On the universality of S<sub>n</sub>-equivariant k-body gates
Sujay Kazi, Martín Larocca, M. Cerezo
Abstract
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mrow><mml:mi mathvariant="fraktur">G</mml:mi></mml:mrow></mml:mrow></mml:math> ), the learning model should respect said symmetry. This can be instantiated via <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mrow><mml:mi mathvariant="fraktur">G</mml:mi></mml:mrow></mml:mrow></mml:math> -equivariant quantum neural networks (QNNs), i.e. parametrized quantum circuits whose gates are generated by operators commuting with a given representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mrow><mml:mi mathvariant="fraktur">G</mml:mi></mml:mrow></mml:mrow></mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mrow><mml:mi mathvariant="fraktur">G</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> -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.