In order to prevent eavesdropping and tampering, the network security protocols use a handshake with an asymmetric cipher to establish a session-specific shared key with which further communication is encrypted using a symmetric cipher. The commonly used asymmetric algorithms include public key encryption, key exchange and identity-based encryption(IBE). However, the network security protocols based on classic identity-based encryption do not have perfect forward security. To solve the problem, we construct the first quantum IBE (QIBE) scheme based on the learning with errors problem, and prove that our scheme is fully secure under the random oracle. Moreover, we construct the quantum circuit of our QIBE scheme and give an estimate of the quantum resource of our circuit including the numbers of Hadamard gate, phase gate, T gate, CNOT gate and the total qubits used in the circuit, and conclude that the quantum resources required by our scheme increase linearly with the number of bits of the encrypted quantum plaintext. Our scheme exhibits the following advantages: (i) The classic key generation center (KGC) system still can be used for our QIBE scheme to generate and distribute the secret identity keys so that the cost can be reduced when the scheme is implemented. The reason why the classic KGC can be used is that the public and private keys are in the form of classic bits. (ii) The network security protocols using a handshake with our QIBE scheme can provide perfect forward security. In our scheme, the ciphertext is transmitted in the form of a quantum state that is unknown to the adversary and therefore cannot be copied and stored. Thus, in the network security protocols based on our QIBE construction, the adversary cannot decrypt the previous quantum ciphertext to threat the previous session keys even if the identity secret key is threatened.