Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems

textit{Normalizer circuits} [1,2] are generalized Clifford circuits that act on arbitrary finite-dimensional systems mathcal{H}_d1otimes ... otimes mathcal{H}_dn with a standard basis labeled by the elements of a finite Abelian group G=mathbb{Z}_d1times... times mathbb{Z}_dn. Normalizer gates implement operations associated with the group G and can be of three types: quantum Fourier transforms, group automorphism gates and quadratic phase gates. In this work, we extend the normalizer formalism [1,2] to infinite dimensions, by allowing normalizer gates to act on systems of the form mathcal{H}_mathbb{Z}^{otimes a}: each factor mathcal{H}_mathbb{Z} has a standard basis labeled by textit{integers} mathbb{Z}, and a Fourier basis labeled by textit{angles}, elements of the circle group mathbb{T}. Normalizer circuits become hybrid quantum circuits acting both on continuous- and discrete-variable systems. We show that infinite-dimensional normalizer circuits can be efficiently simulated classically with a generalized textit{stabilizer formalism} for Hilbert spaces associated with groups of the form mathbb{Z}^atimes mathbb{T}^b times mathbb{Z}_d1times...times mathbb{Z}_dn. We develop new techniques to track stabilizer-groups based on normal forms for group automorphisms and quadratic functions. We use our normal forms to reduce the problem of simulating normalizer circuits to that of finding general solutions of systems of mixed real-integer linear equations [3] and exploit this fact to devise a robust simulation algorithm: the latter remains efficient even in pathological cases where stabilizer groups become infinite, uncountable and non-compact. The techniques developed in this paper might find applications in the study of fault-tolerant quantum computation with superconducting qubits [4,5].