameters

• For each value of r, the following eigenvalue
  equation is valid

If the parameters r change slowly in time, the
adiabatic theorem assures that the system remains
in the eigenstate of the Hamiltonian, i.e.

• What if for same value of tT r(T)r(0)?
  (Cyclic evolution)
• Certainly

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Berry phaseM. Berry, Proc. Roy. Soc. A 392, 45
(1984)
Then the state returns to its initial form but
since eigenstates are defined up to a phase
factor, the state could acquire a phase due to
the adiabatic and cyclic evolution that took
place.
Geometrical phase It is the same expression
derived before for a general curve.
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The single qubit example(Canonical example of
Berry phase)
A typical example is the phase acquired by a
qubit evolving under the following Hamiltonian
Where the B is a 3D vector and
Are the qubit matrixes
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If the direction of the vector B is changed in
time as shown, then the initial eigenstates will
be transformed as follows
Where is the dynamical phase and..
The geometric phase g is equal to half of the
area V enclosed in the path traversed by the
vector B.
Using clever protocols it is possible to get rid
of the dynamical phase d and obtain an evolution
completely determined by the geometric phase.
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And obtain
This can be used for example for
Single qubit gates (Geometric) phase gate
H
c
H
Or analogously for two-qubit gates For example
(Geometric) controlled phase-gate
c
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Why is the geometric phase for a qubit is given
by gV/2?The reason is because the Projective
Hilbert space for a qubit is a sphere (the bloch
sphere) and...
qubit
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Why geometric evolution?

• Geometric phase is robust against classical
  fluctuation of the phase (of the first order)
  see for example
• De Chiara, Palma, PRL03, quant-ph/0303155

• It is independent of sistematical errors.

• Decoherence?

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What about decoherence?
When quantum systems are not anymore isolated
from the environment, their states are spoilt by
the action of the environment and a description
in terms of pure state is not realistic anymore...

• We would need a description of geometric phase
  for mixed states..
• The definition of a geometric phase associated to
  a mixed state scenario is an open problem.
• But there are ways to overcome this problem

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...The Quantum jump approach.
Loosely specking, this approach is based on the
fact that (under certain kind of environment) a
system is affected in such a way, that its
evolution may be regarded as divided into various
branches
At each step of time the action of the
environment can be modeled as a statistical
process that transform the state
With probability
Where is an operator that describes the action of
the environment
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• Using this method is then possible to calculate
  the geometric phase for each branch of the
  evolution using the formula

• In this way it is possible to derive the
  geometric phase for any trajectory..

• Using this quantum jump approach is possible to
  show that under specific kind of actions of the
  environment, the geometric phase is robust.

• There are, in fact, situation where the geometric
  phase associated to each branch is always the
  same, thereby remaining unaffected by the
  different actions of the enviroment.

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Conclusions

• We have reviewed a general method to calculate
  geometric phases from a very general approach
• We showed how the usual definition of geometric
  phase given by Berry is recovered as a specific
  example
• We showed that this approach is particularly
  useful to tackle problems such as the evaluation
  of the geometric phase under decoherence
• It is possible to show that, for special
  decoherence sources, the phase remains unaffected
  for any trajectory.

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• Example Decoherence for a 2 level system

Suppose that the systems evolves under the
Hamiltonian
In case of no jump, (which means that the
environment is not acting on the system for the
all evolution) during the time T2p/? the system
evolves under the Hamiltonian H And the
geometric phase is
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In case of 1 jump, The case of one jump means
that the system evolves freely under the
Hamiltonian H, until (at some time t1) the
environment acts with the operator
Whose action can be pictured on the Bloch sphere
(i.e. the projective Hilbert space for a qubit)
can as shown
In the case of one jump, the complete evolution
will look as follows
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• 1 jump trajectory

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• 1 jump trajectory

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• 1 jump trajectory

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• 1 jump trajectory

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• 1 jump trajectory

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• 1 jump trajectory

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• 1 jump trajectory

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• 1 jump trajectory

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• 1 jump trajectory