



02350nas a2200133   4500008004100000245009000041210006900131260001500200520189100215100002302106700002402129700002602153856003702179       2014                        eng d00aNormalizer circuits and a Gottesman-Knill theorem for  infinite-dimensional systems
0 aNormalizer circuits and a GottesmanKnill theorem for infinitedim  c2014/09/103 a  $\textit{Normalizer circuits}$ [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].
1 aBermejo-Vega, Juan1 aLin, Cedric, Yen-Yu1 aVan den Nest, Maarten  uhttp://arxiv.org/abs/1409.3208v2