  We show in this paper that a strong and easy connection exists between
quantum error correction and Lattice Gauge Theories (LGT) by using the Gauge
symmetry to construct an efficient error-correcting code for Abelian LGTs. We
identify the logical operations on this gauge covariant code and show that the
corresponding Hamiltonian can be expressed in terms of these logical operations
while preserving the locality of the interactions. Furthermore, we demonstrate
that these substitutions actually give a new way of writing the LGT as an
equivalent hardcore boson model. Finally we demonstrate a method to perform
fault-tolerant time evolution of the Hamiltonian within the gauge covariant
code using both product formulas and qubitization approaches. This opens up the
possibility of inexpensive end to end dynamical simulations that save physical
qubits by blurring the lines between simulation algorithms and quantum error
correcting codes.
