# Tag Info

### CX and CZ commutation

$$CX=(I\otimes H) CZ( I\otimes H)$$ So we can express $$CX_{1,3}CZ_{2,3}=(I\otimes I\otimes H) CZ_{1,3}(I\otimes I\otimes H) CZ_{2,3}$$ By the first identity I showed, $CX=(I\otimes H) CZ( I\otimes H)$...
• 1,105
1 vote

### 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 ...
• 60.8k

### 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 ...
• 1,105
Accepted

### Power of Toffoli vs T in quantum logic

I think of the gatesets as equivalent. Clifford+Toffoli can do everything Clifford+T can do, if given access to a single $|T\rangle$ state. Clifford+Toffoli can duplicate the T state, allowing an ...
• 40.8k
Yes, unitaries map orthonormal bases to orthonormal bases. As $|A\rangle$ and $|A^{\perp}\rangle$ are orthogonal they can be extended to an orthonormal basis on the space. Let's call such a basis \$\{|\...