Symmetries in Quantum Circuit Construction

In this presentation, we discuss our latest findings on the mathematical structure of symmetry-invariant quantum circuits. Our research reveals that these circuits, characterized by either discrete symmetries or sets of mutually abelian matrices, form a well-behaved subgroup of the special unitary group. We demonstrate that the mathematical space spanned by these symmetries is not only connected but also equipped with an associated Lie algebra. Quantum circuits can be constructed from the Lie algebra with the symmetry in consideration. With this structured approach, the quantum circuits significantly exceed the performance of standard ansatzes when their symmetries are aligned. Our results indicate that the performance of variational quantum circuits can be improved if the symmetry is known and taken into account when designing the circuit.