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I have a $x \in \mathbb{N}$, and I would like to decompose it in terms of the Pauli matrices $\sigma_x, \sigma_y, \sigma_z$ and the identity. My first steps are as follows:

$$ \begin{align} |x \rangle\langle x| & = |x_n \rangle \langle x_n| \otimes \cdots \otimes |x_1 \rangle \langle x_1| \\ & = \frac{1 + (-1)^{x_n} \sigma_z}{2} \otimes \cdots \otimes \frac{1 + (-1)^{x_1} \sigma_z}{2} \end{align} $$

where $x_n, \ldots, x_1$ are the bits of $x$, starting from the most significant bit. I'm stuck at this point. Do you know if there is a well-known nice expression at the end, or do you know how to go ahead from this point?

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At this point, I would tend to use the notation $Z_x$ for $x\in\{0,1\}^n$ to denote which os the $N$ qubits have had a $\sigma_Z$ applied to them (identity on the othe qubits). Then you'll have $$ |x\rangle\langle x|=\frac{1}{2^n}\sum_{y\in\{0,1\}^n}Z_y(-1)^{x\cdot y}. $$

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