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Quantum spherical codes

Shubham P. Jain, Joseph T. Iosue, Alexander Barg, Victor V. Albert

TL;DR

This work introduces a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes, and applies this framework to bosonic coding, and obtains multimode extensions of the cat codes that can outperform previous constructions but require a similar type of overhead.

Abstract

We introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, obtaining multimode extensions of the cat codes that can outperform previous constructions while requiring a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that at the same time form averaging sets known as spherical designs. We also recast concatenations of CSS codes with cat codes as quantum spherical codes, revealing a new way to autonomously protect against dephasing noise.

Quantum spherical codes

TL;DR

This work introduces a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes, and applies this framework to bosonic coding, and obtains multimode extensions of the cat codes that can outperform previous constructions but require a similar type of overhead.

Abstract

We introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, obtaining multimode extensions of the cat codes that can outperform previous constructions while requiring a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that at the same time form averaging sets known as spherical designs. We also recast concatenations of CSS codes with cat codes as quantum spherical codes, revealing a new way to autonomously protect against dephasing noise.
Paper Structure (8 sections, 20 equations, 3 figures, 5 tables)

This paper contains 8 sections, 20 equations, 3 figures, 5 tables.

Table of Contents

  1. Real-polytope QSCs
  2. Complex-polytope QSCs
  3. CSS-based QSCs
  4. Performance of quKit QSCs
  5. Lindbladian stabilization
  6. General scheme
  7. CSS QSCs
  8. Hessian QSC

Figures (3)

  • Figure 1: Quantum spherical codewords are quantum superpositions of constellations on a sphere. Logical constellations can form the vertices of a polytope and unite to form a code polytope compound. Projections of polytope compounds are shown for the $\mathbf{(a)}$ cat, $\mathbf{(b)}$ Möbius-Kantor, and $\mathbf{(c)}$ Hessian quantum spherical codes, with logical constellation points colored either green, red, or purple.
  • Figure D.1: Comparing cat ($1$-mode) and simplex ($2$-mode) quKit codes for varying values of $K$, it is observed that the simplex family provides more pronounced advantages in code parameters and performance with growing logical dimension. The sweet-spot energy was calculated at the loss rate $\gamma=0.095$.
  • Figure D.2: As shown in (a), the Möbius-Kantor code demonstrates a universal improvement over polygon based codes and outperforms them over a range of energies and loss rates, as exemplified in (b) by choosing $\textnormal{$\bar{\textsc{n}}$}$ corresponding to the sweet spot energy (at $\gamma=0.095$) of the $3\text{-gon}\subset 18\text{-gon}$ code.