Optimal query complexity for fractional quantum evolution

Abstract

Given oracle access to an unknown unitary $U=e^{iH}$ , the fractional query problem asks how many queries are required to implement a noninteger power $U^t=e^{itH}$, $0<t<1$, when the spectrum is separated from the branch cut by a gap $\delta$. Quantum singular value transformation gives an upper bound of $O\left(\frac{1}{\delta }\log \frac{1}{\epsilon }\right)$ queries for approximation error $\epsilon$. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any $N$-query circuit to the approximation of $e^{it\theta }$ by a trigonometric polynomial with degree bounded by $O(N)$, together with Remez inequality. This allows us to establish the lower bound of ${\Omega }_{\tau }\left(\frac{1}{\delta }\log \frac{1}{\epsilon }\right)$. Consequently, the optimal query complexity for fractional query problem is ${\Theta }_{\tau }\left(\frac{1}{\delta }\log \frac{1}{\epsilon }\right)$, showing that the known QSVT construction is asymptotically optimal. We also give an alternative lower bound proof based on constructing a linear functional that annihilates the approximant space, yielding a ${\Omega }_{\tau }\left(\log \frac{1}{\epsilon }\right)$ bound uniform to $\delta$.