  To realize fault-tolerant quantum computing, it is necessary to store quantum
information in logical qubits with error correction functions, realized by
distributing a logical state among multiple physical qubits or by encoding it
in the Hilbert space of a high-dimensional system. Quantum gate operations
between these error-correctable logical qubits, which are essential for
implementation of any practical quantum computational task, have not been
experimentally demonstrated yet. Here we demonstrate a geometric method for
realizing controlled-phase gates between two logical qubits encoded in photonic
fields stored in cavities. The gates are realized by dispersively coupling an
ancillary superconducting qubit to these cavities and driving it to make a
cyclic evolution depending on the joint photonic state of the cavities, which
produces a conditional geometric phase. We first realize phase gates for
photonic qubits with the logical basis states encoded in two quasiorthogonal
coherent states, which have important implications for
continuous-variable-based quantum computation. Then we use this geometric
method to implement a controlled-phase gate between two binomially encoded
logical qubits, which have an error-correctable function.
