A Quantum Computer attack is that very fast computers could be used to find the private key from the Bitcoin address. The general though is that as these computers begin to be developed the Bitcoin can move to a different algorithm that cannot be solved by quantum computers. This will affect many things in addition to Bitcoin as many encryption and cryptography schemes will be broken.

Bitcoin uses Elliptical Curve Cryptography and the problem to be solved is the discrete logarithm problem on elliptic curves.

See https://en.wikipedia.org/wiki/Elliptic_curve_cryptography#Quantum_computing_attacks

For a tutorial on Elliptical Curve Cryptography this video explains it (45 minutes)

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