Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sample Complexity for Embedded Multipartite Entanglement Witness via Pauli and Clifford Classical Shadows

Ziran Zhang

TL;DR

The paper addresses scalable entanglement certification in $N$-qubit systems by estimating a family of subsystem $n$-partite entanglement witnesses using classical shadows. It compares local Pauli and global Clifford measurement ensembles, deriving ensemble-dependent variance bounds that predict a crossover: the snapshot cost scales as $\mathcal{O}(4^n)$ for Pauli and $\mathcal{O}(2^{N-n})$ for Clifford. Numerical simulations on a perturbed GHZ witness confirm that Pauli shadows are more efficient for small, local witnesses, while Clifford shadows excel as the witness becomes more global, with a crossover near $n \approx N/2$. These results offer practical guidance for efficient multipartite entanglement certification in large quantum devices, leveraging the shadow-norm framework to relate measurement strategy to witness locality.

Abstract

Detecting multipartite entanglement in many qubit systems is measurement-intensive, motivating protocols that estimate only selected observables with provable efficiency. In this work we use the classical shadow protocol to study the sample complexity required to estimate a family of subsystem $n$-partite entanglement witness embedded in an larger $N$-qubit system. We derive ensemble dependent variance bounds that lead to qualitatively distinct scaling for the snapshots cost at fixed additive error $ε$ with numerical simulations confirm these trends, exhibiting a clear crossover from Pauli favorable performance for local witness to Clifford favorable performance as the witness becomes more global.

Sample Complexity for Embedded Multipartite Entanglement Witness via Pauli and Clifford Classical Shadows

TL;DR

The paper addresses scalable entanglement certification in -qubit systems by estimating a family of subsystem -partite entanglement witnesses using classical shadows. It compares local Pauli and global Clifford measurement ensembles, deriving ensemble-dependent variance bounds that predict a crossover: the snapshot cost scales as for Pauli and for Clifford. Numerical simulations on a perturbed GHZ witness confirm that Pauli shadows are more efficient for small, local witnesses, while Clifford shadows excel as the witness becomes more global, with a crossover near . These results offer practical guidance for efficient multipartite entanglement certification in large quantum devices, leveraging the shadow-norm framework to relate measurement strategy to witness locality.

Abstract

Detecting multipartite entanglement in many qubit systems is measurement-intensive, motivating protocols that estimate only selected observables with provable efficiency. In this work we use the classical shadow protocol to study the sample complexity required to estimate a family of subsystem -partite entanglement witness embedded in an larger -qubit system. We derive ensemble dependent variance bounds that lead to qualitatively distinct scaling for the snapshots cost at fixed additive error with numerical simulations confirm these trends, exhibiting a clear crossover from Pauli favorable performance for local witness to Clifford favorable performance as the witness becomes more global.
Paper Structure (20 sections, 58 equations, 5 figures, 1 algorithm)

This paper contains 20 sections, 58 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Separable state, product state, and GHZ state
  3. Classical Shadow protocol preview
  4. Classical Shadow
  5. Single shot estimation
  6. Sample complexity
  7. Shadow norms for specific ensembles
  8. Generating Entanglement Witness
  9. Perturbed GHZ entanglement witness
  10. Clifford&Pauli measurement Performance analysis
  11. Shadow norm bound & Sample complexity estimation
  12. Local Pauli measurement
  13. Global Clifford measurement
  14. Discussion
  15. Conclusion
  16. ...and 5 more sections

Figures (5)

  • Figure 1: Comparison of the ideal and reconstructed GHZ states.
  • Figure 2: Theoretical and numerical discrepancy at fixed sanpshots versus observable entanglement entropy
  • Figure 3: Estimation error $|\hat{w}-w_{\text{true}}|$ of Clifford and Pauli shadow at difference snapshot usage
  • Figure 4: Numerical simulation result of 6 qubits system
  • Figure 5: Numerical simulation result of 7 qubits system