Preparing Many Copies of a Quantum State in the Black-Box Model

TL;DR

This paper presents an optimal quantum algorithm for preparing multiple copies of a quantum state efficiently, extending Grover's method and improving over naive repetition, with applications to sampling from quantum-encoded distributions.

Contribution

The paper introduces a refined quantum rejection sampling technique that achieves optimal time complexity for preparing multiple quantum state copies.

Findings

Achieves $ heta(\sqrt{KN})$ time complexity for preparing K copies.

Extends Grover's single-copy preparation to multiple copies efficiently.

Provides a speed-up for sampling from quantum-encoded distributions.

Abstract

We describe a simple quantum algorithm for preparing $K$ copies of an $N$-dimensional quantum state whose amplitudes are given by a quantum oracle. Our result extends a previous work of Grover, who showed how to prepare one copy in time $O(\sqrt{N})$. In comparison with the naive $O(K\sqrt{N})$ solution obtained by repeating this procedure~$K$ times, our algorithm achieves the optimal running time of $\theta(\sqrt{KN})$. Our technique uses a refinement of the quantum rejection sampling method employed by Grover. As a direct application, we obtain a similar speed-up for obtaining $K$ independent samples from a distribution whose probability vector is given by a quantum oracle.