# Optimizing sparse quantum state preparation with measurement and feedforward

## TL;DR

This paper introduces two new algorithms for sparse quantum state preparation that significantly reduce circuit depth by utilizing measurement and feedforward techniques, improving efficiency over previous methods.

## Contribution

The paper presents two depth-optimized SQSP algorithms that leverage measurement and feedforward, reducing circuit depth with limited ancilla qubits compared to prior work.

## Key findings

- One algorithm achieves $O(n ext{log}d)$ depth.
- Another reduces depth to $O(n)$ using measurement and feedforward.
- Both algorithms have size $O(dn)$ and use $O(d)$ ancilla qubits.

## 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 $m=d$.

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Source: https://tomesphere.com/paper/2508.21346