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Encoding quantum information into a set of harmonic oscillators is considereda hardware efficient approach to mitigate noise for reliable quantuminformation processing. Various codes have been proposed to encode a qubit intoan oscillator -- including cat codes, binomial codes andGottesman-Kitaev-Preskill (GKP) codes -- and are among the first to reach abreak-even point for quantum error correction. Though GKP codes are widelyrecognized for their promise in quantum computation, they also facilitatenear-optimal quantum communication rates in bosonic channels and offer theability to safeguard arbitrary quantum states of oscillators. This reviewfocuses on the basic working mechanism, performance characterization, and themany applications of GKP codes -- emphasizing recent experimental progress insuperconducting circuit architectures and theoretical advancements in multimodeGKP qubit codes and oscillators-to-oscillators (O2O) codes. We begin with apreliminary continuous-variable formalism needed for bosonic codes. We thenproceed to the quantum engineering involved to physically realize GKP states.We take a deep dive into GKP stabilization and preparation in superconductingarchitectures and examine proposals for realizing GKP states in the opticaldomain (along with a concise review of GKP realization in trapped-ionplatforms). Finally, we present multimode GKP qubits and GKP-O2O codes, examinecode performance and discuss applications of GKP codes in quantum informationprocessing tasks such as computing, communication, and sensing.

Advances in Bosonic Quantum Error Correction with Gottesman-Kitaev-Preskill Codes: Theory, Engineering and Applications