Commutator graph
======================

This tutorial will illustrate how to use :code:`paulie` to analyse short term dynamics of quantum systems.
The Hamiltonian of the system is described by a linear combination of Pauli strings :math:`H = \sum_{P \in \mathcal{G}} c_P P` where
:math:`\mathcal{G}` is the generator set. The commutator graph of an :math:`n`-qubit system has the :math:`4^n` Pauli strings as
vertices and there exists and edge :math:`(P,Q)` if there exists a generator :math:`G \in \mathcal{G}` s.t.
:math:`[P,G] \propto Q`.
We can plot the commutator graph

.. code-block:: python

    from paulie import get_pauli_string as p
    generators = p(["XI", "ZI", "IX", "IZ", "ZZ"], n=2)
    vertices, edges = generators.get_commutator_graph()
    plot_graph(vertices, edges)


.. image:: ../media/commutator_graph.png

While the anticommutation graph is isomorphic for generator sets with isomorphic DLA's and only capture the
long term dynamics, the commutator graph captures the short term dynamics (see :cite:t:`West_2025`).
In particular, we can determine how chaotic the system behaves. A quantifier of quantum chaos is the
the out-of-time-order correlator (OTOC) between two unitary operators :math:`W` and :math:`V` defined as

.. math::

    F(W, V) = \frac{1}{d}\text{tr}\left[W^{\dagger}V^{\dagger}WV\right]

where :math:`d` is the total Hilbert space dimension. For two initially commuting operators, the OTOC is equal to one.

If we Heisenberg-evolve one of the operators under some dynamics as :math:`V_t = U^{\dagger}VU` where :math:`U=e^{-iHt}`, the operators :math:`W` and :math:`V_t` begin to fail to commute as the operator :math:`V_t` becomes global under the dynamics.
The exponential decay of the OTOC :math:`F(W, V_t)` with time is a probe of quantum choas.

Given a Pauli string dynamical Lie algebra (DLA), we can compute the average OTOC between two Pauli strings where the Heisenberg-evolution is averaged over the dynamics using the function :code:`average_otoc`:

.. code-block:: python

    import numpy as np
    from paulie import average_otoc, get_pauli_string as p

    # These are the generators of n=3 "matchgate" dynamics
    generators = p(["ZII", "IZI", "IIZ", "XXI", "IXX"])
    v = p("XYZ")
    w = p("YZZ")
    print(f"Do {v} and {w} initially commute?", v.commutes_with(w))
    print("Average OTOC of {v} and {w} under matchgate dynamics:", np.round(average_otoc(generators, v, w), 3))

which outputs:

.. code-block:: bash

    Do XYZ and YZZ initially commute? True
    Average OTOC of {v} and {w} under matchgate dynamics: -0.333

As we can see, two initially commuting operators may fail to commute after Heisenberg-evolution under some dynamics.