Dear Editor,
We thank you for arranging reviews for our submitted manuscript, which contained helpful comments. Here’s our response to both referee’s questions / suggestions:
Referee 1
Referee 2
In Fig. 4 the authors present results for 10 independent, repeated runs: what are the differences between these runs? Only the initial conditions? It would be good to make this explicit in the manuscript.
These VQA instances are only distinguishable by initial values of Pauli rotation parameters. As suggested, we have now explicitly stated it in Page 3 as follows: “From the collection of 10 independent, repeated runs for each architecture and depth, the overall trend stands out.” → “We repeated 10 independent VQA runs with randomly initialized parameters for each chosen circuit architecture and depth. The following overall trend stands out in the collection of results: ”
When quantifying operator spreading, the authors consider “random circuit instances”: are these circuits variationally optimized, as in Fig. 4, or are all one-site unitary gates randomly chosen within the given circuit architecture? If the latter, are these realizations representative of the optimized circuits? The same questions apply to the spectral diagnostics based on the modular Hamiltonian.
Plus, are the results for the studied circuits different from expected results on the random brickwork unitary circuits where operator spreading and entanglement growth have already been extensively studied?
The random circuits used for our numerical studies of operator spreading & modular Hamiltonian spectra are not variationally optimized circuits. We don’t expect them to be representative of the optimized circuits.
As correctly mentioned, these studies are analogous to the study of random brickwork unitary circuits (e.g., [27]). The qualitative trend in both operator spreading and entanglement growth must be therefore simular. The main difference lies on that, our random 2-qubit “bricks” are not sampled from Haar unitary ensemble (or approximate t-designs) but rather very low-complexity unitary operators. We’ve observed that a concrete form of the low-complexity random bricks do affect the saturation value of bipartite entanglement entropy, such that it maintains a finite gap from the theoretical maximum that follows the volume law, and/or the type of the emergent random matrix ensemble.
We hope the above responses and accompanying manuscript changes can clarify the raised issues. Please let us know if there’s any questions that should be further addressed.
Best, Yaron Oz