  A pure state of fixed Hamming weight is a superposition of computational
basis states such that each bitstring in the superposition has the same number
of ones. Given a Hilbert space of the form $\mathcal{H} =
(\mathbb{C}_2)^{\otimes n}$, or an $n$-qubit system, the identity operator can
be decomposed as a sum of projectors onto subspaces of fixed Hamming weight. In
this work, we propose several quantum algorithms that realize a coherent
Hamming weight projective measurement on an input pure state, meaning that the
post-measurement state of the algorithm is the projection of the input state
onto the corresponding subspace of fixed Hamming weight. We analyze a
depth-width trade-off for the corresponding quantum circuits, allowing for a
depth reduction of the circuits at the cost of more control qubits. For an
$n$-qubit input, the depth-optimal algorithm uses $O(n)$ control qubits and the
corresponding circuit has depth $O(\log (n))$, assuming that we have the
ability to perform qubit resets. Furthermore, the proposed algorithm
construction uses only one- and two-qubit gates.
