Add ApplyUnitary operation

[fedimser]

This PR adds Q# operation which applies arbitrary gate specified by 2^n x 2^n unitary matrix on a qubit register, as requested in #291.

It's based on an algorithm which I have implemented in Python in this repository.

Implementation consists of 3 parts:

1. Implementing 1-qubit gate specified by 2x2 matrix - in native Q#.
2. Matrix decomposition algorithm, which decomposes unitary matrix as product of two-level unitary matrices with indices differing exactly in 1 bit - in C#.
3. Q# operation which uses decomposition in (2) and implements arbitrary n-qubit unitary via multiple calls to controlled (1) operation and X gates.

Complexity of this algorithm is O(N^3) and size of output is O(N^2), where N is size of matrix (N=2^n, where n is number of qubits).

[ghost]

All CLA requirements met.

[msoeken]

Hi @fedimser, you're initial commits look interesting! I left some initial comments.

[fedimser]

Thanks for the review.
This PR is work in progress, so I converted it to draft.

[fedimser]

@msoeken, can you please advice which linear algebra library I can use in the C# code?
All I need is to be able to represent complex matrices and matrix multiplication. I can implement it from scratch on top of C# arrays, but I would prefer to use some C# equivalent of NumPy.

[msoeken]

@fedimser I don't have much experience with NumPy alternatives in .NET, but you can see if NumSharp has the operations you need.

[msoeken]

Thanks a lot @fedimser, this is great work! I left some comments, mostly formatting and some suggestions.

[cgranade]

Thanks for doing this, we really appreciate your contribution! I've marked as request changes for some of the required modifications (e.g.: copyright headers), but I'm happy to help expand on or implement any of my comments.

Since this PR modifies the Q# API, I'll also go on and schedule it for the next API Review Meeting, and will comment here accordingly following that meeting. Thank you!

[msoeken]

Looks great to me, thanks! I left some few comments.

[fedimser]

Hi Mathias,

I addressed your comments.
Also I have found a bug in my algorithm, which I fixed in my Python repository, as well as here.
I added one more test for 8x8 matrix which is equivalent to 4 controlled gates (this test helped me find a bug).

One more thing to double check is that my algorithm indeed applies matrix to little-endian register. The easiest way to check that is to see what is matrix for CNOT gate, and I have a test for that.
So, if you give this matrix to ApplyUnitary:
[[1, 0, 0, 0],
[0, 0, 0, 1],
[0, 0, 1, 0],
[0, 1, 0, 0]],
then it will apply operation equivalent to CNOT(q[0], q[1]).
We can see that it swaps states 1 and 3, i.e. states with the lowest bit equal to 1. Which means qubit corresponding to thelowest bit is control qubit. And we see that control qubit is q[0]. So, the qubit with index 0 encodes the lowest bit, as it should.

[msoeken]

Just some formatting suggestions.

[cgranade]

This looks really good, thanks for the contribution! I've left a few last comments as suggestions that would be good to bring in before merging this to main, but otherwise I think this is getting pretty close now that the API review has completed.

Thank you again!

[fedimser]

@cgranade, I addressed your comments. I think this PR is now ready to be merged.

[cgranade]

Looks good, thank you for the contribution! ♥

[cgranade]

/azp run

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[cgranade]

/azp run

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