Preparation of Hamming-Weight-Preserving Quantum States with Log-Depth Quantum Circuits

Quantum state preparation is a critical task in quantum computing, particularly in fields such as quantum machine learning, Hamiltonian simulation, and quantum algorithm design. The depth of preparation circuit for the most general state has been optimized to approximately optimal, but the log-depth appears only when the number of ancillary qubits reaches exponential. Actually, few log-depth preparation algorithms assisted by polynomial ancillary qubits have been come up with even for a certain kind of non-uniform state. We focus on the Hamming-Weight-preserving states, defined as $|\psi_{\text{H}}\rangle = \sum_{\text{HW}(x)=k} \alpha_x |x\rangle$, which have leveraged their strength in quantum machine learning. Especially when $k=2$, such Hamming-Weight-preserving states correspond to simple undirected graphs and will be called graph-structured states. Firstly, for the $n$-qubit general graph-structured states with $m$ edges, we propose an algorithm to build the preparation circuit of $O(\log n)$-depth with $O(m)$ ancillary qubits. Specifically for the $n$-qubit tree-structured and grid-structured states, the number of ancillary qubits in the corresponding preparation circuits can be optimized to zero. Next we move to the preparation for the HWP states with $k\geq 3$, and it can be solved in $O(\log{{n \choose k}})$-depth using $O\left({n \choose k}\right)$ ancillary qubits, while the size keeps $O\big( {n \choose k} \big)$. These depth and size complexities, for any $k \geq 2$, exactly coincide with the lower bounds of $\Omega (\log{{n \choose k}})$-depth and $\Omega ({n \choose k})$-size that we prove lastly, which confirms the near-optimal efficiency of our algorithms.