Optimal Circuit Size for Fixed-Hamming-Weight Quantum States Preparation

We study the problem of efficiently preparing fixed-Hamming-weight (HW-$k$) quantum states, which are superpositions of $n$-qubit computational basis states with exactly $k$ ones. We present a quantum circuit construction that prepares any $n$-qubit HW-$k$ state with a circuit size of $O(\binom{n}{k})$ using at most $\max\{0, n-3\}$ ancillary qubits. This is the first construction that achieves the theoretical lower bound on circuit size while using only a small number of ancillary qubits. We believe that the techniques presented in this work can be extended to other quantum state preparation algorithms based on decision diagrams, potentially reducing the reliance on ancillary qubits or lowering the overall circuit size.