Space-time tradeoff for sparse quantum state preparation

TL;DR

This paper explores the optimal balance between circuit depth and ancillary qubits for preparing sparse quantum states, achieving the best trade-off with fewer resources than previous methods.

Contribution

It introduces a new quantum circuit construction that optimally balances depth and ancilla qubits for sparse state preparation, improving resource efficiency.

Findings

Achieves the best known depth-ancilla trade-off for sparse quantum states.

Recovers the optimal circuit depth with significantly fewer gates and ancillas.

Provides a scalable method for efficient sparse state preparation.

Abstract

In this work, we investigate the trade-off between the circuit depth and the number of ancillary qubits for preparing sparse quantum states. We prove that any $n$-qubit $d$-spare quantum state (i.e., it has only $d$ non-zero amplitudes) can be prepared by a quantum circuit with depth $O\left(\frac{nd \log m}{m \log m/n} + \log nd\right)$ using $m\geq 6n$ ancillary qubits, which achieves the current best trade-off between depth and ancilla number. In particular, when $m = \Theta({\frac{nd}{\log d}})$, our result recovers the optimal circuit depth $\Theta(\log nd)$ given in \hyperlink{cite.zhang2022quantum}{[Phys. Rev. Lett., 129, 230504(2022)]}, but using significantly fewer gates and ancillary qubits.