We present a polynomial-time quantum algorithm making a single query (in superposition) to a classical oracle, such that for every state $|\psi\rangle$ there exists a choice of oracle that makes the algorithm construct an exponentially close approximation of $|\psi\rangle$. Previous algorithms for this problem either used a linear number of queries and polynomial time [arXiv:1607.05256], or a constant number of queries and polynomially many ancillae but no nontrivial bound on the runtime [arXiv:2111.02999]. As corollaries we do the following: - We simplify the proof that statePSPACE $\subseteq$ stateQIP [arXiv:2108.07192] (a quantum state analogue of PSPACE $\subseteq$ IP) and show that a constant number of rounds of interaction suffices. - We show that QAC$\mathsf{_f^0}$ lower bounds for constructing explicit states would imply breakthrough circuit lower bounds for computing explicit boolean functions. - We prove that every $n$-qubit state can be constructed to within 0.01 error by an $O(2^n/n)$-size circuit over an appropriate finite gate set. More generally we give a size-error tradeoff which, by a counting argument, is optimal for any finite gate set.

翻译：我们提出一种多项式时间量子算法，该算法仅需对经典预言机进行单次查询（叠加态），使得对于任意态 $|\psi\rangle$，总存在一个预言机选择使得算法构建出 $|\psi\rangle$ 的指数级精确近似。此前解决该问题的算法要么使用线性次数查询和多项式时间 [arXiv:1607.05256]，要么使用常数次查询和多项式数量辅助比特但不对运行时间做有效约束 [arXiv:2111.02999]。作为推论，我们完成以下工作：- 简化了 statePSPACE $\subseteq$ stateQIP [arXiv:2108.07192]（即 PSPACE $\subseteq$ IP 的量子态类比）的证明，并表明仅需常数轮交互即可实现。- 证明构建显式态的 QAC$\mathsf{_f^0}$ 下界将蕴含计算显式布尔函数的突破性电路下界。- 证明任意 $n$ 量子比特态均可通过一个在适当有限门集上的 $O(2^n/n)$ 规模电路在 0.01 误差内构建。更一般地，我们给出规模-误差权衡，该权衡通过计数论证对任意有限门集均为最优。