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There was a similar question asked here, but I feel like mine is even more basic.

What's the easiest way to implement a circuit $U$ corresponding to a matrix-vector multiplication modulo 2?

$$ |x_1x_2x_3\ldots x_n\rangle \mapsto |y_1y_2y_3\ldots y_n\rangle=U|x_1x_2x_3\ldots x_n\rangle $$ where $$ \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ \ldots \\ y_n \end{pmatrix} = M \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ \ldots \\ x_n \end{pmatrix} \mod 2 \ . $$ Matrix $M$ entries are $0$ and $1$, the multiplication is done modulo 2.

UPDATE

The optimal solution in given here.

For a simple solution see my answer below.

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The answer to my question is given here. The solution from the provided paper is extremely efficient (for it requires $O(n^2 / \log n)$ CNOTs instead of the naïve $O(n^2)$ estimate), and is somewhat nontrivial to implement.

An easy-to-implement solution with $O(n^2)$ CNOTs can be, however, obtained in just a few lines. First of all, one has to perform the LU-decomposition of the desired matrix $M$. This can be done as follows:

P, L, U = scipy.linalg.lu(M)
L = np.mod(L, 2)
U = np.mod(U, 2)

Luckily, for an invertible matrix $M$, the resulting $P,L,U$ matrices will contain binary entries. Here $P$ is a permutation matrix, and $L$ and $U$ are the lower- and upper-triangular matrices, whose product is the original matrix, $M = P \ L \ U$.

Permutations $P$ can be implemented using SWAPs. Multiplication by a triangular matrix is also quite straightforward, and is done using CNOTs starting from the most occupied row. In other words, for $U$ one should start from the first row: once the first qubit is calculated, it is never used in the following calculation, and so on. If you, say, wanted to transform qubits according to $$ M \begin{pmatrix} q_0 \\ q_1 \\ q_2 \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} q_0 \\ q_1 \\ q_2 \end{pmatrix} \ , $$

the circuit would look like

enter image description here

A typical scenario in which one may need to implement such a matrix transformation on qubit registers arises in quantum simulation. Namely, in order to encode a state of a fermionic system using the Bravyi-Kitaev encoding, one needs to perform a matrix transformation of the original qubit registers (which can equivalently be thought of as being Jordan-Wigner encoded).

Interestingly enough, the Bravyi-Kitaev encoder matrix is lower-triangular from the very beginning [1], so in this particular case there is no need to perform the LU-decomposition.

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