Q3L11a.txt

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Simulation of physical systems is an attractive application
for quantum computers.
The challenge of simulation may be expressed as the desire
to solve certain differential equations.
For example, we may have Newton's equation of motion
perhaps with a spatially dependent mass,
or we may have Maxwell's equation such as a wave
equation with time dependent material parameters epsilon
and mu.
Or we may have a diffusion equation,
which is second order in space and first order in time.
Three steps are typically taken in solving such simulation
problems.
First, the state of the system is typically approximated
with a digital representation.
Second, the equations of motion are
discretized in space and time.
For example, the derivative is made discrete.
To first order, this may be simply the typical difference
formula.
The discretized equations are then solved with bounded error.
Ideally, these equations are a linear set of equations,
but that is not necessarily the case.
Of course, quantum systems can also be simulated this way,
and perhaps it was Richard Feynman
who first noted in 1981 that quantum systems could
efficiently simulate classical systems.
But the reverse of this is apparently not true.
Classical systems have a hard time simulating
quantum systems efficiently.
Consider for example Schrodinger's equation which
describes quantum mechanics.
This is a first order differential equation in time.
And the key difference in complexity
is that for a system of n quantum bits,
one has 2 to the n coefficients.
That leads to 2 to the n differential equations
which must be simulated.
For a fair comparison though, we need to consider the goal,
and that is to answer certain questions
such as-- how do solids melt?
What are their phase diagrams?
What molecular structure of atoms are stable?--
a chemistry question.
What are the optical, thermal, and electrical properties
of a given physical system?
How fast do two or more systems react if they
are put together and so forth?
These are our questions about emergent behavior,
and simulation is a compelling way to answer them.
In order to quantify the benefits of quantum simulation
versus classical, let us consider some strategies
and issues related to the simulation of quantum systems.
Suppose that the system we wish to simulate
is described by the goal Hamiltonian, H sys.
We would describe a quantum simulation of a quantum system
and let the simulator be described
by these Hamiltonians, the quantum computer
Hamiltonians indexed by k.
The first thing to check is that the quantum computer
can even in principle simulate the goal Hamiltonian.
And we do this by seeing if the system Hamiltonian being
simulated can be constructed from the quantum computer
Hamiltonian.
And this mathematically means that H sys
must be in the algebra generated by Hqc k.
The construction allows for all commutators
of these Hamiltonian elements and linear combinations
of them.
The mechanism for combining them is known as trotterization,
as we shall see in a moment.
The second issue is what to measure.
After all, if we have a quantum state as the output, psi,
this state collapses after a single measurement.
And so if we desire something like the expectation
value of an observable or a whole set of features-- say,
a sample of some phase diagram structure-- then
we should be concerned about the resource requirements
as a function of the precision desired in the measurement
result.
A third issue to consider is whether an analog simulation
or a digital simulation is more appropriate.
Analog simulations deform the quantum computer Hamiltonian
to become the system Hamiltonian.
Analog simulations may offer more optimal mappings.
Also, one may want to use one open quantum system to simulate
another open quantum system.
A fourth issue to consider is whether classical simulation
might actually be better than quantum.
After all, approximate answers may be officially obtainable
and may be sufficient for the desired result.
There have been many successful approximate classical
algorithms including renormalization group, matrix
products state techniques, PEPS techniques, and recently MERA
type techniques, all of which you're welcome to look up.
These use quantum information ideas
to improve the classical algorithms,
and that is a very important approach
to keep in mind as one does quantum simulations.