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Finding a unitary transformation to swap the control bit

Unitaries $U$ are diagonalisable. That means there exists a unitary $V$ such that $VUV^\dagger=P$, where $P$ is a phase gate (diagonal matrix). This means that we can think of controlled-$U$ as the ...
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How would you write $e^{i \frac{\pi}{4} \text{CNOT}}$ in ZX calculus?

Is there any simple method to write gates of the form $e^{i \frac{\pi}{4} \sigma_z^{(1)} \sigma_x^{(2)}}$? This graph should perform $\exp(i \theta \cdot (Z \otimes X))$: There may be a factor of 2 ...
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Finding a unitary transformation to swap the control bit

Given a 2 qubit controlled unitary $U$, one generally expresses it as $$|0\rangle\langle0|\otimes I + |1\rangle\langle1|\otimes P,$$ where in this example $P$ can be any unitary operator. In your ...
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Is it possible to approximately compile Toffoli using H and CSWAP?

You can exactly implement a Toffoli using CSWAPs catalyzed by a singlet state: The singlet state is the -1 eigenstate of the swap gate, and you can get phase kickback from a swap by using cswap, so ...
• 40.7k