= Collisional phase gate
{title2=$U_\phi=\exp(-i\phi|01\rangle\langle01|)$}
A <two-qubit gate> in which an interaction shifts the <energy> of a selected joint <computational basis> state and thereby accumulates a <relative quantum phase>. With <Hamiltonian operator> $H=\Omega|01\rangle\langle01|$, where $\Omega$ has units of <energy>, an interaction time $\tau$ gives $U_\phi=\operatorname{diag}(1,e^{-i\phi},1,1)$ with $\phi=\Omega\tau/\hbar$. The selected basis state is part of the convention: this gate differs from the usual <Controlled-Z gate> at $\phi=\pi$. Conjugation by a <Pauli X gate> on the first <qubit> gives $(X\otimes I)U_\pi(X\otimes I)=\mathrm{CZ}$, because the conjugation interchanges $|01\rangle$ and $|11\rangle$.
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