Modern mainframes have an optional cryptographic coprocessor addon. The hardware, or cryptocard [1], is specialized to provide high-throughput for common cryptographic functions, including SHA-256. (It does lots more than that too.)

The latest cryptocard available is the Crypto Express5S [2] and is compatible with z13s.

And since modern mainframes can run Linux natively (s390x), you could probably get popular CPU miners to run. Albeit, you'd have to recompile for s390x and provide patches to utilize CPACF [3].

I could never find any SHA-256 hash rates for the Crypto Express5S, but if you really wanted to mine bitcoins on a mainframe, this would be the way to go.

[1]: http://www-03.ibm.com/security/cryptocards/pciecc/overview.s...

[2]: http://www-03.ibm.com/systems/z/hardware/z13_specs.html

[3]: https://www-01.ibm.com/support/knowledgecenter/linuxonibm/co...