# Encoding a qudit in an oscillator

**Authors:** Daniel Gottesman, Alexei Kitaev, John Preskill  
**arXiv:** [quant-ph/0008040](https://arxiv.org/abs/quant-ph/0008040)  
**Date:** 2000-08-08  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/quant-ph--0008040v1/

## Key points

- Construction of quantum error-correcting codes embedding finite-dimensional code space in infinite-dimensional Hilbert space
- Codes protect against errors shifting values of canonical variables q and p
- Noncommutative geometry of phase space used to safeguard against errors
- Fault-tolerant universal quantum computation executed on protected code subspace using techniques in quantum optics (linear optical operations, squeezing, homodyne detection, photon counting)
- Nonlinear mode coupling necessary for preparing encoded states
- Finite-dimensional versions of codes protect against shifts in amplitude or phase of d-state system
- Continuous-variable codes establish lower bounds on quantum capacity of Gaussian quantum channels
- Implications beyond error correction - contribute to understanding fundamental limits in transmitting information through certain types of quantum channels

## Summary

In their paper "Encoding a qudit in an oscillator," authors Daniel Gottesman, Alexei Kitaev, and John Preskill discuss the construction of quantum error-correcting codes that embed a finite-dimensional code space within the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes are designed to protect against errors that shift the values of the canonical variables q and p. The authors explain that these codes take advantage of the noncommutative geometry of phase space to safeguard against such errors. In particular, they demonstrate how fault-tolerant universal quantum computation can be executed on the protected code subspace using various techniques in quantum optics. This includes linear optical operations, squeezing, homodyne detection, and photon counting. However, they note that nonlinear mode coupling is necessary for preparing the encoded states. Furthermore, Gottesman, Kitaev, and Preskill highlight that finite-dimensional versions of these codes can also be constructed to protect encoded quantum information against shifts in the amplitude or phase of a d-state system. This extension allows for more versatile applications across different types of quantum systems. Additionally, the authors mention that continuous-variable codes can be utilized to establish lower bounds on the quantum capacity of Gaussian quantum channels. This implies that these codes have implications beyond error correction and can contribute to understanding fundamental limits in transmitting information through certain types of quantum channels. Overall, this paper presents an important advancement in the field of quantum computing by introducing error-correcting codes specifically tailored for systems described by continuous quantum variables. The authors' findings pave the way for more robust and reliable computations in various physical implementations while shedding light on broader aspects related to channel capacities in quantum communication.