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Efficient graph-diagonal characterization of noisy states distributed over quantum networks via Bell sampling

Zherui Jerry Wang, Joshua Carlo A. Casapao, Naphan Benchasattabuse, Ananda G. Maity, Jordi Tura, Akihito Soeda, Michal Hajdušek, Rodney Van Meter, David Elkouss

TL;DR

This work introduces Bell Sampling for Quantum Networks (BSQN), a protocol that uses two-copy Bell measurements to efficiently estimate all diagonal elements in the graph-state basis of noisy graph states distributed across a network. The key result is a sample complexity that scales linearly with the number of qubits, $\mathcal{O}\big((n+\log(1/\delta))/\varepsilon^4\big)$, for recovering the entire graph-diagonal vector, representing a dramatic reduction from the $\mathcal{O}(2^n)$ scaling of direct estimation. Moreover, fidelity (a global property) can be estimated with a sample complexity that is independent of network size, $\mathcal{O}\big(\log(1/\varepsilon^2\delta)/\varepsilon^4\big)$, making the method robust to system growth. Numerical results corroborate the theory, showing practical efficiency gains over direct diagonal estimation and revealing favorable scaling in realistic noise scenarios. BSQN thereby provides a scalable, implementable tool for error detection and verification of noisy graph states in large quantum networks.

Abstract

Graph states are an important class of entangled states that serve as a key resource for distributed information processing and communication in quantum networks. In this work, we propose a protocol that utilizes a Bell sampling subroutine to characterize the diagonal elements in the graph basis of noisy graph states distributed across a network. Our approach offers significant advantages over direct diagonal estimation using unentangled single-qubit measurements in terms of scalability. Specifically, we prove that estimating the full vector of diagonal elements requires a sample complexity that scales linearly with the number of qubits ($\mathcal{O}(n)$), providing an exponential reduction in resource overhead compared to the best known $\mathcal{O}(2^n)$ scaling of direct estimation. Furthermore, we demonstrate that global properties, such as state fidelity, can be estimated with a sample complexity independent of the network size. Finally, we present numerical results indicating that the estimation in practice is more efficient than the derived theoretical bounds. Our work thus establishes a promising technique for efficiently estimating noisy graph states in large networks under realistic experimental conditions.

Efficient graph-diagonal characterization of noisy states distributed over quantum networks via Bell sampling

TL;DR

This work introduces Bell Sampling for Quantum Networks (BSQN), a protocol that uses two-copy Bell measurements to efficiently estimate all diagonal elements in the graph-state basis of noisy graph states distributed across a network. The key result is a sample complexity that scales linearly with the number of qubits, , for recovering the entire graph-diagonal vector, representing a dramatic reduction from the scaling of direct estimation. Moreover, fidelity (a global property) can be estimated with a sample complexity that is independent of network size, , making the method robust to system growth. Numerical results corroborate the theory, showing practical efficiency gains over direct diagonal estimation and revealing favorable scaling in realistic noise scenarios. BSQN thereby provides a scalable, implementable tool for error detection and verification of noisy graph states in large quantum networks.

Abstract

Graph states are an important class of entangled states that serve as a key resource for distributed information processing and communication in quantum networks. In this work, we propose a protocol that utilizes a Bell sampling subroutine to characterize the diagonal elements in the graph basis of noisy graph states distributed across a network. Our approach offers significant advantages over direct diagonal estimation using unentangled single-qubit measurements in terms of scalability. Specifically, we prove that estimating the full vector of diagonal elements requires a sample complexity that scales linearly with the number of qubits (), providing an exponential reduction in resource overhead compared to the best known scaling of direct estimation. Furthermore, we demonstrate that global properties, such as state fidelity, can be estimated with a sample complexity independent of the network size. Finally, we present numerical results indicating that the estimation in practice is more efficient than the derived theoretical bounds. Our work thus establishes a promising technique for efficiently estimating noisy graph states in large networks under realistic experimental conditions.

Paper Structure

This paper contains 28 sections, 8 theorems, 60 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph states
  4. Bell sampling
  5. Estimating graph states over a network
  6. Predicting the graph-diagonal elements
  7. Sample complexity
  8. Measurements and classical post-processing
  9. Estimation with few stabilizer elements
  10. Pauli error detection
  11. Numerical Results
  12. Simulation setup
  13. Errors of estimation
  14. Scalability with the number of qubits
  15. Performance of estimation with random sampling
  16. ...and 13 more sections

Key Result

Lemma 1

Elements of $\mathbf{a}$ can be expressed as Here, the function $f_b\colon \mathcal{S}_G^+ \to \mathbb{F}_2$ is defined as where $U_b \equiv \bigotimes_{i=1}^{n} Z^{b_i}$ and $b \equiv \overline{b_1 \cdots b_{1<i<n} \cdots b_{n}}\in \mathbb{F}_2^{n}$ is an $n$-bit string.

Figures (7)

  • Figure 1: Bell Sampling for Quantum Networks (BSQN). Two copies of the noisy graph state are distributed across the network nodes. The initial stage of BSQN is a Bell sampling subroutine: each node performs a qubit-wise CNOT operation, followed by applying a Hadamard gate to the control qubit and measuring in the computational basis. The measurement statistics are then collected and post-processed to estimate the diagonal elements of the distributed noisy graph state.
  • Figure 2: Effects of general Pauli channels acting on graph states can be captured by dephasing errors only. For example, the effect of a single $Z_3$ error acting on a linear graph state is equivalent to eight distinct Pauli errors.
  • Figure 3: Performance comparison between BSQN and DGE for an $8$-node complete graph state. The top row (a, b) shows the $2$-norm error $\Vert \Delta \mathbf{a} \Vert_2$ and fidelity error $\abs{\Delta F}$ as a function of the number of copies $N_s$, for a fixed fidelity of $F=0.9$. The bottom row (c, d) shows the same errors as a function of fidelity $F$, for a fixed $N_s=10^4$. The markers show the mean while the envelopes (not visible in (c)) show the standard deviation.
  • Figure 4: Scalability comparison of BSQN and DGE with fixed number of copies of the graph state, $N_s = 2 \times 10^4$. The plot shows the $2$-norm estimation error $\Vert \Delta \mathbf{a} \Vert_2$ as a function of the number of qubits $n$. The states $\rho$ are noisy $n$-node complete graph states with fidelity $F = 0.9$ under depolarizing noise $\mathcal{N}_1$.
  • Figure 5: Scalability of the random sampling fidelity estimation protocol (Theorem \ref{['thm:fidelity_random']}). The plots show the mean estimated fidelity (solid line) and standard deviation (envelope) as a function of the number of qubits $n$ with the number of sampled stabilizer elements $M = 2n$ and fixed fidelity $F = 0.53$, under different noise models.
  • ...and 2 more figures

Theorems & Definitions (15)

  • Definition 1: Graph states
  • Definition 2: Graph-state basis
  • Definition 3: Bell sampling protocol
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma \ref{lem:exactrep}
  • proof
  • Theorem \ref{thm:signs}
  • ...and 5 more