PDEs: theses (physical) bounds are made for solvin'

This is the second post of the series about partial differential equations (PDEs). In this post, I will dive with you in an issue noted in the first post but left unexplored… until now!

The problem of heat waves

Let’s consider a simple bedroom: a cube with 3 meters sides.

In my earlier blog post, I wrote the following:

But does it makes sense to consider the temperature of everything around you in order to compute the temperature of your bedroom? Realistically, the temperature of this huge anticyclonic storm on Jupiter is not that important to know if you’re going to be cold in bed tonight.

You probably agreed about Jupiter, but what about the temperature in your bedroom on a cold winter day? Surely, it will be different from the temperature during a heat wave in August right?

So considering the bedroom to be

completly isolated from the outside world

does not seem right.

And you are right, the bedroom is not completely isolated from the outside world as the walls temperature will be different depending on:

• whether or not the wall is on the boundary of your house (it might be a wall between your bedroom and the kitchen for example),
• the temperature at the other side of the wall,
• the neighbours behind the wall having an air conditioner,
• etc.

Boundary conditions, or knowing the limits

As we have seen in the last section, the world is complex and there are a lot of factors that might impact the boundaries of your domain of definition.

As a simplification, all the external contributions to the domain of definition are encoded as boundary conditions.

This means that instead of considering the whole universe in our simulation, e.g.,

The wall is made of concrete with a 2cm glass wool packed both sides by a 1cm wooden planck and a 3cm concrete, its outside side has been at 23ºC (~73ºF) during 2 hours yesterday and slowly decayed to 20ºC (68ºF) until now, with the thermal conductivity of the wall we get a temperature of [exercise left to the reader]ºC on its inside side.

we will consider a simplification such as:

The wall is currently and uniformly at 19ºC (~66ºF).

In other words, we set each boundary point to a pre-defined value, that does not have to be a constant, and that will hopefully not be too far away from the true value.

Values on boundary: again a potential issue for quantum computers

Do you remember what I wrote in my previous post?

For most of the classical data, loading it into a quantum computation is not efficient (i.e., has a cost that grows exponentially with the number of qubits used).

Yup, that’s about it, boundary conditions require to encode classical values on the boundary of your domain of definition. And these classical values are hard to input into a quantum computation in general.

So just like the domain of definition of your problem, boundary conditions might also limit the capabilities of your quantum PDE solver.

Conclusion

In this post I showed how any kind of PDE will, in practice, need to define values on the boundaries of its domain of definition.

Defining these boundary conditions requires to encode classical data into the quantum computation, which is not an easy task.

Any quantum algorithm that aims at solving a given PDE will have to take this data encoding into account in order to exhibit any kind of advantage over a classical computer.

What’s next?

In today’s post I wrote about the spatial boundaries of your PDEs and how they might impact the ability of quantum computers to efficiently solve these kind of problems.

Next week, I will write about a special boundary condition that was not included in today’s post: the boundary of the time dimension!