Given a series of facts for known inputs and outputs of the form:
$U^\dagger.input.U = output$
(where $U^\dagger$ represents the conjugate transpose of unitary matrix $U$)
which procedure can be applied to determine $U$?
An example fact is:
$U^\dagger.(\hat \sigma_i \otimes \hat \sigma_z).U = \hat \sigma_x \otimes \hat \sigma_z$
In Matrix form this is:
$U^\dagger.\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1\end{pmatrix}.U = \begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & -1\\ 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\end{pmatrix}$
Here is the full set of facts for this example:
- input $\hat \sigma_x \otimes \hat \sigma_i$ output $\hat \sigma_z \otimes \hat \sigma_x$
- input $\hat \sigma_y \otimes \hat \sigma_i$ output $-\hat \sigma_y \otimes \hat \sigma_x$
- input $\hat \sigma_z \otimes \hat \sigma_i$ output $\hat \sigma_x \otimes \hat \sigma_i$
- input $\hat \sigma_i \otimes \hat \sigma_x$ output $\hat \sigma_i \otimes \hat \sigma_x$
- input $\hat \sigma_i \otimes \hat \sigma_y$ output $\hat \sigma_x \otimes \hat \sigma_y$
- input $\hat \sigma_i \otimes \hat \sigma_z$ output $\hat \sigma_x \otimes \hat \sigma_z$
The answer for $U$ in this manufactured example is:
$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 0 & 1 & 0 & -1\\ 1 & 0 & -1 & 0\end{pmatrix} $
In Quantum Computing it represents an inverse bell measurement, which is a Hadamard gate followed by a CNot operation. The unitary transformation takes state $|00\rangle$ to state $|00\rangle + |11\rangle$. The objective of trying to find $U$ in the general case is to take a Heisenburg represenation of a state, which describes the relationship between qubits, and convert it to the ket state.