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, or a constant number of queries and polynomially many ancillae but no nontrivial bound on the runtime. As corollaries we do the following: - We simplify the proof that statePSPACE $\subseteq$ stateQIP (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的量子算法，该算法仅需对经典预言机进行一次（叠加态）查询，使得对于任意态$|\psi\rangle$，都存在一种预言机选择，使算法能够构造出$|\psi\rangle$的指数精度近似。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，要么使用常数次查询和多项式个辅助比特但未给出运行时非平凡界。作为推论，我们实现了以下结果： - 简化了statePSPACE $\subseteq$ stateQIP（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成立。 - 证明构造显式量子态的QAC$\mathsf{_f^0}$下界将隐含着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，由基于适当有限门集的$O(2^n/n)$规模电路构造。更一般地，我们给出一种规模-误差权衡，根据计数论证，该权衡对任何有限门集均是最优的。