Deriving formulations to estimate worst-case size bounds for conjunctive queries under various constraints has been at the core of theoretical database research. If the problem has no constraints or has a single functional dependency, tight worst-case size bounds are computable. If the problem has more than one functional dependency, computing tight bounds can be difficult in practice and may even require an infinite number of linear inequalities in its optimization formulation. While these challenges have been addressed with varying methods, no prior research has employed quantum information theory to address this problem. In this work, we establish a connection between earlier classical information theory-based works and quantum information theory. We propose replacing the classical Shannon entropy formulation with the quantum Rényi entropy of order $α\in (0,1)$ whose entropy cone is characterized simply by non-negativity. The first key result is to express the bound in terms of optimizing over quantum states and Rényi entropy. Optimizing with respect to quantum states rather than classical distributions transfers the hardness into the problem of characterizing classical states, yielding a sound but generally not tight upper bound. We further quantify this hardness explicitly by proposing a dichotomy theorem: if the query satisfies a head absorption rule, then the Rényi program value is at most one and $|Q(D)| \leq \mathrm{rmax}(D)$. Otherwise, the program is unbounded.
Deriving formulations to estimate worst-case size bounds for conjunctive queries under various constraints has been at the core of theoretical database research. If the problem has no constraints or has a single functional dependency, tight worst-case size bounds are computable. If the problem has more than one functional dependency, computing tight bounds can be difficult in practice and may even require an infinite number of linear inequalities in its optimization formulation. While these challenges have been addressed with varying methods, no prior research has employed quantum information theory to address this problem. In this work, we establish a connection between earlier classical information theory-based works and quantum information theory. We propose replacing the classical Shannon entropy formulation with the quantum Rényi entropy of order α∈(0,1) whose entropy cone is characterized simply by non-negativity. The first key result is to express the bound in terms of optimizing over quantum states and Rényi entropy. Optimizing with respect to quantum states rather than classical distributions transfers the hardness into the problem of characterizing classical states, yielding a sound but generally not tight upper bound. We further quantify this hardness explicitly by proposing a dichotomy theorem: if the query satisfies a head absorption rule, then the Rényi program value is at most one and ∣Q(D)∣≤rmax(D). Otherwise, the program is unbounded.
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.
Estimation of Shannon and Rényi entropies of unknown discrete distributions is a fundamental problem in statistical property testing. In this paper, we give the first quantum algorithms for estimating α-Rényi entropies (Shannon entropy being 1-Rényi entropy). In particular, we demonstrate a quadratic quantum speedup for Shannon entropy estimation and a generic quantum speedup for α-Rényi entropy estimation for all α ≥ 0 values, including tight bounds for the Shannon entropy, the Hartley entropy (α = 0), and the collision entropy (α = 2). We also provide quantum upper bounds for estimating min-entropy (α = +∞) as well as the Kullback-Leibler divergence. We complement our results with quantum lower bounds on α-Rényi entropy estimation for all α ≥ 0 values. Our approach is inspired by the pioneering work of Bravyi, Harrow, and Hassidim (BHH); however, with many new technical ingredients: 1) we improve the error dependence of the BHH framework by a fine-tuned error analysis together with Montanaro's approach to estimating the expected output of quantum subroutines for α = 0, 1; 2) we develop a procedure, similar to cooling schedules in simulated annealing, for general α ≥ 0, and 3) in the cases of integer α ≥ 2 and α = +∞, we reduce the entropy estimation problem to the α-distinctness and [log n]-distinctness problems, respectively.
We investigate the computational hardness of estimating the quantum α-Rényi entropy SαR(ρ)=1−αlnTr(ρα) and the quantum q-Tsallis entropy SqT(ρ)=q−11−Tr(ρq), both of which converge to the von Neumann entropy as the order approaches 1. The promise problems Quantum α-Rényi Entropy Approximation (RényiQEAα) and Quantum q-Tsallis Entropy Approximation (TsallisQEAq) ask whether $ {\rm S}^ {\tt R}_α(ρ)$ or SqT(ρ), is at least τ1 or at most τ2, where τ1−τ2 is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order 1) and some cases of the quantum q-Tsallis entropy, while existing approaches do not readily extend to other orders.
We establish that for all positive real α and q, and also for α=∞, the rank-2 variants Rank2RényiQEAα and Rank2TsallisQEAq are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply:
- For all real orders α>0 or α=∞, and for all real orders 0<q≤1, LowRankRényiQEAα and LowRankTsallisQEAq are BQP-complete, where both are restricted versions of RényiQEAα and TsallisQEAq with ρ of polynomial rank.
- For all real order q>1, TsallisQEAq is BQP-complete.
Our hardness results stem from reductions based on new inequalities relating the α-Rényi or q-Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.
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.
Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting Q(f∘g)=Θ(Q(f)Q(g)) for all Boolean functions f and g; a direct sum theorem asserting that computing k copies of a function (or search problem) costs Ω(k) times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires Ω(k) times the cost of computing one copy to bounded error.
However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution μ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing k copies of f when the input comes from the product distribution μk instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions.
In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the γ2 norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.
Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime.
Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) single-letter expressions for the exact error exponents of quantum state merging in the low-entanglement-cost and high-entanglement-distillation-rate regimes, in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for achievable error exponents of entanglement distillation and quantum channel coding in terms of Petz-Rényi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes.