New answers tagged circuit-construction
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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, $CX=(I\otimes H) CZ( I\otimes H)$...
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CX and CZ commutation
$E$ is the map given by $E|i,j\rangle |\psi \rangle = |i,j\rangle Z^j X^i |\psi \rangle$.
$E'$ is the map given by $E'|i,j\rangle |\psi \rangle = |i,j\rangle X^i Z^j |\psi \rangle = (-1)^{ij}|i,j\...
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Creating a uniform superposition of a subset of basis states
Just adding to Quantum_Brains' answer, the algorithm by Alok Shukla et al. has been implemented as UniformSuperPositionGate in Qiskit release 1.2. Therefore, you can directly use it in Qiskit. Here is ...
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Block encoding of a diagonal matrix with equidistant eigenvalues
$\newcommand{ket}[1]{|#1\rangle}\newcommand{bra}[1]{\langle#1|}$
We can construct a block-encoding with optimal scale factor $\alpha = 2^n-1$ and using only a single ancilla qubit as follows:
Let $...
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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 ...
- 60.8k
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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 ...
- 1,105
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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
3
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Accepted
Swapping amplitudes of orthogonal components
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 $\{|\...
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