The entanglement-assisted stabilizer formalism is a generalized form of the stabilizer formalism. This framework allows the code designer to take advantage of a significantly wider range of classical error-correcting codes by using pairs of qubits in a maximally entangled state (or ebits). Low-density parity-check (LDPC) codes are among the best known error-correcting codes in terms of error correction performance and decoding complexity in the classical domain and can also be imported to the quantum domain in a simple manner through the entanglement-assisted stabilizer formalism. From a practical viewpoint, it is desirable to rely on fewer ebits while keeping the error correction ability inherited from classical LDPC codes. We present necessary and sufficient conditions for the existence of quantum LDPC codes consuming only one ebit which are obtainable from pairs of identical LDPC codes, and show relations of entanglement-assisted quantum LDPC codes to some fundamental classes of combinatorial designs.