Estimating the volume of a high-dimensional convex body is a fundamental problem in theoretical computer science. Given a quantum membership oracle for a convex body $K\subset\mathbb{R}^d$, we show that estimating $\mathrm{vol}(K)$ to relative error $\varepsilon$ takes $\widetilde O(d^{5/2}+d^{3/2}/\varepsilon)$ queries. This improves the previous quantum upper bound $\widetilde O(d^3+d^{9/4}/\varepsilon)$ of Chakrabarti et al. (ACM TQC 2023) and Cornelissen--Hamoudi (SODA 2023), and achieves a larger quantum speedup over the currently best classical upper bound $\widetilde O(d^{7/2}+d^3/\varepsilon^2)$ of Jia et al. (JACM 2026). Our quantum algorithm combines two new ingredients: an input-dependent effective spectral-gap analysis of quantum walks based on uniform classical warm-start mixing bounds, and a single quantum estimator for products of normalizer ratios in simulated annealing based on classical bridge sampling. We also prove an $Ω(d)$ quantum query lower bound for constant relative error, improving the previous $Ω(\sqrt d)$ lower bound.
Estimating the volume of a high-dimensional convex body is a fundamental problem in theoretical computer science. Given a quantum membership oracle for a convex body K⊂Rd, we show that estimating vol(K) to relative error ε takes O(d5/2+d3/2/ε) queries. This improves the previous quantum upper bound O(d3+d9/4/ε) of Chakrabarti et al. (ACM TQC 2023) and Cornelissen--Hamoudi (SODA 2023), and achieves a larger quantum speedup over the currently best classical upper bound O(d7/2+d3/ε2) of Jia et al. (JACM 2026). Our quantum algorithm combines two new ingredients: an input-dependent effective spectral-gap analysis of quantum walks based on uniform classical warm-start mixing bounds, and a single quantum estimator for products of normalizer ratios in simulated annealing based on classical bridge sampling. We also prove an Ω(d) quantum query lower bound for constant relative error, improving the previous Ω(d) lower bound.
Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an n -dimensional convex body within multiplicative error ε using Õ(n 3 + n 2.5 /ε ) queries to a membership oracle and Õ(n 5 +n 4.5 /ε) additional arithmetic operations. For comparison, the best known classical algorithm uses Õ(n 3.5 +n 3 /ε 2 ) queries and Õ(n 5.5 +n 5 /ε 2 ) additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of “Chebyshev cooling,” where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requires Ω (√ n+1/ε) quantum membership queries, which rules out the possibility of exponential quantum speedup in n and shows optimality of our algorithm in 1/ε up to poly-logarithmic factors.
Entropy estimation of discrete probability distributions is a fundamental problem in statistical inference and quantum property estimation. In this paper, we study the problem of estimating Rényi entropy Hα(p) for any distribution p=(pi)i=1n given its purified quantum query-access. For non-integer α>1, our algorithm estimates Hα(p) to within additive error ε using only O(nmax{1/2,1−1/α}/ε) queries, which improves the best known upper bound for dependence on both n and 1/ε. For 0<α<1, our algorithm has query complexity O(n1/(2α)/εmax{1,1/(2α)}), which improves the dependence on 1/ε of the current best upper bound without incresing its n dependence. Our approach adapts the multi-level estimation framework to relative approximation of the power sums and leverage binary search to remove the requirement of prior knowledge of them. We also strengthen the precision-dependent lower bound to Ω(1/εmax{1,1/(2α)}) for 0<α<1, which identifies the optimal precision dependence of our algorithm.
We establish a linear quantum query lower bound for high-accuracy convex optimization over an explicit family of n-dimensional ellipsoids. We focus on linear optimization with an explicitly given objective, where the feasible set is accessed through a membership oracle. We show that any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error Θ(n−1/2) requires Ω(n) membership queries, where the ellipsoids are contained in the unit ball. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within Θ(n−3/2) distance from the feasible set. This resolves an open question posed by Chakrabarti, Childs, Li, and Wu~(\textit{Quantum}, 2020) and by van Apeldoorn, Gilyén, Gribling, and de Wolf~(\textit{Quantum}, 2020). Coupled with the upper bounds in these papers, the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The proof is built around a lower bound for determinant computation that is derived via a novel polynomial method based on Fourier-rank. In the continuous matrix phase-query model, computing the determinant of a real n×n matrix requires at least n/2 matrix-vector product queries. The construction also yields an Ω(n) phase-query lower bound for estimating the minimum eigenvalue of a real symmetric n×n matrix to additive accuracy Θ(n−2). These results extend the determinant and minimum-eigenvalue lower bounds of Childs, Hung, and Li~(ICALP 2021) from finite fields to the real-valued setting. Based on the same constructions, we also prove an Ω(min{κ,n}) gradient-query lower bound for constant-accuracy optimization of 1-strongly convex and κ-smooth functions, which is optimal up to a logarithmic factor.
Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the query complexity of volume estimation to a membership oracle has been a longstanding open question. Most of the previous work focuses on the high-dimensional limit. In this work, we tightly characterize the deterministic, randomized and quantum query complexity of this problem in the high-precision limit, i.e., when the dimension is constant.
Quantum counting is a fundamental quantum algorithm that estimates the fraction of marked elements using a membership oracle, achieving a quadratic speedup over classical sampling. The membership oracle, however, assumes exact labeling of each element, but this assumption fails when the labels are inherently probabilistic. We study quantum counting with \emph{Bernoulli oracles}, where given m Bernoulli distributions with unknown biases p1,…,pm and a gap parameter Δ, the goal is to estimate the fraction ρ of \emph{positive} distributions (pi≥1/2+Δ) to within additive error ε. We prove an upper bound of O~(Δερ+Δε1) queries, achieving a quadratic speedup over the classical sample complexity. % of Θ(ρ/Δ2ε2).
Our algorithm first uses the Quantum Singular Value Transformation (QSVT) to coherently amplify the bias gap without collapsing the superposition over distributions, and then applies two-stage adaptive amplitude estimation. We complement this upper bound with a near-matching lower bound of Ω(ρ/(Δε)) via a new composition theorem for the quantum adversary method in the Boolean-over-average-case direction. For the special case of a constant gap Δ=Θ(1), which corresponds to the bounded-error oracle where each query returns the correct label with constant probability, our bounds specialize to O~(ερ+ε1) and Ω(ερ), thereby characterizing the query complexity of quantum counting with bounded-error oracles.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most q oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model:
(i) (Unbiased) Phase estimation up to a precision ε requires Θ(qε21) queries;
(ii) (Unbiased) Amplitude estimation up to a precision ε requires Θ(qε2p(1−p)) queries;
(iii) Search among N items requires Θ(qN) queries;
(iv) Two level AND-OR tree (AND of m ORs, with n inputs to each OR) requires Θ(qnm) queries.
All of these bounds are optimal up to a constant factor, for all q from 1, corresponding to the classical complexity, to Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor.
Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.