A Quantum Algorithm Framework for Discrete Probability Distributions with Applications to Rényi Entropy Estimation

TL;DR

The paper presents a unified quantum algorithm framework for estimating properties of discrete probability distributions, notably the $\alpha$-Rényi entropy, by integrating quantum singular value transformation, annealing, and variable-time amplitude estimation. It encodes a distribution via a block-encoded operator whose singular values are $\sqrt{p_i}$ and uses polynomial transforms to encode target statistics into amplitudes, followed by amplitude estimation to extract estimates. The authors derive refined quantum query complexities that improve over prior results for both $α>1$ and $0<α<1$, and extend the framework to density matrices, low-rank settings, and quantum Rényi divergences. The approach supports multiple access models, including pure-state preparation and purified-query-access, and demonstrates broad applicability to quantum entropy estimation and related information-theoretic quantities. Overall, the framework advances the understanding of parameter- and instance-dependent quantum speedups for statistical property estimation in both classical and quantum settings.

Abstract

Estimating statistical properties is fundamental in statistics and computer science. In this paper, we propose a unified quantum algorithm framework for estimating properties of discrete probability distributions, with estimating Rényi entropies as specific examples. In particular, given a quantum oracle that prepares an $n$-dimensional quantum state $\sum_{i=1}^{n}\sqrt{p_{i}}|i\rangle$, for $α>1$ and $0<α<1$, our algorithm framework estimates $α$-Rényi entropy $H_α(p)$ to within additive error $ε$ with probability at least $2/3$ using $\widetilde{\mathcal{O}}(n^{1-\frac{1}{2α}}/ε+ \sqrt{n}/ε^{1+\frac{1}{2α}})$ and $\widetilde{\mathcal{O}}(n^{\frac{1}{2α}}/ε^{1+\frac{1}{2α}})$ queries, respectively. This improves the best known dependence in $ε$ as well as the joint dependence between $n$ and $1/ε$. Technically, our quantum algorithms combine quantum singular value transformation, quantum annealing, and variable-time amplitude estimation. We believe that our algorithm framework is of general interest and has wide applications.

Figures (2)

• Figure 1: Comparison between our algorithms and the algorithm in Li and Wu li2019entropy.
• Figure 2: Circuit of $\text{C}_{\widetilde{\Pi }\otimes|+\rangle\langle+|}\text{NOT} = \widetilde{\Pi }\otimes |+\rangle\langle+|_Q\otimes X_F + (I-\widetilde{\Pi }\otimes |+\rangle\langle+|_Q)\otimes I_F$ if $n$ is a power of 2. $n_A,n_B,n_C,n_F$ are the sizes of the registers $A,B,C,F$, respectively. The CNOT gate between two registers with the same size is an abbreviation of a sequence of CNOT gates between qubits in different registers with the same index and the CNOT gate targeting a qubit conditional on a register will flip the qubit when the regisiter is an all-0/all-1 state.

Theorems & Definitions (61)

• Theorem 1: Main theorem
• Corollary 1
• Definition 1: Pure-state preparation access to classical distribution
• Definition 2: Discrete quantum query-access to classical distribution
• Definition 3: Purified quantum query-access
• Definition 4: Quantum sampling
• Theorem 2: Fixed-point amplitude amplification gilyen2019singular
• Theorem 3: Amplitude estimation brassard2002quantum
• Definition 5: Singular value transformation gilyen2019singular
• Theorem 4: gilyen2019singular