Quantum and classical model reduction

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Quantum and classical model reduction
Andrew Doherty
Quantum Control Summer School, Caltech August 2005
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Outline
Model reduction, simplifying models in physics
and control
Classical Brownian motion as an example
Determining the significant degrees of freedom in
an input-output system gramians
Balancing and truncation an algorithmic method
for simplifying models with error bounds.
Outline of applications to quantum
systems/adiabatic elimination.
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Background
Finding simple approximations to complex systems
is a common theme in both theoretical physics and
control. Physicists are more willing to accept
uncontrolled approximations in order to obtain
simple analytical descriptions that will provide
intuition about the physics (e.g. differences in
timescales). Control theorists are often more
interested in algorithmic numerical techniques
with rigorous error bounds. Brownian motion is
a good example of this.
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A coupled oscillator model
Free to choose bath frequencies and couplings.
But do we need to work so hard to work out the
response to a force?
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Ohmic Bath
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Time Dynamics
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Input-Output systems
Consider a linear (stable) input-output system
Taking a Laplace transform we find the transfer
function
Where the transfer function (cf. resolvent,
Greens fcn) is
We wish to approximate this linear map. The
process will be analogous to singular value based
matrix approximation. (Described in notes)
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Balanced truncation physical picture
The specific method we are interested in is
called balanced truncation. The physical picture
for balanced truncation is that we identify the
degrees of freedom that only weakly affect the
output or are only slightly changed by the input
and remove those degrees of freedom from the
problem. The objects we use to do this are
quadratic forms known as the gramians and they
were introduced in Krishnaprasads talk earlier
in the school.
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State transformations
We can change the state variables while
preserving the transfer function
Where T is a nonsingular matrix. This includes
unitary transformations and scaling
transformations. We get the system
So the overall transformation is
This linear system is equivalent to the last one.
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Observability Gramian
Balanced truncation proceeds by determining which
degrees of freedom are most easily excited by the
input signal and which contribute most energy to
the output signal. Consider the output signal
Where we have defined a matrix known as the
observability gramian Y. The observability
gramian Y is such that the energy in the signal
resulting from the initial state x0 is just
x0TYx0 and thus weights the state space according
to how strongly a given degree of freedom affects
the output. Check that So find Y is standard
linear algebra problem.
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Controllability Gramian
Just as some degrees of freedom have a greater
effect on the output, some degrees of freedom are
easier to excite with the input. Maybe it would
be nice to have a controllability gramian X where
larger eigenvalues corresponded to greater ease
in exciting those degrees of freedom. Perhaps we
would like the energy required in the input
signal to prepare the state x0 to be equal to
x0TX-1x0
Perhaps you could guess just from the dimensions
of the matrices what the correct definition to
be Likewise there is an algebraic
characterization Lets check that it is possible
to reach the state x0 with the predicted amount
of energy.
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Energy cost for state preparation
We can ask ourselves what is the state that may
be reached at t0 from an initial condition of
x0 as t! 1 using a control at earlier times.
Consider (assuming (A,B) controllable) Then, by
substitution, we get The energy in this input
signal is This is the energy that we predicted
would be required supporting our interpretation
of the gramian. We should also show that it is
not possible to prepare x0 using less energy see
the notes.
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Balancing Transformation
We have found gramians that determine the extent
to which different degrees of freedom affect the
output signal and the extent to which the
different degrees of freedom can be excited by
the input. Conceptually an approximate
description should discard degrees of freedom
that are neither controllable nor observable.
However there is not necessarily any agreement
between the two weightings we have on state
space. This can be resolved by considering state
transformations
under which The gramians undergo a
transformation known as a congruence
transformation. These transformations often arise
for quadratic forms, such as in the principal
axis transformation for rigid bodies in classical
mechanics.
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Balancing Transformation
It is a remarkable fact that since the gramians
are Hermitian and positive it is possible to
diagonalise both simultaneously under these
congruence transformations (recall that T is not
necessarily unitary). Given two positive
semidefinite matrices X and Y, there exists a
nonsingular matrix T such that Where S is
diagonal, positive semidefinite and has entries
equal to the singular values of Y1/2X1/2. Using
T as a state transformation then we find the
equivalent state space system for which the above
result implies that the gramians are equal and
diagonal. This system of equations is known as
the balanced realization. In the balanced
realisation the degrees of freedom that are most
controllable are also most observable.
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Balanced Truncation
You may be ready to guess a useful approximation
to our dynamical system. One picks some singular
value that is sufficiently small it may be
regarded as negligible and then simply removes
the corresponding degrees of freedom in the
balanced realisation. We partition the gramian
into blocks (singular values in order) We wish
to remove the degrees of freedom corresponding to
the second block that all have small singular
values. The nonzero elements of S1 are s1,s2 ,
sr where rltn and we assume srltsr1. Likewise we
may partition the other matrices in the problem
as follows Then our simplified dynamical
system is
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Error bounds
Note that our procedure is completely algorithmic
except for the choice of how many degrees of
freedom to remove. The other attractive feature
of balanced truncation is the fact that it is
possible to bound the error of the approximation.
We can characterize the size of the error by
considering the error signal
Our measure of the size of the error will be
the total energy in this error signal Since
both the true signal and the approximate signal
depend linearly on the input, so does the error.
So we are looking for a bound on the error in
terms of the norm of the input. In the notes I
will discuss the proof of the following upper
bound on the error
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Example Ohmic Bath
If we go back to our example of the classical
oscillator bath model we saw that we were able to
find a simple damped oscillator approximation to
the full system. Before continuing if you were
sharp you would have noticed that the original
model is not in fact stable. One way to proceed,
as here, is to argue that we wish to look for
approximations that are good over some specific
time-horizon T. Then we have to ask how to do
balanced truncation over this horizon. One way of
doing that is to artificially make the system
stable. Where we anticipate that a'1/T. Now we
measure error over 0,T
and can show
M. Barahona, A. C. Doherty, M. Sznaier, H.
Mabuchi and J. C. Doyle, Proceedings of the 41st
IEEE Conference of Decision and Control (2002).
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Example Ohmic Bath
Now we may apply our method once again, now there
are two free choices, the relevant time horizon
and the number of singular values retained. In
this case we get two large singular values
(reflecting the damped oscillator reduced model)
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Example non-Ohmic bath
Typically the model reduction technique will
retain degrees of freedom not directly reflected
in the output. This corresponds to a
non-Markovian model. For example we can change
the frequency distribution of the bath
oscillators and obtain 4 dimensional model.
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Applications to Quantum Systems?
What are some input-output systems relevant to
quantum mechanics? How does this relate to
established methods, such as adiabatic
elimination? One answer to this question is
given by considering the density matrix as a
state vector just take the columns of r and
stack them on top of each other as a vector that
we will call vec r. (There are other more
sensible ways of doing this). If we do this then
it is easy to show that From which we have that
both expectation values and correlation functions
are impulse responses of the form
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Applications to Quantum Systems?
Consider a master equation for a coupled quantum
system (two atoms 1 and 2 say) First term acts
only on system 1, the second is the coupling and
the third acts only on the second system. Well
imagine that there is steady state rss for the
first system so For comparison with adiabatic
elimination we will look for solutions of the
form We can implement this by defining a
projector You should check that
likewise we can find the complementary
projector having Then
look at the notes to convince yourself that we
have or can assume Finally we define
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Applications to Quantum Systems?
Now we may rewrite our master equation in the
form Given that all we are interested in is
the dynamics of v there is a mapping between the
problem of reducing the degrees of freedom in w
and the general model reduction problem we have
defined above. To highlight this note that we
could consider the system with Ignoring
transients we may consider the Laplace transform
of these equations to find where We are free to
apply model reduction to G(s) as described
above. Adiabatic elimination a commonly used
technique in quantum optics is just the
replacement
A. C. Doherty, J. Doyle, H. Mabuchi, K. Jacobs
and S. Habib, Proceedings of the 39th IEEE
Conference on Decision and Control, p949 (2000).