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Classical shadows for non-iid quantum sources

Leonardo Zambrano

Abstract

Classical shadow tomography has emerged as a powerful framework for predicting properties of quantum many-body systems with favorable sample complexity. Standard theoretical guarantees, however, rely on the assumption that experimental rounds are independent and identically distributed (i.i.d.). This idealization is often violated in practice, where parameter drift, environmental noise, and active feedback generate history-dependent sequences of states or channels. To address this, we introduce a robust classical shadow protocol based on a truncated mean estimator. We prove that its sample complexity for predicting properties of the time-averaged state or channel matches the standard i.i.d. scaling governed by the shadow norm, even when experimental rounds depend arbitrarily on the past. Our results establish the robustness of the shadow formalism beyond the i.i.d. regime.

Classical shadows for non-iid quantum sources

Abstract

Classical shadow tomography has emerged as a powerful framework for predicting properties of quantum many-body systems with favorable sample complexity. Standard theoretical guarantees, however, rely on the assumption that experimental rounds are independent and identically distributed (i.i.d.). This idealization is often violated in practice, where parameter drift, environmental noise, and active feedback generate history-dependent sequences of states or channels. To address this, we introduce a robust classical shadow protocol based on a truncated mean estimator. We prove that its sample complexity for predicting properties of the time-averaged state or channel matches the standard i.i.d. scaling governed by the shadow norm, even when experimental rounds depend arbitrarily on the past. Our results establish the robustness of the shadow formalism beyond the i.i.d. regime.
Paper Structure (14 sections, 5 theorems, 63 equations)

This paper contains 14 sections, 5 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical shadows estimator
  4. Martingales and filtrations
  5. Spatial unraveling and virtual sequences
  6. Results
  7. Data acquisition
  8. Observable estimation
  9. Main theorem
  10. Proof of the main theorem
  11. Extension to Quantum Processes
  12. Discussion and conclusions
  13. Proof of corollary: global Clifford shadows
  14. Proof of corollary: Pauli shadows

Key Result

Theorem 1

Let $\{Z_t\}_{t=1}^N$ be a martingale difference sequence with respect to a filtration $\{\mathcal{F}_t\}$, satisfying $|Z_t| \le R$ almost surely. Define the predictable quadratic variation Then, for any $x > 0$ and $v > 0$,

Theorems & Definitions (8)

  • Theorem 1: Freedman’s Inequality freedman1975tail
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • proof
  • Theorem 3
  • proof
  • proof