Quantum Cubature Codes

TL;DR

Quantum Cubature Codes (QCCs) provide a mathematically grounded framework for bosonic codes by linking coherent-state constellations to cubature formulas. This unifies existing cat codes and Quantum Spherical Codes (QSCs) as special cases while expanding the design space to non-uniform weights and multi-shell constellations based on Euclidean designs. The approach yields a degree-$t$ error-correction theory: QCCs can correct up to $\lfloor t/2\rfloor$ photon-loss errors and detect higher-order loss/gain errors, with performance enhanced by exploiting radial (multi-shell) structure. Numerical benchmarks show multi-shell QCCs can surpass single-shell QSCs at fixed energy, offering improved robustness under pure-loss channels and opening avenues for optimized, hardware-efficient bosonic error correction.

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.

Figures (2)

• Figure 1: (a) Coherent-state logical constellation of a QCC constructed from the weighted union of (b) 8-cell and (c) 16-cell polytopes (see Eq. \ref{['eqn:8-16_cell_constellation']}). The 8-cell (blue) and 16-cell (red) components occupy coherent states with different mean photon numbers (energy). All panels are orthogonal projections onto the $B_4$ Coxeter plane.
• Figure 2: (a) Entanglement fidelity $\mathcal{F}$ of QSCs and QCCs as a function of coherent-state amplitude scale $\tilde{\alpha}$ for fixed pure-loss rate $\gamma = 0.1$. Markers indicate the optimal $\tilde{\alpha}_{\mathrm{op}}$ for each code. (b) Entanglement fidelity $\mathcal{F}$ as a function of the pure-loss rate $\gamma$, evaluated at these optimized amplitudes (inset: zoom for small $\gamma$). (c) Relative improvement of entanglement infidelity $R_{\mathrm{infidelity}} = (1-\mathcal{F})/(1-\mathcal{F}')$ of QCCs (primed) over QSCs (unprimed) with the same number of constellation points; the black dashed line $R_{\mathrm{infidelity}} = 1$ denotes equal performance. In all panels, blue, orange, and green curves correspond to constellations with $8$, $12$, and $24$ points, respectively.

Theorems & Definitions (6)

• Theorem 1
• Proposition 1: Bounds on Constellation Size
• Lemma 1: Complex extension of cubature formulas
• proof
• Theorem : Fisher-type bound stroud1971approximatesawa2019euclidean
• Theorem : Möller bound moller1979lowersawa2019euclidean