Is any multi-qubit unitary operation a rotation about a specific unit vector?

Question

I understand that any single qubit unitary operation can be expressed as a rotation around a three dimensional unit vector. Is it possible to do the same for multi-qubit unitaries? Can I express an $n$-qubit unitary operation as a rotation about some $m$-dimensional vector, where (I assume) $m$ would depend on $n$?

Answer 1

No.

This works for a single qubit because a single-qubit unitary can be written as $$ U=e^{i\theta}|\psi\rangle\langle\psi|+e^{i\phi}|\phi\rangle\langle\phi| $$ where $\langle\phi|\psi\rangle=0$. Since a global phase doesn't make a difference, this might as well be $$ e^{i\theta'}|\psi\rangle\langle\psi|+e^{-i\theta'}|\phi\rangle\langle\phi| $$ where $\theta'=(\theta-\phi)/2$. The eigenvectors specify the axis to rotate about, and the $\theta'$ specifies the angle of the rotation. The mapping to the Bloch sphere is particularly neat because $|\psi\rangle$ and $|\phi\rangle$ map to the same axis.

When you go to a general multi-qubit scenario, you have many eigenvectors, and many different phases/rotation angles/eigenvalues. There's too much data to map to a single axis and a single rotation angle.

On the other hand, if you want to visualise it as a real space rotation, you can (it's just not a single axis/single angle). Let me divide the unitary into real and imaginary components: $$ U=R+iI $$ such that $UU^\dagger=1$, which means that $RR^\dagger+II^\dagger=1$ and $RI^\dagger-IR^\dagger=0$. The matrix $$ M=\begin{bmatrix} R & I \\ -I & R \end{bmatrix} $$ is real, and is a rotation matrix in the sense that $MM^T=1$, which you can verify just by multiplying everything out.

Answer 2

When we say that a single-qubit unitary "represents a rotation" in this context what we mean is that such unitaries are rotations in the adjoint representation. That is, representing qubits $\rho$ as vectors in the Bloch representation, the map $\rho\mapsto U\rho U^\dagger$ is a rotation. In other words, $\operatorname{Ad}(\mathbf{U}(2))\simeq \mathbf{SO}(3)$. See also Can we rotate Bloch vectors for qudits like we do with qubits in the Bloch sphere? and Is every single-qubit unitary just a rotation around some unit vector on the Bloch sphere? for more discussion on this point.

It remains true that $\mathbf U(N)$ unitaries act in the adjoint representation (i.e. in the Bloch representation) as $\mathbf{SO}(N^2-1)$ matrices. So in this sense yes, even in higher dimensions unitaries act like "rotations" in the Bloch representation. But note that this strongly depends on what is meant by "rotation". Generic $\mathbf{SO}(N)$ matrices always preserve distances, but notably do not generally correspond to an operation that fixes a single axis. You can generally think of them as a series of simultaneous rotations (in the standard sense) on orthogonal two-dimensional subspaces of $\mathbb R^N$, possibly with other directions kept fixed. See here and here for more details on the relation between orthogonal matrices and "rotations".