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

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.

7+15 pages, 3+5 figures, updated to published version

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• Barren Plateaus in Variational Quantum Computing
• A Lie Algebraic Theory of Barren Plateaus for Deep Parameterized Quantum Circuits
• Symmetry breaking in geometric quantum machine learning in the presence of noise
• Analyzing the quantum approximate optimization algorithm: ansätze, symmetries, and Lie algebras
• Geometric Quantum Machine Learning with Horizontal Quantum Gates
• Architectures and random properties of symplectic quantum circuits
• Analyzing the free states of one quantum resource theory as resource states of another
• A quantum tug of war between randomness and symmetries on homogeneous spaces
• Non-Universality from Conserved Superoperators in Unitary Circuits