Abstract

We present a polynomial-time quantum algorithm making a single query (in superposition) to a classical oracle, such that for every state |ψ⟩ there exists a choice of oracle that makes the algorithm construct an exponentially close approximation of |ψ⟩. 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 ⊆ stateQIP (a quantum state analogue of PSPACE ⊆ IP) and show that a constant number of rounds of interaction suffices. • We show that QACf0 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. To appear in SODA 2024 . Paper available at https://arxiv.org/abs/2306.01723.