In classical cryptography, security proofs are often based on the (assumed) computational hardness of some mathematical problem. Using the principles of quantum mechanics might provide means to design cryptographic protocols that can classically not be implemented (information-theoretically) securely. But is there also a notion of computational security in quantum cryptography (assuming a polynomial-time quantum adversary)? Why does or doesn't make this notion of security sense?
But is there also a notion of computational security in quantum cryptography (assuming a polynomial-time quantum adversary)?
No, not really, or at least, none that has been explored. The goal of Quantum Cryptography is to be secure, even if the adversary has a Quantum Computer and that they are computationally unbounded; that is, the goal is to rely (as much as possible) on security-by-the-laws-of-physics alone .
One could include the assumption that they are computationally bounded; however if you did that, you would have a number of competing options available, including large key symmetric cryptography and postquantum (public key) cryptography. These existing solutions already solve the problem, and are considerably cheaper and more versatile. Hence, there appears to be little need to lower the security bar on Quantum Cryptography.
Kelalaka brought up the paper "Computational Security of Quantum Encryption"; however a close review shows that the security targets it makes do not rely on the security assumptions of Quantum Mechanics. Instead, it examines the extension of classical cryptography into the realm of Qubits (and how that differs in nontrivial ways from the cryptography on bits). This may be semantics, however I do not believe that falls into the realm of "Quantum Cryptography".
: Of course, that's not the only assumption; they always need to assume that there are no exploitable side channels, that the equipment is operating as designed (and not in a way that looks correct, but is exploitable) and also for many QKD systems, the shared keys are used to AES encrypt the actual traffic (and hence those systems need to assume AES is strong as well).