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Criteria for unbiased estimation: applications to noise-agnostic sensing and learnability of quantum channel

Hyukgun Kwon, Kento Tsubouchi, Chia-Tung Chu, Liang Jiang

TL;DR

By interpreting unbiased estimation as learnability, the result extends to the fundamental learnability of parameters in general quantum channels, and investigates the learnability of noise affecting non-Clifford gates via cycle benchmarking.

Abstract

We establish the necessary and sufficient conditions for unbiased estimation in multi-parameter estimation tasks. More specifically, we first consider quantum state estimation, where multiple parameters are encoded in a quantum state, and derive simple and intuitive necessary and sufficient conditions for an unbiased estimation based on the derivatives of the encoded state. To demonstrate the utility of our framework, we consider phase estimation under unknown Pauli noise. We show that while unbiased phase estimation is infeasible with a naive scheme, employing an entangled probe with a noiseless ancilla enables unbiased estimation. Next, we extend our analysis to quantum channel estimation (equivalently, quantum channel learning), where the goal is to estimate parameters characterizing an unknown quantum channel. We establish the necessary and sufficient condition for unbiased estimation of these parameters. Notably, by interpreting unbiased estimation as learnability, our result extends to the fundamental learnability of parameters in general quantum channels. As a concrete application, we investigate the learnability of noise affecting non-Clifford gates via cycle benchmarking.

Criteria for unbiased estimation: applications to noise-agnostic sensing and learnability of quantum channel

TL;DR

By interpreting unbiased estimation as learnability, the result extends to the fundamental learnability of parameters in general quantum channels, and investigates the learnability of noise affecting non-Clifford gates via cycle benchmarking.

Abstract

We establish the necessary and sufficient conditions for unbiased estimation in multi-parameter estimation tasks. More specifically, we first consider quantum state estimation, where multiple parameters are encoded in a quantum state, and derive simple and intuitive necessary and sufficient conditions for an unbiased estimation based on the derivatives of the encoded state. To demonstrate the utility of our framework, we consider phase estimation under unknown Pauli noise. We show that while unbiased phase estimation is infeasible with a naive scheme, employing an entangled probe with a noiseless ancilla enables unbiased estimation. Next, we extend our analysis to quantum channel estimation (equivalently, quantum channel learning), where the goal is to estimate parameters characterizing an unknown quantum channel. We establish the necessary and sufficient condition for unbiased estimation of these parameters. Notably, by interpreting unbiased estimation as learnability, our result extends to the fundamental learnability of parameters in general quantum channels. As a concrete application, we investigate the learnability of noise affecting non-Clifford gates via cycle benchmarking.

Paper Structure

This paper contains 30 sections, 10 theorems, 164 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Quantum state estimation and noise agnostic sensing
  3. Unbiased quantum state estimation
  4. Application to noise-agnostic sensing
  5. Unbiased quantum channel estimation
  6. Application to learnability
  7. Conclusion
  8. Methods
  9. Proof of Theorem \ref{['theorem1']}
  10. proof of if direction
  11. proof of only if direction
  12. Proof of Corollary \ref{['corollary1']}
  13. Details of Noise Learnability in Cycle Benchmarking for the Pauli-Z Rotation Gate
  14. Acknowledgments
  15. Fully Generalized multi-parameter Cramér-Rao matrix inequality
  16. ...and 15 more sections

Key Result

Lemma 1

There exists an unbiased estimator of $\phi=\vb*{w}^{\mathrm{T}}\vb*{\theta}$ with finite estimation error if and only if $\vb*{w} \in \mathrm{supp}\left(\mathbf{J}\right)$. In other words When Eq. NScondit1 is satisfied, the achievable lower bound of the estimation error of $\phi=\vb*{w}^{\mathrm{T}}\vb*{\theta}$ is given by generalized QCRB: Here, $\mathbf{J}^{+}$ is the Moore-Penrose pseudo i

Figures (2)

  • Figure 1: (a) Schematic of the naive estimation. (b) Schematic of the estimation that exploits an entangled state with a noiseless ancilla.
  • Figure 2: Schematic of cycle benchmarking. We aim to estimate the noise $\mathcal{N}$ affecting the $R_Z(\phi)$ gate by repeatedly applying the noisy $R_Z(\phi)$ gate. Here, we consider the case where state preparation of the initial state $\rho_0$ and measurement are affected by unknown Pauli noise $\mathcal{N}_S$ and $\mathcal{N}_M$.

Theorems & Definitions (21)

  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof
  • proof
  • Corollary 2
  • proof
  • proof
  • proof
  • ...and 11 more