# Is there a general way to parametrize 2-qubit unitaries?

So in the single-qubit case, we can write any unitary operation as an instance of the following parametrized unitary:

$$U(\theta, \phi, \lambda) = \begin{bmatrix} \cos(\theta) & -e^{i\lambda}\sin(\theta) \\ e^{i\phi}\sin(\theta) & e^{i(\lambda+\phi)}\cos(\theta) \\ \end{bmatrix}$$

What's the extension of this idea into 2-qubit operations? Can we write any 2-qubit unitary operation as an instance of some parametrized unitary?

There is some parametrized matrix form for a 2-qubit unitary, but it would be extremely inconvenient to work with. An $$n$$-qubit gate is an element of the group $$SU(2^n)$$, which has dimension $$2^{2n} - 1$$ (in other words, the number of required parameters). This is manageable for 1-qubit unitaries because there are only 3 parameters to manage. Going to 2 qubits, this number balloons to 15; no matter how the unitary is parametrized, this is going to be quite unwieldy. This is probably why you won't find such a gate actually implemented.

• Is there a formula for how many of those parameters control norms of the elements in the matrix and how many control the phases? So in the case of $U(2)$ only $\theta$ defines the norms of the elements and the others define the phases? Jun 25, 2022 at 21:41
• I'm honestly not sure that there even is a fixed formula for all parametrizations Jun 25, 2022 at 22:05
• @CodyWang Why do you say that an $n$ qubit gate is an element of $SU(2^n)$? For me it is simply an element of $U(2^n)$. Why do you add the extra condition that the determinant of the matrix is exactly equal to $1$? Jun 26, 2022 at 8:32
• That's true; I guess I'm really talking about equivalence classes of gates that differ only by a global phase. Jun 28, 2022 at 4:13

What I propose here could possibly be simplified further, but that's at least a first direction.

First, any unitary can be written as $$U=e^{-it/\hbar \widetilde{H}}$$ for some $$t$$ and $$\widetilde{H}$$ (the Hamiltonian) hermitian which we can rewrite $$U=e^{-i H}$$ where $$H$$ is also Hermitian.

Then, the $$n$$-Pauli matrices form a basis for any operator acting on $$n$$ qubits.

For this reason, $$H=\sum_{i_1...i_n} \alpha_{i_1...i_n} \sigma_{i_1} \otimes ... \otimes \sigma_{i_n}$$ with $$\alpha_{i_1...i_n}$$ a real coefficient, and $$\sigma_k$$ being equal to the identity matrix in two dimensions for $$k=0$$, or $$X$$ for $$k=1$$, $$Y$$ for $$k=2$$, $$Z$$ for $$k=3$$.

In general, an $$n$$-qubit unitary matrix can then be written as:

$$U=e^{-i \sum_{i_1...i_n} \alpha_{i_1...i_n} \sigma_{i_1} \otimes ... \otimes \sigma_{i_n}}$$

Your question is a particular case for $$n=2$$.

Now it might not be exactly the question you ask as you would probably like to have it in an non exponentiated form. Also, I guess that some symmetries could be exploited to simplify my answer (for instance a global phase doesn't matter).

But maybe that's still somewhat usefull for you, I don't know.