  We propose a linear-optical implementation of a hyperentanglement-assisted
quantum error-correcting code. The code is hyperentanglement-assisted because
the shared entanglement resource is a photonic state hyperentangled in
polarization and orbital angular momentum. It is possible to encode, decode,
and diagnose channel errors using linear-optical techniques. The code corrects
for polarization "flip" errors and is thus suitable only for a
proof-of-principle experiment. The encoding and decoding circuits use a
Knill-Laflamme-Milburn-like scheme for transforming polarization and orbital
angular momentum photonic qubits. A numerical optimization algorithm finds a
unit-fidelity encoding circuit that requires only three ancilla modes and has
success probability equal to 0.0097.
