Question: I was looking into applications of Quantum Computing for machine learning and encountered the following pre-print from 2003. [Quantum Convolution and Correlation Algorithms are Physically Impossible](https://arxiv.org/abs/quant-ph/0309070). The article doesn't appear to have been published in any journals, but it has been cited a few dozen times.

The article author makes the case that it is impossible to compute discrete convolution over quantum states. Intuitively this seems incorrect to me, since I know we can perform quantum matrix multiplication, and I know that discrete convolution can be framed simply as multiplication with a Toeplitz (or circulant) matrix.

The crux of his argument seems to be that there is no realizable composition of unitary operators for the elementwise (Hadamard) product of two vectors.

Where is my disconnect? Is there any reason we cannot in general construct a Toeplitz matrix for discrete convolution in a quantum computer?

Or is the article simply incorrect? I have worked through the contradiction that the author presents in his proof of Lemma 14, and it seems to make sense to me.

Answer: 
As I can relate in my experience, I will say yes. One can indeed design algorithms without physics knowledge. For me it is so far Maths concepts. 
I remember once I watched a course about quantum computing from Scott Aaronson and he quoted : 

> 
> Quantum computing is really "easy" when you take the physics out of it.
> 
> 
> 

However, if you are going to work on applications in physics or chemistry, it will always be useful to have a background of what you are going to work on.

The field is open to many backgrounds (Maths, Physics, Computer Science...).
I think it is just a challenge sometimes communicating between different backgrounds but it is not impossible. Indeed, I would say it is constructive and it can be beneficial to collaborate together. But one can always relate to his preferred interpretation/concepts.

As a career, it is again according to your point of view. I think there is much work to be done in this field so do not worry about it. Do it if you feel you like it. Plus working in this field does not mean you have to restrain yourself. You will still have to work with classical algorithms and you will need coding skills. 

If you are interested in learning it from a computer scientist point of view,
there is this book that may be helpful:
[https://www.amazon.com/Quantum-Computing-Computer-Scientists-Yanofsky/dp/0521879965](https://rads.stackoverflow.com/amzn/click/0521879965)

Good luck on your quantum trip !
Quantum computing is totally understandable without physics background for several reasons:

1. Quantum computing is sufficiently written in abstract mathematical language (Linear Algebra).
2. Quantum computing does not include dynamical equations such as Schrodinger equations. Moreover, all operations can be understood as logical gates instead of physical entities.
3. Quantum mechanics itself made a bridge between physics and information in a such way that information scientist can understand it from his/her perspective.

However, quantum mechanics and quantum computing includes new concepts that are hard to accept by everyone including physicists. So don't worry.