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arXiv · 2306.01723

Efficient Quantum State Synthesis with One Query

Gregory Rosenthal
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Abstract

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.

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Gregory Rosenthal. 2023-06-02. Efficient Quantum State Synthesis with One Query. https://arxiv.org/abs/2306.01723

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