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As discussed in this question, the expected security of 1024-bit RSA is 80-bits:

NIST SP 800-57 §5.6.1 p.62–64 specifies a correspondence between RSA modulus size $n$ and expected security strength $s$ in bits:

Strength  RSA modulus size
  80        1024
 112        2048
 128        3072
 192        7680
 256       15360

According to Wikipedia, we now have a 20-qubit quantum computer:

IBM Q System One is a 20-qubit computer.

Question: If we tried to use a 20-qubit computer, e.g. IBM Q System One, to calculate the $\sim {2}^{80}$ keys in the 1024-bit RSA keyspace, how long would it take?

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"If we tried to use a 20-qubit computer to calculate the ∼280 keys in the 1024-bit RSA keyspace, how long would it take?"

That depends on the speed of the gates, but you are unlikely to do anything useful associated with RSA cryptography with only 20 qubits, and you are even less likely to do anything useful with IBM's gate and measurement fidelities.

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