  In this paper, we propose a new way to approach qudit systems using toric
geometry and related topics including the local mirror symmetry used in the
string theory compactification. We refer to such systems as (n,d) quantum
systems where $n$ and $d$ denote the number of the qudits and the basis states
respectively. Concretely, we first relate the (n,d) quantum systems to the
holomorphic sections of line bundles on n dimensional projective spaces CP^{n}
with degree n(d-1). These sections are in one-to-one correspondence with d^n
integral points on a n-dimensional simplex. Then, we explore the local mirror
map in the toric geometry language to establish a linkage between the (n,d)
quantum systems and type II D-branes placed at singularities of local
Calabi-Yau manifolds. (1,d) and (2,d) are analyzed in some details and are
found to be related to the mirror of the ALE space with the A_{d-1} singularity
and a generalized conifold respectively.
