The paper by Gottesman, Kitaev, and Preskill introduces quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables \( q \) and \( p \). The authors demonstrate how fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting. However, nonlinear mode coupling is required for the preparation of the encoded states. The paper also discusses the construction of finite-dimensional versions of these codes, which protect encoded quantum information against shifts in the amplitude or phase of a \( d \)-state system. Continuous-variable codes are used to establish lower bounds on the quantum capacity of Gaussian quantum channels. Key contributions include: 1. **Shift-Resistant Quantum Codes**: The authors describe codes that protect a finite-dimensional quantum system (or "qubit") encoded in an infinite-dimensional system. These codes can be useful for implementing quantum computation and communication protocols using harmonic oscillators or rotors. 2. **Fault-Tolerant Manipulation**: The paper outlines the implementation of fault-tolerant manipulation of encoded quantum information using linear optics, squeezing, and homodyne detection. Photon counting is used to complete the universal gate set. 3. **Continuous-Variable Codes**: The authors construct continuous-variable codes by considering the large-\( d \) limit of shift-resistant codes. These codes can correct shifts in position and momentum, with the ability to handle both diffusive phenomena and amplitude damping. 4. **Error Models**: The codes are designed to protect against errors that shift the values of the canonical variables \( p \) and \( q \). The paper discusses various error models, including decoherence, amplitude damping, and unitary errors. 5. **Gaussian Quantum Channel**: The authors establish lower bounds on the quantum capacity of the Gaussian quantum channel, providing insights into the rate at which error-free digital information can be conveyed by a noisy continuous signal. Overall, the paper provides a comprehensive framework for constructing and understanding quantum error-correcting codes for continuous-variable systems, with potential applications in quantum computing and communication.The paper by Gottesman, Kitaev, and Preskill introduces quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables \( q \) and \( p \). The authors demonstrate how fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting. However, nonlinear mode coupling is required for the preparation of the encoded states. The paper also discusses the construction of finite-dimensional versions of these codes, which protect encoded quantum information against shifts in the amplitude or phase of a \( d \)-state system. Continuous-variable codes are used to establish lower bounds on the quantum capacity of Gaussian quantum channels. Key contributions include: 1. **Shift-Resistant Quantum Codes**: The authors describe codes that protect a finite-dimensional quantum system (or "qubit") encoded in an infinite-dimensional system. These codes can be useful for implementing quantum computation and communication protocols using harmonic oscillators or rotors. 2. **Fault-Tolerant Manipulation**: The paper outlines the implementation of fault-tolerant manipulation of encoded quantum information using linear optics, squeezing, and homodyne detection. Photon counting is used to complete the universal gate set. 3. **Continuous-Variable Codes**: The authors construct continuous-variable codes by considering the large-\( d \) limit of shift-resistant codes. These codes can correct shifts in position and momentum, with the ability to handle both diffusive phenomena and amplitude damping. 4. **Error Models**: The codes are designed to protect against errors that shift the values of the canonical variables \( p \) and \( q \). The paper discusses various error models, including decoherence, amplitude damping, and unitary errors. 5. **Gaussian Quantum Channel**: The authors establish lower bounds on the quantum capacity of the Gaussian quantum channel, providing insights into the rate at which error-free digital information can be conveyed by a noisy continuous signal. Overall, the paper provides a comprehensive framework for constructing and understanding quantum error-correcting codes for continuous-variable systems, with potential applications in quantum computing and communication.