$\textit{Normalizer circuits}$ [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].
