$\textit{Normalizer circuits}$ [1,2] are generalized Clifford circuits that

act on arbitrary finite-dimensional systems $\mathcal{H}_{d_1}\otimes ...

\otimes \mathcal{H}_{d_n}$ with a standard basis labeled by the elements of a

finite Abelian group $G=\mathbb{Z}_{d_1}\times... \times \mathbb{Z}_{d_n}$.

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}^a\times \mathbb{T}^b \times

\mathbb{Z}_{d_1}\times...\times \mathbb{Z}_{d_n}$. 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].