Are there any SSL cipher suites that will not be affected by the post-quantum world? $ openssl version
 OpenSSL 1.0.1 14 Mar 2012
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$ openssl ciphers
 ECDHE-RSA-AES256-GCM-SHA384:ECDHE-ECDSA-AES256-GCM-SHA384:ECDHE-RSA-AES256-SHA384:ECDHE-ECDSA-AES256-SHA384:ECDHE-RSA-AES256-SHA:ECDHE-ECDSA-AES256-SHA:SRP-DSS-AES-256-CBC-SHA:SRP-RSA-AES-256-CBC-SHA:SRP-AES-256-CBC-SHA:DHE-DSS-AES256-GCM-SHA384:DHE-RSA-AES256-GCM-SHA384:DHE-RSA-AES256-SHA256:DHE-DSS-AES256-SHA256:DHE-RSA-AES256-SHA:DHE-DSS-AES256-SHA:DHE-RSA-CAMELLIA256-SHA:DHE-DSS-CAMELLIA256-SHA:ECDH-RSA-AES256-GCM-SHA384:ECDH-ECDSA-AES256-GCM-SHA384:ECDH-RSA-AES256-SHA384:ECDH-ECDSA-AES256-SHA384:ECDH-RSA-AES256-SHA:ECDH-ECDSA-AES256-SHA:AES256-GCM-SHA384:AES256-SHA256:AES256-SHA:CAMELLIA256-SHA:PSK-AES256-CBC-SHA:ECDHE-RSA-DES-CBC3-SHA:ECDHE-ECDSA-DES-CBC3-SHA:SRP-DSS-3DES-EDE-CBC-SHA:SRP-RSA-3DES-EDE-CBC-SHA:SRP-3DES-EDE-CBC-SHA:EDH-RSA-DES-CBC3-SHA:EDH-DSS-DES-CBC3-SHA:ECDH-RSA-DES-CBC3-SHA:ECDH-ECDSA-DES-CBC3-SHA:DES-CBC3-SHA:PSK-3DES-EDE-CBC-SHA:ECDHE-RSA-AES128-GCM-SHA256:ECDHE-ECDSA-AES128-GCM-SHA256:ECDHE-RSA-AES128-SHA256:ECDHE-ECDSA-AES128-SHA256:ECDHE-RSA-AES128-SHA:ECDHE-ECDSA-AES128-SHA:SRP-DSS-AES-128-CBC-SHA:SRP-RSA-AES-128-CBC-SHA:SRP-AES-128-CBC-SHA:DHE-DSS-AES128-GCM-SHA256:DHE-RSA-AES128-GCM-SHA256:DHE-RSA-AES128-SHA256:DHE-DSS-AES128-SHA256:DHE-RSA-AES128-SHA:DHE-DSS-AES128-SHA:DHE-RSA-SEED-SHA:DHE-DSS-SEED-SHA:DHE-RSA-CAMELLIA128-SHA:DHE-DSS-CAMELLIA128-SHA:ECDH-RSA-AES128-GCM-SHA256:ECDH-ECDSA-AES128-GCM-SHA256:ECDH-RSA-AES128-SHA256:ECDH-ECDSA-AES128-SHA256:ECDH-RSA-AES128-SHA:ECDH-ECDSA-AES128-SHA:AES128-GCM-SHA256:AES128-SHA256:AES128-SHA:SEED-SHA:CAMELLIA128-SHA:PSK-AES128-CBC-SHA:ECDHE-RSA-RC4-SHA:ECDHE-ECDSA-RC4-SHA:ECDH-RSA-RC4-SHA:ECDH-ECDSA-RC4-SHA:RC4-SHA:RC4-MD5:PSK-RC4-SHA:EDH-RSA-DES-CBC-SHA:EDH-DSS-DES-CBC-SHA:DES-CBC-SHA:EXP-EDH-RSA-DES-CBC-SHA:EXP-EDH-DSS-DES-CBC-SHA:EXP-DES-CBC-SHA:EXP-RC2-CBC-MD5:EXP-RC4-MD5
 $ 


Our question is: are there any current ciphers that can be used in an HTTPS connection that will not be affected after quantum computers can beat most of the PKI?
Symmetric algorithms are secure post-quantum, only with less bits of security (usually about half). That means you only need to care about the authentication and key exchange parts of the cipher suite. Suites that don't use public key authentication or key exchange, i.e. preshared key suites, are post-quantum secure, but not useful in most usecases.

There are post-quantum ways to do key exchange, but SSL/TLS does not currently support any.

(In practice, at least. There is an old draft for NTRU in TLS, and apparently even some implementation(s), but the usual SSL libraries do not support it.)