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 Pauli strings. A Pauli string 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 Pauli strings, the closure under the commutator defines a Lie algebra.

In Aguilar et al. [2024], an efficient algorithm for classifying which Lie algebra is generated is given. PauLie implements a modified version of this algorithm. 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 Wiersema et al. [2024]:

from paulie import get_pauli_string as p

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

outputs

algebra = so(3)

The algorithm is based on the concept of an anticommutation graph. Given a set of n-qubit Pauli strings \(\mathcal{G} = \{P_1,\dots ,P_{n_G}\}\), the anticommutation graph has as a vertex set \(\mathcal{G}\) and edges between all vertices that do not commute. The fundamental operation we use is a contraction between two Pauli strings \(P_i\) and \(P_j\) which maps \(P_i \mapsto \pm \frac{1}{2} i [P_i,P_j] = P_i^\star\). Crucially, it can be shown that this operation leaves the Lie algebra invariant. Now if \(P_i^\star\) is already in \(\mathcal{G}\), the size of the generator set has been reduced. Otherwise, the operation results the complementation of the edge set between \(P_i^\star\) and vertices in the neighbourhood of \(P_j\). Simply put, we can use such contractions to manipulates the edges of the anticommutation graph so as to bring it to a canonical form.

For any generator set consisting of Pauli strings, the anticommutation graph can be transformed to four canonical types (Theorem 1 in Aguilar et al. [2024]). Each canonical type corresponds to a particular algebra. There is an exception if \(\mathcal{P}\) only has one Pauli string. A single Pauli string generates the algebra \(\mathfrak{u}(1)\).

Canonical types and associated Lie algebras for a starlike graph with \(n_1 + 1\) \(\left(n_1 \geq 0\right)\) legs of length \(1\) and \(n_2\) \(\left(n_2 \geq 1\right)\) legs of length \(2\). Note that some graphs may belong to multiple canonical types, which reflects the existence of certain exceptional isomorphisms between Lie algebras.

Canonical type	Structure	Lie algebra
A	\(\left|N\right| = n_1\), \(\left|T\right| = 0\), \(\left|L^{\mathcal{B}}\right| = n\)	\(\bigoplus_{i=1}^{2^{n_1}}\mathfrak{so}(n + 3)\)
B1	\(\left|N\right|=n_1\), \(\left|T\right| = n_2\), \(\left|L^{\mathcal{B}}\right|=0\)	\(\bigoplus_{i=1}^{2^{n_1}}\mathfrak{sp}(2^{n_2})\)
B2	\(\left|N\right|=n_1\), \(\left|T\right| = n_2\), \(\left|L^{\mathcal{B}}\right|=4\)	\(\bigoplus_{i=1}^{2^{n_1}}\mathfrak{so}(2^{n_2 + 3})\)
B3	\(\left|N\right|=n_1\), \(\left|T\right| = n_2\), \(\left|L^{\mathcal{B}}\right|=3\)	\(\bigoplus_{i=1}^{2^{n_1}}\mathfrak{su}(2^{n_2 + 2})\)

Classification of A-type canonical graph

Let’s try to classify a generator set that corresponds to an A-type canonical graph. This algebra is generated by \(\mathcal{P}=\{IYZI,IIXX,IIYZ,IXXI,XXII,YZII\}\).

generators = p(["IYZI", "IIXX", "IIYZ", "IXXI", "XXII", "YZII"])
algebra = generators.get_algebra()
print(f"algebra = {algebra}")

outputs

algebra = 4*so(5)

According to the table, the resultant graph corresponds to \(\mathfrak{so}(5)\oplus \mathfrak{so}(5)\oplus \mathfrak{so}(5)\oplus \mathfrak{so}(5)\). But it is worth noting that it also corresponds to \(\mathfrak{sp}(2)\oplus \mathfrak{sp}(2)\oplus \mathfrak{sp}(2)\oplus \mathfrak{sp}(2)\). This shows that there is an exceptional isomorphism between \(\mathfrak{so}(5)\) and \(\mathfrak{sp}(2)\).

Classification of B-type canonical graph

Let’s try to classify a generator set that corresponds to a B-type canonical graph, that is a anticommutation graph that is a 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 = generators.get_algebra()
print(f"algebra = {algebra}")

outputs

algebra = sp(4)

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 functionalities of paulie.