Post-quantum candidates for asymmetric cryptography with small key sizes
It appears that SIDH, or Supersingular Isogeny Diffie-Hellman, is the post-quantum candidate with the smallest key size. When using compression, public key size can be reduced to 330 bytes (2640 bits) to achieve the 128-bit security level. For classical Diffie-Hellman, this value is often quoted at 3072 bits. From the above-linked paper describing the attractiveness of the small public key size of SIDH:


  Not only are high-security SIDH public keys smaller than their lattice- and code-based counterparts, they are even smaller than some of the traditional (i.e., finite field) Diffie-Hellman public keys.


The SIDH-based public key encryption and key encapsulation scheme submitted to the NIST post-quantum cryptography process is called SIKE, for Supersingular Isogeny Key Exchange, and specifies parameters called SIKEp503 to obtain a 378-byte (3024-bit) public key for the 128-bit security target.