# CQG: Geometry of Gaussian Circuit Complexity

## Main Content

Computational circuit complexity has drawn increased interest from various fields, including quantum computation, holography and tensor networks. The question is what is the minimal number of elementary unitary operations required to transform an unentangled reference state into a physical target state, such as a highly correlated field theory ground state. Defining circuit complexity of states in a quantum field theory is still an open problem, but I will present recent progress towards this goal for free field theories and Gaussian states. Our results are based on Nielsen's approach: the complexity of a circuit U is defined as the geodesic distance from the identity to U with respect to some right-invariant metric. In particular, I will discuss the geometry of states as the quotient of two groups and group theoretic methods to find Nielsen's circuit complexity for systems with many degrees of freedom.