On the universality of $S_n$-equivariant $k$-body gates

TL;DR

The paper investigates how symmetry constraints and locality (up-to-$k$-body gates) affect the expressiveness of $S_n$-equivariant QNNs. Using the dynamical Lie algebra framework, it shows that 1- and 2-body $S_n$-equivariant gates yield semi-universal behavior, enabling arbitrary unitaries within symmetry isotypic subspaces but not control over relative phases across subspaces. To achieve universality, one must include up-to-$n$-body gates for even $n$ or up-to-$(n-1)$-body gates for odd $n$, with the central projections theorem explaining why center elements cannot appear unless higher-body gates are included. Extending to general $k$, the DLAs decompose as $rak{su}^{S_n}_{ m cless}(d)oxplus Q_k(igoplus_{r=1}^{loor{k/2}} rak{u}(1))$, i.e., at most $loor{k/2}$ independent central directions can be realized, thus bounding universality unless $k$ reaches the parity-dependent threshold. Overall, the work clarifies fundamental limits for symmetry-preserving, locally generated QNNs and corrects prior claims, providing a precise link between locality, symmetry, and universality via Schur–Weyl duality.

Abstract

The importance of symmetries has recently been recognized in quantum machine learning from the simple motto: if a task exhibits a symmetry (given by a group $\mathfrak{G}$), the learning model should respect said symmetry. This can be instantiated via $\mathfrak{G}$-equivariant Quantum Neural Networks (QNNs), i.e., parametrized quantum circuits whose gates are generated by operators commuting with a given representation of $\mathfrak{G}$. In practice, however, there might be additional restrictions to the types of gates one can use, such as being able to act on at most $k$ qubits. In this work we study how the interplay between symmetry and $k$-bodyness in the QNN generators affect its expressiveness for the special case of $\mathfrak{G}=S_n$, the symmetric group. Our results show that if the QNN is generated by one- and two-body $S_n$-equivariant gates, the QNN is semi-universal but not universal. That is, the QNN can generate any arbitrary special unitary matrix in the invariant subspaces, but has no control over the relative phases between them. Then, we show that in order to reach universality one needs to include $n$-body generators (if $n$ is even) or $(n-1)$-body generators (if $n$ is odd). As such, our results brings us a step closer to better understanding the capabilities and limitations of equivariant QNNs.

Figures (8)

• Figure 1: Summary of our main results. a) Recent results in the field of GQML indicate that if the problem has a given relevant symmetry $\mathfrak{G}$, then the gates in the QNN should be $\mathfrak{G}$-equivariant nguyen2022atheory. In this work we consider the case where $\mathfrak{G}=S_n$, the permutation group, and study how the expressiveness of the QNN changes when one imposes the additional constraint of few-bodyness on the QNN's gates. In the circuit, gates with the same color share a common parameter. b) The effect of restricting the set of available elementary gates in the QNN is to restrict its expressiveness, i.e., how much of the unitary space it can cover by varying its parameters. Imposing $\mathfrak{G}$-equivariance can appropriately reduce the QNN's expressiveness to a region of unitaries respecting the task's symmetry. Imposing additional restrictions, such as few-bodyness, further restricts its expressiveness.
• Figure 1: Barycentric coordinates. a) Here we show the barycentric coordinates for $n=3$, where the levels $0$, $1$, $2$ and $3$ shown separately for ease of visualization. b) One can imagine these levels stacked on top of each other to form a regular tetrahedron.
• Figure 2: Important Lie algebras and the irrep structure of the elements in the associated Lie groups. In the main text we have defined three important subalgebras. a) The first is the maximal $\mathfrak{G}$-symmetric subalgebra $\mathfrak{u}^{\mathfrak{G}}(d)$. As schematically shown, the associated Lie group contains the set of matrices where an arbitrary unitary can be prepared in each isotypic component. b) Next, we define the special subalgebra of $\mathfrak{u}^{\mathfrak{G}}(d)$, which we denote as $\mathfrak{su}^{\mathfrak{G}}(d)$. Now the associated Lie group contains all the matrices where an arbitrary unitary can be prepared in each isotypic component, under the additional restriction that the determinant of the overall matrix must be equal to one. c) Finally, we define the centerless subalgebra of $\mathfrak{u}^{\mathfrak{G}}(d)$, which we denote as $\mathfrak{su}^{\mathfrak{G}}_{\rm{cless}}(d)$. In this case, the Lie group contains all matrices where an arbitrary special unitary can be prepared in each isotypic component. Note that the difference between $\mathfrak{su}^{\mathfrak{G}}_{\rm{cless}}(d)$ and $\mathfrak{u}^{\mathfrak{G}}(d)$ is that the former does not allow us to control the relative phases between the isotypic components.
• Figure 2: Hopping by one site. In the figure we schematically show how one can hop in the barycentric lattice by one site within the same level by taking specific commutators. Shown is the $k=6$ level. The red (horizontal) arrows show what happens when taking the commutator of a symmetrized Pauli with $P_{(1,0,0)}=\sum_{j=1}^n X_j$. Similarly, the green (top-left to bottom-right) and the orange (bottom-left to top-right) arrows respectively show the connections made when taking the commutator with $P_{(0,1,0)}=\sum_{j=1}^n Y_j$ and $P_{(0,0,1)}=\sum_{j=1}^n Z_j$.
• Figure 3: $S_n$-equivariance, few-bodyness, and DLA. Here we review the main results of our work. In particular, we consider the case where the single-body operators in $\mathcal{G}$ are $P_{(1,0,0)}$ and $P_{(0,1,0)}$ and where the $k$-body operators are of the form $P_{(0,0,k)}$.

Theorems & Definitions (43)

• Definition 1: Equivariant operators
• Definition 2
• Theorem 1
• Definition 3: Symmetrized Pauli strings
• Definition 4
• Proposition 1: Center of the $S_n$-equivariant Lie algebra
• Theorem 2: Central Projections. Result 1 in zimboras2015symmetry, Restated
• Theorem 3
• Theorem 4
• Theorem 5