Great question. Made me to think a bit on a possible solution. Below is just my high-level thought without getting into too much of feasibility check.

Quantum computer possesses two major threats to bitcoin Network.

Challenge 1. Quantum computer can rig up blockchain by centralizing its hashing power and

Challenge 2. It can crack to cryptography key and the encryption.

Both are valid security concerns. Something to worry about for sure. But let's understand this basic that it is not just a security threat to bitcoin, but also a security threat to all software's that use encryption see for example it is the threat to all the WhatsApp messages that you send to your friends, it is a threat for you to secure your login credentials for bitcointalk, it is a thread for you as a customer of a bank to prevent your money in the bank account and so on and so forth.

Having established that it is not just a Bitcoin Network security concerned also a security concern for every software that uses encryption technique, let's evaluate two possible solutions to the problem. Once again let me clarify that double two solutions are just my high-level thoughts.

Solution 1.  DAG based blockchain network will grow in future and Proof of Work based blockchain network systems will possibly be eliminated. This might take care of challenge number 1 above

Solution 2. One time password (OTP) or additional security layer to get access to the account will rise and might take care of challenge number 2 above. Two-factor Authentication may become more advanced by the time. Face recognition or voice recognition maybe used along with biometrics in order to get an access to an account in next few years maybe. But this challenge is not hard to overcome. In this manner The Challenge number can be solved.