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Quantum Advantage in Learning Mixed Unitary Channels

Yue Tu, Liang Jiang

TL;DR

The work targets learning mixed unitary quantum channels via a Fisher-information framework under an IOMS protocol. It derives FI expressions for both non-concatenating and concatenating strategies and proves a sample-complexity lower bound $N_1 \ge \Omega\left(\frac{r}{k d \varepsilon^2}\right)$, highlighting the role of ancilla and rank in learnability. The paper also shows that random Haar-distributed unitaries render mixed-unitary channels easy to learn with a PGM-based method, achieving $\mathcal{O}(1/\varepsilon^2)$ samples and concrete constants, which demonstrates the tightness of the bound in high dimensions. These results have practical implications for quantum error mitigation and establish a baseline for resource trade-offs between ancilla, concatenation, and measurement strategies. The analysis paves the way for extending Fisher-information based guarantees to broader channel classes and finite-sample regimes.

Abstract

We study the task of learning mixed unitary channels using Fisher information, under different quantum resource assumptions including ancilla and concatenation. Our result shows that the asymptotic sample complexity scales as $\frac{r}{d\varepsilon^2}$, where $r$ is the rank of the channel (i.e.\ the number of different unitaries), $d$ is the dimension of the system, and $\varepsilon^2$ is the mean-square error. Thus the critical resource is the ancilla, which mirrors the result in~\cite{chen2022quantum} but in a more precise form, as we point out that $r$ is also important. Additionally, we demonstrate the practical potential of mixed unitary channels by showing that random mixed unitary channels are easy to learn.

Quantum Advantage in Learning Mixed Unitary Channels

TL;DR

The work targets learning mixed unitary quantum channels via a Fisher-information framework under an IOMS protocol. It derives FI expressions for both non-concatenating and concatenating strategies and proves a sample-complexity lower bound , highlighting the role of ancilla and rank in learnability. The paper also shows that random Haar-distributed unitaries render mixed-unitary channels easy to learn with a PGM-based method, achieving samples and concrete constants, which demonstrates the tightness of the bound in high dimensions. These results have practical implications for quantum error mitigation and establish a baseline for resource trade-offs between ancilla, concatenation, and measurement strategies. The analysis paves the way for extending Fisher-information based guarantees to broader channel classes and finite-sample regimes.

Abstract

We study the task of learning mixed unitary channels using Fisher information, under different quantum resource assumptions including ancilla and concatenation. Our result shows that the asymptotic sample complexity scales as , where is the rank of the channel (i.e.\ the number of different unitaries), is the dimension of the system, and is the mean-square error. Thus the critical resource is the ancilla, which mirrors the result in~\cite{chen2022quantum} but in a more precise form, as we point out that is also important. Additionally, we demonstrate the practical potential of mixed unitary channels by showing that random mixed unitary channels are easy to learn.

Paper Structure

This paper contains 7 sections, 8 theorems, 57 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Main Results
  4. Definition of Fisher Information
  5. Sample Complexity Lower Bound
  6. Learning Random Mixed Unitary Channels
  7. Conclusions

Key Result

Lemma 1

For a non-concatenating IOMS protocol, the Fisher information matrix for a single measurement defined in Eq. eq:FI is given by where $K_{ij} = \operatorname{Tr}(E_i\rho_j)$, $\rho_j = U_j \rho U_j^\dagger$, $D(p)$ is a diagonal matrix with entries $D(p)_{ii} = p_i = \sum_{j=1}^r \operatorname{Tr}(E_i\rho_j)\theta_j$, and $P_s = I - u u^\top/d$ is the projector enforcing $\sum_i \theta_i = 1$.

Figures (1)

  • Figure 1: Schematic illustration of the independent one-measurement-per-state (IOMS) protocol.

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Definition 1: Effective mixed unitary channel of $k$-fold concatenation
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • Lemma 4: Sample complexity lower bound via Van Trees' inequality
  • ...and 7 more