  Quantum error correcting codes protect quantum computation from errors caused
by decoherence and other noise. Here we study the problem of designing logical
operations for quantum error correcting codes. We present an automated
procedure which generates logical operations given known encoding and
correcting procedures. Our technique is to use variational circuits for
learning both the logical gates and the physical operations implementing them.
This procedure can be implemented on near-term quantum computers via quantum
process tomography. It enables automatic discovery of logical gates from
analytically designed error correcting codes and can be extended to error
correcting codes found by numerical optimizations. We test the procedure by
simulation on classical computers on small quantum codes of four qubits to
fifteen qubits and show that it finds most logical gates known in the current
literature. Additionally, it generates logical gates not found in the current
literature for the [[5,1,2]] code, the [[6,3,2]] code, and the [[8,3,2]] code.
