  In this paper, we explore the possibility of building a quantum memory that
is robust to thermal noise using large $N$ matrix quantum mechanics models.
First, we investigate the gauged $SU(N)$ matrix harmonic oscillator and
different ways to encode quantum information in it. By calculating the mutual
information between the system and a reference which purifies the encoded
information, we identify a transition temperature, $T_c$, below which the
encoded quantum information is protected from thermal noise for a memory time
scaling as $N^2$. Conversely, for temperatures higher than $T_c$, the
information is quickly destroyed by thermal noise. Second, we relax the
requirement of gauge invariance and study a matrix harmonic oscillator model
with only global symmetry. Finally, we further relax even the symmetry
requirement and propose a model that consists of a large number $N^2$ of
qubits, with interactions derived from an approximate $SU(N)$ symmetry. In both
ungauged models, we find that the effects of gauging can be mimicked using an
energy penalty to give a similar result for the memory time. The final qubit
model also has the potential to be realized in the laboratory.
