# Benchmarks of Nonclassicality for Qubit Arrays

## TL;DR

This paper introduces practical, resource-efficient benchmarks for testing multi-qubit nonclassicality and entanglement in qubit arrays, using fewer measurement settings than full tomography, and verifies their effectiveness through numerical simulations.

## Contribution

It develops measurement-efficient benchmarks based on geometric structures from the $N$-qubit Pauli group, enabling verification of nonclassicality with minimal measurement settings.

## Key findings

- Benchmarks require at most N+1 measurement settings.
- Simulations confirm the bounds are tight and detect nonclassicality.
- Results demonstrate the benchmarks' applicability to realistic superconducting qubit arrays.

## Abstract

We present a set of practical benchmarks for $N$-qubit arrays that economically test the fidelity of achieving multi-qubit nonclassicality. The benchmarks are measurable correlators similar to 2-qubit Bell correlators, and are derived from a particular set of geometric structures from the $N$-qubit Pauli group. These structures prove the Greenberger-Horne-Zeilinger (GHZ) theorem, while the derived correlators witness genuine $N$-partite entanglement and establish a tight lower bound on the fidelity of particular stabilizer state preparations. The correlators need only $M \leq N+1$ distinct measurement settings, as opposed to the $2^{2N}-1$ settings that would normally be required to tomographically verify their associated stabilizer states. We optimize the measurements of these correlators for a physical array of qubits that can be nearest-neighbor-coupled using a circuit of controlled-$Z$ gates with constant gate depth to form $N$-qubit linear cluster states. We numerically simulate the provided circuits for a realistic scenario with $N=3,...,9$ qubits, using ranges of $T_1$ energy relaxation times, $T_2$ dephasing times, and controlled-$Z$ gate-fidelities consistent with Google's 9-qubit superconducting chip. The simulations verify the tightness of the fidelity bounds and witness nonclassicality for all nine qubits, while also showing ample room for improvement in chip performance.