  Qubit regularization is a procedure to regularize the infinite dimensional
local Hilbert space of bosonic fields to a finite dimensional one, which is a
crucial step when trying to simulate lattice quantum field theories on a
quantum computer. When the qubit-regularized lattice quantum fields preserve
important symmetries of the original theory, qubit regularization naturally
enforces certain algebraic structures on these quantum fields. We introduce the
concept of qubit embedding algebras (QEAs) to characterize this algebraic
structure associated with a qubit regularization scheme. We show a systematic
procedure to derive QEAs for the O(N) lattice spin models and the SU(N) lattice
gauge theories. While some of the QEAs we find were discovered earlier in the
context of the D-theory approach, our method shows that QEAs are far more
richer. A more complete understanding of the QEAs could be helpful in
recovering the fixed points of the desired quantum field theories.
