# PDEs: All I want for Christmas is... my result!

## Table of Contents

This is the fourth post of the series about partial differential equations (PDEs). In the first, second and third posts, we explored why data **input** was an issue for quantum PDE solvers. In this post, I want to show you why data **output** is not to be forgotten.

`data_encoding_Vfinal.final.FINAL.pdf`

Yes, I know, I am writing too much about data encoding… But this time I do not want to speak about **classical** data encoding, I swear!

This time, I am writing about **quantum** data encoding and its potential impacts on the efficiency of our quantum PDE solvers. Relieved? Sadly, you should probably not…

First, let me tell you a story.

### The story of M-theory woman

Once upon a time, there was a superheroine called “M-theory woman” that was capable of doing extraordinary things by expanding herself into 11 dimensions and going back in our lame 4-dimensional world (3 dimensions of space, 1 dimension for the time) to see the results of her actions.

But M-theory woman had an issue with her powers: people she helped often did not realize she was helping them. The results of her actions in the 11-dimensional realm were often too complex and subtle to be witnessed by mere 4-dimensional humans. This lack of understanding from humans put her in a difficult position when she had to prove to the Hero Tax Administration (HTA) that she was indeed working hard as an heroine.

In order to have humans certify to the HTA that she helped them and avoid a hefty fine, she had to teach them how to look differently at their lives and recognize the impacts of actions on 11 dimensions even though they were limited to 4.

Of course, humans were not capable of understanding the full picture of her actions, but acknowledging that something was done by M-theory woman was sufficient for her to avoid the HTA “regularization”.

### Looking differently at your life: the secret of happiness?

You might ask yourselves “but **how** are we supposed to look differently at our lives to witness things in 11 dimensions?”. Let’s use another example and change dimensions to illustrate this strange statement.

You, as a human, can only see images in 2-dimensions. Each one of your eyes can only see a flat 2-dimensional image of the world we live in. Nevertheless, you can “view” in 3 dimensions thanks to stereoscopic vision (also called stereopsis).

In simpler terms, the fact that your eyes are located at two different places on your face allow them to view two slightly different images. The two 2-dimensional images, both taken from a slightly different point of view, are then processed by your brain to extract data about the third dimension.

What happens with your eyes and your brain when you look at something is exactly what M-theory woman taught to humans: how to look at several 4-dimensional snapshots of our world and reconcile them together to get an idea of what is happening in 11 dimensions.

### M-theory woman is a quantum computer

The story of M-theory woman can be re-phrased by replacing the superheroine by a quantum computer, tax administration set aside. The quantum computer (like M-theory woman) can perform actions on a space that is in high dimension, harnessing the full power of that space. For M-theory woman, the space is an hypothetical 11-dimensional space. For a quantum computer, it is an Hilbert space of dimension $2^n$, with $n$ the number of qubits.

But just like for M-theory woman actions, humans cannot have the full knowledge of the results of a quantum computations. They can only have access to a tiny fraction of the $2^n$ dimensions, limiting their ability to deeply understand the results of the computation.

Nevertheless, by looking in a specific way at the results of a quantum computers, humans can still extract useful information. Not the full result, but “summaries” or “part” of it.

## What is the link with PDEs?

All the quantum algorithms that can be used to solve a PDE and that I know about encode the solution of the problem into this $2^n$-dimensional-space-you-can-only-extract-summaries-of.

And they cannot do otherwise in order to be efficient.

This means that once the algorithm is over, the result of your problem is located in the $2^n$-dimensional-space-you-can-only-extract-summaries-of. So in the end, you will not have access to the full solution.

And if your use-case requires you to have access to the full solution vector, then you are basically sentenced to use classical computing as reading the full result out of a quantum computer:

- is exponentially costly with respect to the number of qubits (i.e., no “runtime quantum advantage” possible anymore) and
- requires huge classical post-processing that also scales exponentially with the number of qubits.

## So I will never get my result back?

It depends what you call a “result”. If you want the full solution vector to your PDE, it is doomed to fail (or be inefficient). But if you only need a specific information about the result, you might have a chance!

Let’s say you only want a “summary” of the full solution, your only challenge will be to find how to measure this “summary”.

I know that the word “summary” seems overly vague, but you will have to dig into the details of the theory of “observables” to get a better understanding about what can possibly be measured and what cannot.

Also, my wording

your only challenge will be to find how to measure this “summary”

seems to implie that this is an easy task. It is **NOT**.

It is also important to note that most of the industrial applications do not require the full result of the PDE. Getting the full simulation result of a combustion chamber is interesting by itself, but might not be the end goal as industrials will likely rather be interested by:

- its efficiency,
- its ability to not explode during operation,
- its noise output,
- etc.

## Conclusion

After dealing with data input in the previous three posts, I wanted to write about data output that brings its own challenges on the table.

If you require the full solution to your PDE then quantum computing is likely not the right tool to solve your problem. If you do not, then *maybe* you can devise a way of recovering the result “summary” that will allow you to solve your problem.

## What’s next?

This was the last post about data input / output as well as the last “easy” post of the series. The next posts will dive into more involved subjects and will require more experience about PDEs and scientific computing.

To mark the change in difficulty and to let you process the 4 previous posts on data input / output, there will be no blog post next week. The weekly schedule will resume on Thursday 26^{th} with an nice exploration of variational algorithms, ansätzs, and one crucial limitation of such algorithms for PDEs.

If you want to be notified when each post is out, you can add me on LinkedIn, I will do a LinkedIn post for each new blog post.

Also, if this post picked your interest or raised some questions in your mind, please feel free to send me a message 😉