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Near optimal quantum algorithm for estimating Shannon entropy

Myeongjin Shin, Kabgyun Jeong

TL;DR

The paper resolves the quantum query complexity of estimating Shannon entropy $H(p)$ in the quantum probability oracle model. By integrating quantum singular value separation with quantum amplitude amplification and quantum singular value transformation, it achieves a near-optimal upper bound of $\tilde{O}\left(\frac{\sqrt{n}}{\epsilon}\right)$ queries, matching a yet-proven lower bound up to logarithmic factors. The lower bound is established via constructions encoding probability mass with Hamming weights, showing no quantum algorithm can do asymptotically better. The work thus demonstrates Heisenberg-limited scaling for entropy estimation and opens avenues for extending these techniques to other entropy measures and information-theoretic quantities.

Abstract

We present a near-optimal quantum algorithm, up to logarithmic factors, for estimating the Shannon entropy in the quantum probability oracle model. Our approach combines the singular value separation algorithm with quantum amplitude amplification, followed by the application of quantum singular value transformation. On the lower bound side, we construct probability distributions encoded via Hamming weights in the oracle, establishing a tight query lower bound up to logarithmic factors. Consequently, our results show that the tight query complexity for estimating the Shannon entropy within $ε$-additive error is given by $\tildeΘ\left(\tfrac{\sqrt{n}}ε\right)$.

Near optimal quantum algorithm for estimating Shannon entropy

TL;DR

The paper resolves the quantum query complexity of estimating Shannon entropy in the quantum probability oracle model. By integrating quantum singular value separation with quantum amplitude amplification and quantum singular value transformation, it achieves a near-optimal upper bound of queries, matching a yet-proven lower bound up to logarithmic factors. The lower bound is established via constructions encoding probability mass with Hamming weights, showing no quantum algorithm can do asymptotically better. The work thus demonstrates Heisenberg-limited scaling for entropy estimation and opens avenues for extending these techniques to other entropy measures and information-theoretic quantities.

Abstract

We present a near-optimal quantum algorithm, up to logarithmic factors, for estimating the Shannon entropy in the quantum probability oracle model. Our approach combines the singular value separation algorithm with quantum amplitude amplification, followed by the application of quantum singular value transformation. On the lower bound side, we construct probability distributions encoded via Hamming weights in the oracle, establishing a tight query lower bound up to logarithmic factors. Consequently, our results show that the tight query complexity for estimating the Shannon entropy within -additive error is given by .

Paper Structure

This paper contains 17 sections, 17 theorems, 83 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum amplitude amplification and estimation
  4. Quantum singular value transformation (QSVT)
  5. Upper bound
  6. Quantum singular value separation algorithm
  7. Main algorithm
  8. Lower bound
  9. Discussion
  10. Proof for Theorems and Lemmas
  11. Proof of Lemma \ref{['lem:W_operator_formal']}
  12. Proof of Theorem \ref{['thm:sv-separation']}
  13. Proof of Lemma \ref{['lem: mag']}
  14. Proof of Lemma \ref{['lem: Uke']}
  15. Proof of Theorem \ref{['thm: v']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $p$ be a $n$-dimensional probability distribution. Given a quantum probability oracle $O_p$ for $p$, estimating the Shannon entropy $H(p)$ within $\epsilon$-additive error involves $\tilde{\Theta}(\frac{\sqrt{n}}{\epsilon})$ queries of $O_p$ and $O_p^\dagger$.

Figures (2)

  • Figure 1: Circuit architecture for the iterative QSVS algorithm. The abstract $\text{QSVS}_k$ operation is decomposed into a sequence of $k$ cycles. Each cycle consists of a state copying gate $T_j$ followed by a singular value separation gate $W_j$, both conditioned on the control register $C$. Horizontal wires represent the coherent evolution across the control ($C$), phase ($P$), index ($I$), and system ($A, B$) registers.
  • Figure 2: Schematic workflow of Algorithm \ref{['alg:Shannon']} for Shannon entropy estimation. The upper path computes $v_k'$ by sequentially applying QSVS (Theorem \ref{['thm:sv-separation']}), QAA (Lemma \ref{['lem: qaa']}), QSVT (Lemma \ref{['lem: QSVT-gilyen']}), and QAE (Lemma \ref{['lem: qae']}) to the input state $|\psi\rangle_{AB}$. The lower path estimates $\text{Sum}(k)$ through QSVS and QAE. The product of these two outputs yields $v_k$, which is finally used to approximate the Shannon entropy as $H(p) \approx -2 + \sum_{k=1}^m 8v_k$.

Theorems & Definitions (29)

  • Definition 1: Quantum probability oracle
  • Theorem 1: Main Theorem
  • Lemma 1: Quantum amplitude amplification
  • Lemma 2: Quantum amplitude estimation
  • Definition 2: Singular value transformation
  • Lemma 3: Ref gilyen2019quantum, Corollary 18
  • Lemma 4: Ref wang2024new, Lemma 3.3
  • Lemma 5
  • Lemma 6: Ref wang2024quantum, Lemma 5
  • Lemma 7
  • ...and 19 more