Quantum computers only provide a significant advantage at breaking a very specific class of asymmetric ciphers (those where the trapdoor function is either based on the discrete logarithm problem or the factorization problem) which we already have replacements for that are quantum-resistant (the trapdoor function is replaced with one based on the lattice problem). If quantum computers became a serious threat, it would not be difficult to just swap out those ciphers. The main issue would be people who have collected encrypted messages and held onto them with the hopes of cracking them in the future.