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：我们提出了一种多项式时间的量子算法，该算法对经典预言机仅进行一次（叠加态上的）查询，使得对于每个态$|\psi\rangle$，存在一种预言机选择使算法能构建出$|\psi\rangle$的指数级近似。此前针对该问题的算法要么使用线性次查询和多项式时间[arXiv:1607.05256]，要么使用常数次查询和多项式数量的辅助量子比特但无运行时非平凡界限[arXiv:2111.02999]。作为推论，我们实现以下结果： - 简化了statePSPACE $\subseteq$ stateQIP[arXiv:2108.07192]（即PSPACE $\subseteq$ IP的量子态类比）的证明，并表明常数轮交互即可满足要求。 - 表明构建显式态的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。 - 证明每个$n$量子比特态可在0.01误差内由适当有限门集上的$O(2^n/n)$规模电路构建。更一般地，我们给出了规模-误差权衡关系，通过计数论证，该关系对任何有限门集均为最优。