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: 
Speaking as a computer scientist without any physics background making contributions to quantum computing: yes, computer scientists without any physics background can make contributions to quantum computing. Though I think it was always that way; it has nothing to do with the field being "mature".

If you understand the postulates of quantum mechanics (operations are unitary matrices, states are unit vectors, measurement is a projection), and know how to apply those in the context of a computation, you can create quantum algorithms. The fact that these concepts were originally derived from physics is historically interesting, but not really relevant when optimizing a quantum circuit. As a concrete example: quantum physics is very heavy on calculus, but quantum computation isn't.

Quantum physics does become relevant if you are trying to design algorithms for simulating quantum systems. And some of the concepts you would learn in a quantum physics course should also appear in a quantum computation course. But overall [I agree with Scott Aaronson](https://www.scottaaronson.com/democritus/lec9.html):

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> [My] way to teach quantum mechanics [...] starts directly from the conceptual core -- namely, a certain generalization of probability theory to allow minus signs. Once you know what the theory is actually about, you can then sprinkle in physics to taste [...]
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> [quantum mechanics is] not a physical theory in the same sense as electromagnetism or general relativity. [...] Basically, quantum mechanics is the operating system that other physical theories run on as application software [...]
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> [...] [quantum mechanics is] about information and probabilities and observables, and how they relate to each other.
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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 : 

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