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Inverse-free quantum state estimation with Heisenberg scaling

Kean Chen

TL;DR

This work investigates inverse-free quantum state and amplitude estimation under query access, showing that Heisenberg-scaling results can be achieved using only forward queries. The authors construct a probe state based on Gamma-shaped Young tableaux and extract the last-column information of the unknown unitary $U$ via a pretty good measurement, achieving an error bound that scales as $O(d^3/(n^2+nd^2))$ and translating to a forward-query complexity of $O\big(\min\{d^{3/2}/\varepsilon, d/\varepsilon^2\}\big)$ for estimating $U|d\rangle$ within $\varepsilon$. They further derive inverse-free bounds for amplitude estimation, namely $O\big(\min\{d^{3/2}/\varepsilon,1/\varepsilon^2\}\big)$, improving previous results and contradicting a conjecture that inverse access is necessary for optimal inverse-free scaling. The results illuminate how inverse access affects different quantum-estimation tasks and demonstrate that Heisenberg scaling can be achieved without incurring a quadratic-dimension overhead in the inverse-free setting.

Abstract

In this paper, we present an inverse-free pure quantum state estimation protocol that achieves Heisenberg scaling. Specifically, let $\mathcal{H}\cong \mathbb{C}^d$ be a $d$-dimensional Hilbert space with an orthonormal basis $\{|1\rangle,\ldots,|d\rangle\}$ and $U$ be an unknown unitary on $\mathcal{H}$. Our protocol estimates $U|d\rangle$ to within trace distance error $\varepsilon$ using $O(\min\{d^{3/2}/\varepsilon,d/\varepsilon^2\})$ forward queries to $U$. This complements the previous result $O(d\log(d)/\varepsilon)$ by van Apeldoorn, Cornelissen, Gilyén, and Nannicini (SODA 2023), which requires both forward and inverse queries. Moreover, our result implies a query upper bound $O(\min\{d^{3/2}/\varepsilon,1/\varepsilon^2\})$ for inverse-free amplitude estimation, improving the previous best upper bound $O(\min\{d^{2}/\varepsilon,1/\varepsilon^2\})$ based on optimal unitary estimation by Haah, Kothari, O'Donnell, and Tang (FOCS 2023), and disproving a conjecture posed in Tang and Wright (2025).

Inverse-free quantum state estimation with Heisenberg scaling

TL;DR

This work investigates inverse-free quantum state and amplitude estimation under query access, showing that Heisenberg-scaling results can be achieved using only forward queries. The authors construct a probe state based on Gamma-shaped Young tableaux and extract the last-column information of the unknown unitary via a pretty good measurement, achieving an error bound that scales as and translating to a forward-query complexity of for estimating within . They further derive inverse-free bounds for amplitude estimation, namely , improving previous results and contradicting a conjecture that inverse access is necessary for optimal inverse-free scaling. The results illuminate how inverse access affects different quantum-estimation tasks and demonstrate that Heisenberg scaling can be achieved without incurring a quadratic-dimension overhead in the inverse-free setting.

Abstract

In this paper, we present an inverse-free pure quantum state estimation protocol that achieves Heisenberg scaling. Specifically, let be a -dimensional Hilbert space with an orthonormal basis and be an unknown unitary on . Our protocol estimates to within trace distance error using forward queries to . This complements the previous result by van Apeldoorn, Cornelissen, Gilyén, and Nannicini (SODA 2023), which requires both forward and inverse queries. Moreover, our result implies a query upper bound for inverse-free amplitude estimation, improving the previous best upper bound based on optimal unitary estimation by Haah, Kothari, O'Donnell, and Tang (FOCS 2023), and disproving a conjecture posed in Tang and Wright (2025).

Paper Structure

This paper contains 23 sections, 6 theorems, 63 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. State estimation with Heisenberg scaling.
  5. Power of inverse queries.
  6. Overview of techniques
  7. Discussion
  8. Preliminaries
  9. Young diagrams and Young tableaux
  10. Gelfand-Tsetlin basis
  11. Branching rule of $\mathbb{U}_d$.
  12. Clebsch-Gordan coefficients
  13. Estimation protocol
  14. A coherent system on Gamma-shaped tableaux
  15. Probe state
  16. ...and 8 more sections

Key Result

Theorem 1.2

There is a quantum protocol that uses $n$ forward queries to $U$ and outputs a classical description of a unit vector $\lvert\psi\rangle$. The output $\lvert\psi\rangle$ is random and satisfies

Figures (2)

  • Figure 1: Definition of the semistandard Young tableaux $\Gamma_i$ and $\Gamma_i^+$.
  • Figure 2: The decomposition of $\lvert\Gamma_i\rangle\otimes \lvert d\rangle$

Theorems & Definitions (17)

  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.5
  • proof
  • Definition 3.1: $\Gamma$-shaped tableaux
  • Definition 3.2: The probe state
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 7 more