new_token_emb = torch.randn(1, dim_model)
extended_input = torch.cat([input_ids_emb, new_token_emb], dim=0)

Q_ext = extended_input @ W_q
K_ext = extended_input @ W_k
V_ext = extended_input @ W_v

# (output_ext would be computed using Q_ext, K_ext, V_ext + masking)