Quantum Cubature Codes

TL;DR

Quantum Cubature Codes provide a unified, geometric framework for designing bosonic quantum error correction codes using superpositions of coherent states, enabling the creation of new codes with improved performance under photon loss.

Contribution

This work introduces Quantum Cubature Codes, a novel formalism connecting phase space geometry with code design, unifying existing codes and enabling new, optimized bosonic codes.

Findings

QCCs unify cat codes and quantum spherical codes.

Multi-shell QCCs outperform single-shell codes under photon loss.

Design space for bosonic codes is significantly expanded.

Abstract

Bosonic codes utilize the infinite-dimensional Hilbert space of harmonic oscillators to encode quantum information, offering a hardware-efficient approach to quantum error correction. Designing these codes requires precise geometric arrangements of quantum states in the phase space. Here, we introduce Quantum Cubature Codes (QCCs), a powerful and generalized framework for constructing bosonic codes based on superpositions of coherent states. This formalism utilizes cubature formulas from multivariate approximation theory, which connect the continuous geometry of the phase space to discrete, weighted point sets, ensuring the conditions for error correction are met. We demonstrate that this framework provides a unifying perspective, revealing that well-established codes, such as cat codes and the recently proposed quantum spherical codes (QSCs), are specific instances of QCCs corresponding to uniform weights on a single energy shell. The QCC formalism unlocks a vast new design space, encompassing non-uniform superpositions and multi-shell configurations. We leverage this framework to discover several new families of codes derived from Euclidean designs, allowing for greater geometric separation between logical states, which correlates with improved performance under photon loss. Numerical simulations under a pure-loss channel show that our multi-shell QCCs can outperform their single-shell counterparts by maximizing geometric separation with optimal energy at fixed pure-loss rate.