  Estimating the state preparation fidelity of highly entangled states on noisy
intermediate-scale quantum (NISQ) devices is an important task for benchmarking
and application considerations. Unfortunately, exact fidelity measurements
quickly become prohibitively expensive, as they scale exponentially as $O(3^N)$
for $N$-qubit states, using full state tomography with measurements in all
Pauli bases combinations. However, Somma and others [PhysRevA.74.052302]
established that the complexity could be drastically reduced when looking at
fidelity lower bounds for states that exhibit symmetries, such as Dicke States
and GHZ States. For larger states, these bounds still need to be tight enough
to provide reasonable estimations on NISQ devices.
  For the first time and more than 15 years after the theoretical introduction,
we report meaningful lower bounds for the state preparation fidelity of all
Dicke States up to $N=10$, and all GHZ states up to $N=20$ on Quantinuum H1
ion-trap systems using efficient implementations of recently proposed scalable
circuits for these states. Our achieved lower bounds match or exceed previously
reported exact fidelities on superconducting systems for much smaller states.
This work provides a path forward to benchmarking entanglement as NISQ devices
improve in size and quality.
