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GKP states, introduced by Gottesman, Kitaev, and Preskill, are continuousvariable logical qubits that can be corrected for errors caused by phase spacedisplacements. Their experimental realization is challenging, in particularusing propagating fields, where quantum information is encoded in thequadratures of the electromagnetic field. However, travelling photons areessential in many applications of GKP codes involving the long-distancetransmission of quantum information. We introduce a new method for encoding GKPstates in propagating fields using single photons, each occupying a distinctauxiliary mode given by the propagation direction. The GKP states are definedas highly correlated states described by collective continuous modes, as timeand frequency. We analyze how the error detection and correction protocolscales with the total photon number and the spectral width. We show that theobtained code can be corrected for displacements in time-frequency phase space- which correspond to dephasing, or rotations, in the quadrature phase space -and to photon losses. Most importantly, we show that generating two-photon GKPstates is relatively simple, and that such states are currently produced andmanipulated in several photonic platforms where frequency and time-bin biphotonentangled states can be engineered.

Gottesman-Kitaev-Preskill encoding in continuous modal variables of single photons