Optimizing sparse quantum state preparation with measurement and feedforward

Abstract: Quantum state preparation (QSP) is a key component in many quantum algorithms. In particular, the problem of sparse QSP (SQSP) $\unicode{x2013}$ the task of preparing the states with only a small number of non-zero amplitudes $\unicode{x2013}$ has garnered significant attention in recent years. In this work, we focus on reducing the circuit depth of SQSP with limited number of ancilla qubits. We present two SQSP algorithms: one with depth $O(n\log d)$, and another that reduces depth to $O(n)$. The latter leverages mid-circuit measurement and feedforward, where intermediate measurement outcomes are used to control subsequent quantum operations. Both constructions have size $O(dn)$ and use $O(d)$ ancilla qubits. Compared to the state-of-the-art SQSP algorithm in arXiv:2108.06150, which allows an arbitrary number of ancilla qubits $m>0$, both of our algorithms achieve lower circuit depth when $m=d$.