paulie.application.otoc.pauli_instability#
- paulie.application.otoc.pauli_instability(u, *, method='exact', num_samples=10000, seed=None, base=None, check_unitary=True, rtol=1e-08, atol=1e-08)#
Pauli instability (Definition 1):
\[I(U) = -\log \mathbb{E}_{P_1,P_2}\left[ \left\lvert\mathrm{OTOC}(U, P_1, P_2)\right\rvert\right],\]expectation over independent uniform \(P_1,P_2\) on \(\{I,X,Y,Z\}^{\otimes n}\).
- Parameters:
u – Unitary of shape
(2^n, 2^n).method – Passed to
mean_abs_otoc_uniform().num_samples – Passed to
mean_abs_otoc_uniform()whenmethod="monte_carlo".seed – Passed to
mean_abs_otoc_uniform()whenmethod="monte_carlo"(see that function for when it affects randomness).base – Logarithm base;
Nonemeans natural log (numpy.log).check_unitary – Passed to
mean_abs_otoc_uniform()/otoc_fixed_unitary().rtol – Passed through.
atol – Passed through.
- Returns:
Finite \(I(U)\) when the expectation is strictly positive.
- Raises:
ValueError – If the expectation is numerically zero (
logundefined),uis invalid, ormethod="exact"is used with \(n\) above the implemented cap.
Note
Denoted \(\mathbb{I}(U)\) in arXiv:2408.01663 (Phys. Rev. Research 7, 033271 (2025)). It is a magic monotone: for example faithfulness gives \(\mathbb{I}(U)=0\) iff \(U\) is Clifford. Scaling with T gates gives \(\mathbb{I}(T^{\otimes k}\otimes I^{\otimes n-k})=k\log(4/3)\). Theorem~1 and Corollary~1 relate \(\mathbb{I}(U)\) to how many random Pauli pairs are needed to estimate this quantity accurately in the Monte Carlo setting.