Description
Develop an example/tutorial script to numerically demonstrate the typicality result (Proposition 6, Eq. 35) from arXiv:2502.16404. This experiment should show that for a random Hamiltonian ( U ) from the ensemble, the single-shot out-of-time-order correlator (OTOC) is exponentially likely to be close to its analytical average.
Tasks
-
Create a new documentation/tutorial docs/source/tutorials/typicality_experiment.rst
-
Write a script that:
- Samples unitaries ( U ) from the ensemble (using
get_commutator_graph() and available ensemble methods).
- Computes a single-shot evaluation ( f(U) ) (e.g., an averaged OTOC using the provided function).
- Compares ( f(U) ) with the analytical average ( E[f] ) (computed from e.g.
average_otoc function).
- Plots the distribution or deviation ( |f(U)-E[f]| ) and demonstrates that the probability of large deviations is exponentially suppressed.
-
Include clear comments and documentation that explain the physical meaning and link to Proposition 6.
-
A tutorial that includes figures/plots demonstrating the concentration of measure: ideally, the plot should indicate that deviations ( |f(U)-E[f]| ) are highly unlikely as stated in the inequality
$$
P_{U \sim \mu_G} \left( |f(U)-E[f]| \ge \epsilon \right) \le 2e^{-c\epsilon^2}
$$
- The tutorial should be comprehensive, allowing a researcher to reproduce the numerical validation step by step.
Additional Notes
- This will not be part of the core library but will serve as an example of how the analytical functions tie into the physical phenomenon of typicality.
- It is a great candidate for inclusion in the project’s documentation to attract users and provide guidance on the application of our tools.
Description
Develop an example/tutorial script to numerically demonstrate the typicality result (Proposition 6, Eq. 35) from arXiv:2502.16404. This experiment should show that for a random Hamiltonian ( U ) from the ensemble, the single-shot out-of-time-order correlator (OTOC) is exponentially likely to be close to its analytical average.
Tasks
Create a new documentation/tutorial
docs/source/tutorials/typicality_experiment.rstWrite a script that:
get_commutator_graph()and available ensemble methods).average_otocfunction).Include clear comments and documentation that explain the physical meaning and link to Proposition 6.
A tutorial that includes figures/plots demonstrating the concentration of measure: ideally, the plot should indicate that deviations ( |f(U)-E[f]| ) are highly unlikely as stated in the inequality
Additional Notes