paulie.application.otoc.mean_abs_otoc_uniform

paulie.application.otoc.mean_abs_otoc_uniform(u, *, method='exact', num_samples=10000, seed=None, check_unitary=True, rtol=1e-08, atol=1e-08)

Uniform expectation of \(\lvert\mathrm{OTOC}(U,P_1,P_2)\rvert\) over independent \(P_1,P_2 \in \{I,X,Y,Z\}^{\otimes n}\) (each uniform on \(4^n\) strings).

\[\mathbb{E}_{P_1,P_2}\left[\left\lvert\mathrm{OTOC}(U, P_1, P_2)\right\rvert\right],\]

using the same OTOC convention as otoc_fixed_unitary().

Parameters:

• u – Unitary of shape (2^n, 2^n).

• method – "exact" — sum over all \(16^n\) pairs, allowed only up to a fixed small \(n\) (see error message); "monte_carlo" — sample \(P_1,P_2\) i.i.d. uniform (any \(n\) supported by dense arithmetic).

• num_samples – Number of i.i.d. pairs for method="monte_carlo".

• seed – For method="monte_carlo", if not None, calls random.seed() before sampling so get_random (stdlib random) is reproducible across calls with the same seed.

• check_unitary – Forwarded to otoc_fixed_unitary().

• rtol – Forwarded to otoc_fixed_unitary().

• atol – Forwarded to otoc_fixed_unitary().

Returns:

Estimated or exact expectation in \([0,1]\) (typically).

Raises:

ValueError – Invalid u or method, or method="exact" with \(n\) above the implemented cap.

Note

This expectation is the quantity inside the logarithm in Pauli instability (Definition~1, arXiv:2408.01663): \(\mathbb{I}(U)=-\log(\cdots)\) uses \(\mathbb{E}|\mathrm{OTOC}|\) exactly in the "exact" branch and estimates it by an empirical mean in "monte_carlo". Theorem~1 there controls how many samples are needed so that pauli_instability() with method="monte_carlo" is close to \(\mathbb{I}(U)\); Corollary~1 contrasts efficient vs intractable scaling of \(\mathbb{I}(U)\) with system size (\(\Theta(\log n)\) vs \(\Theta(n)\)).