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arXiv:2610.01940

Optimal query complexity for fractional quantum evolution

Anthony Yuezhang Liu, Adam Wesołowski, Jayne Thompson, Mile Gu, Lirandë Pira

quant-phSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
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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 $δ$. Quantum singular value transformation gives an upper bound of $O\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$ queries for approximation error $\varepsilon$. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any $N$-query circuit to the approximation of $e^{itθ}$ by a trigonometric polynomial with degree bounded by $O(N)$, together with Remez inequality. This allows us to establish the lower bound of $Ω_τ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$. Consequently, the optimal query complexity for fractional query problem is $Θ_τ\!\left(\frac{1}δ\log\frac{1}{\varepsilon}\right)$, 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 $Ω_τ\!\left(\log\frac{1}{\varepsilon}\right)$ bound uniform to $δ$.

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An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2n-dimensional, Hilbert space, which is a major source of quantum speed-ups. We develop a new “Quantum singular value transformation” algorithm that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure - typically using only a constant number of ancilla qubits - leading to optimal algorithms with appealing constant factors. We show that our framework allows describing many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to various quantum machine learning applications. We also devise a new singular vector transformation algorithm, describe how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum, and show how to efficiently implement principal component regression. Finally, we also prove a quantum lower bound on spectral transformations.
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