Toward optimal circuit size for sparse quantum state preparation
Rui Mao, Guojing Tian, Xiaoming Sun · Physical Review A · 2024
Compared to general quantum states, the sparse states arise more frequently in the field of quantum computation. In this work we consider the preparation for $n$-qubit sparse quantum states with $s$ nonzero amplitudes and propose two algorithms. The first algorithm uses $O(ns/{log}_{2}n+n)$ gates, improving upon previous methods by $O({log}_{2}n)$. Moreover, the classical runtime of this algorithm is optimal. We further establish a matching lower bound for any algorithm that is not amplitude aware and employs at most $poly(n)$ ancillary qubits. The second algorithm is tailored for binary strings that exhibit a short Hamiltonian path. An application is encoding the input data into a state with specified Hamming weight $k$ in quantum machine learning, for which our algorithm constructs a circuit of size $O(\left(\genfrac{}{}{0pt}{}{n}{k}\right){log}_{2}n)$. This surpasses previous results by $O(k/{log}_{2}n)$ and is close to the lower bound $O(\left(\genfrac{}{}{0pt}{}{n}{k}\right))$. The classical runtime is also nearly optimal. Both algorithms shrink the existing gap theoretically and provide increasing advantages numerically.