Classification of Pauli DLAs

This tutorial will illustrate how to use paulie to classify the dynamical Lie algebra of a circuit given the generators consisting of Paulistrings. A Paulistring is a tensor product of Pauli matrices

\[P = \bigotimes_i \sigma_i , \, \sigma_i \in \{I,X,Y,Z\}\]

and is represented as a string indicating the Pauli matrices successively. Given a set of Paulistrings, the closure under the commutator defines a Lie algebra.

In “Full classification of Pauli Lie algebras” [1]. an efficient algorithm for classifying which Lie algebra is generated is given. The function get_algebra returns exactly which algebra is generated when given the generator set which can be extended with identities to arbitrary qubit numbers specified. We can reproduce Example I.5 in “Classification of dynamical Lie algebras of 2-local spin systems on linear, circular and fully connected topologies” [2]:

from paulie.application.classify import get_algebra
from paulie.common.pauli_string_factory import get_pauli_string as p

generators = p(["XY"])
algebra = get_algebra(generators)
print(f"algebra = {algebra}")

outputs

algebra = u(1)

whereas changing to a three qubit system, results in another algebra:

size = 3
generators = p(["XY"], n=size)
algebra = get_algebra(generators)
print(f"algebra = {algebra}")

outputs

algebra = so(3)

The algorithms is based on the concept of an anticommutation graph. Given a set of n-qubit Paulistrings \(\mathcal{G} = \{P_1,\dots ,P_{n_G}\}\), the anticommutation graph has as a vertex set \(\mathcal{G}\) and edges between all vertices that do not commute. Now the edge between \(P_i\) and \(P_j\) can be contracted by mapping \(P_i \mapsto \pm \frac{1}{2} i [P_i,P_j] = P_i^\star\). Now if \(P_i^\star\) is already in \(\mathcal{G}\), the size of the generator set has been reduced while leaving the Lie algebra invariant. We can illustrate this for the generating set ["XYI", "IXY", "XZY"] via the function animation_anti_commutation_graph(generators). As the third Paulistring is proportional to the commutator of the first two Paulistrings it can be contracted.

For any generator set consisting of Paulistrings, the anticommutation graph can be transformed to four canonical types (Theorem 1, [1]). In the example the resulting graph is a line graph with two vertices and no single vertices. Therefore it is of type A and the according to Theorem 2 [1], it corresponds to \(\mathfrak{so}(3)\).

We can try it also for classifying a generator set that does correspond to a B-type canonical graph, that is a anticommutation graph that is star graph. We demonstrate it by the algebra \(\mathfrak{a}_9\) [1], generated by \(XY\) and \(XZ\).

n_qubits = 4
generators = p(["XY", "XZ"], n=n_qubits)
algebra = get_algebra(generators)
print(f"algebra = {algebra}")

outputs

algebra = sp(4)

We can also animate the transformation to a star graph:

The Lie algebra plays a pivotal role in quantum control theory to understand the reachability of states. Also measures of operator spread complexity rely on this concept. Furthermore, determining moments of circuits can be significantly simplified when the Lie algebra is known. All these applications are to be added as functionalities of paulie in the future.

Finding an efficient way to transform the initial graph into a canonical graph is non-trivial. The current implementation in worst case has a exponential scaling in the number of independent generators, but we hope to improve it to a polynomial scaling.