Encoding quantum information into a set of harmonic oscillators is considered a hardware efficient approach to mitigate noise for reliable quantum information processing. Various codes have been proposed to encode a qubit into an oscillator—including cat codes, binomial codes and Gottesman-Kitaev-Preskill (GKP) codes—and are among the first to reach a break-even point for quantum error correction. Though GKP codes are widely recognized for their promise in quantum computation, they also facilitate near-optimal quantum communication rates in bosonic channels and offer the ability to safeguard arbitrary quantum states of oscillators. This review focuses on the basic working mechanism, performance characterization, and the many applications of GKP codes—emphasizing recent experimental progress in superconducting circuit architectures and theoretical advancements in multimode GKP qubit codes and oscillators-to-oscillators (O2O) codes. We begin with a preliminary continuous-variable formalism needed for bosonic codes. We then proceed to the quantum engineering involved to physically realize GKP states. We take a deep dive into GKP stabilization and preparation in superconducting architectures and examine proposals for realizing GKP states in the optical domain (along with a concise review of GKP realization in trapped-ion platforms). Finally, we present multimode GKP qubits and GKP-O2O codes, examine code performance and discuss applications of GKP codes in quantum information processing tasks such as computing, communication, and sensing.

将量子信息编码到一组谐振子中被认为是一种硬件有效的方法，可以减轻噪声以实现可靠的量子信息处理。人们已经提出了各种代码来将量子位编码到振荡器中，包括猫代码、二项式代码和 Gottesman-Kitaev-Preskill (GKP) 代码，并且是最先达到量子纠错盈亏平衡点的代码。尽管 GKP 代码因其在量子计算中的前景而得到广泛认可，但它们还有助于在玻色子通道中实现接近最佳的量子通信速率，并提供保护振荡器的任意量子态的能力。本综述重点关注 GKP 代码的基本工作机制、性能表征和许多应用，重点介绍超导电路架构的最新实验进展以及多模式 GKP 量子位代码和振荡器到振荡器 (O2O) 代码的理论进展。我们从玻色码所需的初步连续变量形式开始。然后我们继续进行涉及物理实现 GKP 状态的量子工程。我们深入研究了超导架构中的 GKP 稳定性和准备工作，并研究了在光域中实现 GKP 状态的建议（以及对俘获离子平台中 GKP 实现的简明回顾）。最后，我们提出了多模式 GKP 量子位和 GKP-O2O 代码，检查代码性能并讨论 GKP 代码在计算、通信和传感等量子信息处理任务中的应用。