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Asymptotically good bosonic Fock state codes: Exact and approximate

Dor Elimelech, Arda Aydin, Alexander Barg

Abstract

We examine exact and approximate error correction for multi-mode Fock state codes protecting against the amplitude damping noise. Based on a new formalization of the truncated amplitude damping channel, we show the equivalence of exact and approximate error correction for Fock state codes against random photon losses. Leveraging the recently found construction method based on classical codes with large distance measured in the $\ell_1$ metric, we construct asymptotically good (exact and approximate) Fock state codes. These codes have an additional property of bounded per-mode occupancy, which increases the coherence lifetime of code states and reduces the photon loss probability, both of which have a positive impact on the stability of the system. Using the relation between Fock state code construction and permutation invariant (PI) codes, we also obtain families of asymptotically good qudit PI codes as well as codes in monolithic nuclear state spaces.

Asymptotically good bosonic Fock state codes: Exact and approximate

Abstract

We examine exact and approximate error correction for multi-mode Fock state codes protecting against the amplitude damping noise. Based on a new formalization of the truncated amplitude damping channel, we show the equivalence of exact and approximate error correction for Fock state codes against random photon losses. Leveraging the recently found construction method based on classical codes with large distance measured in the metric, we construct asymptotically good (exact and approximate) Fock state codes. These codes have an additional property of bounded per-mode occupancy, which increases the coherence lifetime of code states and reduces the photon loss probability, both of which have a positive impact on the stability of the system. Using the relation between Fock state code construction and permutation invariant (PI) codes, we also obtain families of asymptotically good qudit PI codes as well as codes in monolithic nuclear state spaces.
Paper Structure (23 sections, 29 theorems, 172 equations, 1 figure)

This paper contains 23 sections, 29 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Exact and approximate quantum codes
  5. Bosonic Fock state codes and the AD channel
  6. The AD channel
  7. The truncated AD channel and AQEC
  8. Asymptotically good Fock state codes from codes
  9. The construction of
  10. Exact asymptotically good Fock state codes
  11. Asymptotically good approximate Fock state codes
  12. Asymptotically good PI and spin codes
  13. PI codes
  14. Spin codes
  15. Classical codes
  16. ...and 8 more sections

Key Result

Theorem 1

Let ${\cal N}$ be a channel with Kraus operators ${\cal E}=\left\{A_k\right\}_{k=0}^{M-1}$. A code space $C\subset {\cal H}$ is an $\varepsilon$-AQECC if and only if there exists a density matrix $\lambda=\left\{\lambda_{k,l}\right\}_{k,l}$ such that $d(\Lambda, \Lambda+{\cal B})\leq \varepsilon$, w where $P$ is the projection on $C$, and $\left\{\ket{k}\right\}_{k=0}^{M-1}$ is an orthonormal bas

Figures (1)

  • Figure 1: Bounds $R_{\mathrm GV}(\delta)$ (the upper curve) and $R_{\mathrm U}(\delta)$ for $\alpha=1$.

Theorems & Definitions (61)

  • Definition 1: Approximate quantum error correction
  • Remark 1
  • Theorem 1: AQEC conditions beny2010general
  • Proposition 2: Approximate KL conditions
  • proof
  • Definition 2
  • Definition 3
  • Definition 4: Code distance
  • Definition 5: Asymptotically good Fock state codes
  • Proposition 3
  • ...and 51 more