> Except that even the first classical computers were built with goals and applications in mind. [...] Instead, when asking a quantum computing company what they're trying to achieve, they'll gesture vaguely at "chemistry, finance, ecology".

I think the problem is a little bit more subtle:

To finance a lot of innovations, better also some intermediate step towards the far goal should already be very useful, otherwise the company that builds it will go bankrupt.

If this is not the case, it's typically not commercially viable, some product category is typically basic research (which is very important, but it typically means that the commercial potential will only come up in some future).

There do exist problems where a quantum computer gives an extreme advantage in the sense that we have no idea how a fast classical algorithm could look like. So, the only viable approaches for these problems are:

1. work on a huge algorithmic breakthrough (to be able to solve these problems fast on a classical computer)

2. build a quantum computer

What are these problems?

They are basically all special cases of the abelian hidden subgroup problem:

> https://en.wikipedia.org/w/index.php?title=Hidden_subgroup_p...

In particular cf. the table at the end of this Wikipedia article:

> https://en.wikipedia.org/w/index.php?title=Hidden_subgroup_p...

If you do have such a problem to solve, 1 and 2 are the only viable approaches.

So, there do exist goals and applications for which a quantum computer is insanely useful (assuming no huge algorithmic breakthrough happens).

The questions are thus:

- Is the abelian hidden subgroup problem sufficient for being able to carry a whole potential industry?

- (To come back to my introduction) What use does a quantum computer that is only capable of solving very small instances of this problem have for the user?

I am familiar with these applications. Indeed, your questions are very relevant. I've personally decided a long time ago that no, these applications are insufficiently useful to justify the billions invested in quantum computing, and the billions more that will be required to build anything remotely capable. So far, nothing's come up to make me reconsider.