Quantum Spherical Codes

The work addresses how to generalize and unify bosonic quantum error-correcting codes by formulating them as quantum spherical codes (QSCs), i.e., quantum superpositions over point constellations on spheres. Bosonic (oscillator) encodings in continuous-variable platforms can offer hardware efficiency and noise-bias advantages but lack a unifying, qubit-like coding structure. The paper proposes QSCs as a common framework that encompasses and extends cat codes and other bosonic encodings, aiming to leverage geometric separation (for robustness to rotations/bit-flips) and averaging properties (for robustness to phase processes) of spherical constellations. The purpose is to construct new multimode codes with improved protection against loss and dephasing, clarify connections to classical spherical codes and designs, and show practical routes for autonomous error protection.

The study builds on bosonic code literature including cat codes and their autonomous stabilization (e.g., Leghtas et al., Touzard et al., Grimm et al.), grid/GKP encodings and large-spin or molecular encodings. It draws from classical spherical codes and designs (Conway & Sloane; Ericson & Zinoviev; Delsarte-Goethals-Seidel), and from complex/real polytopes (Coxeter, Shephard) as structured constellations. It connects with group-orbit codes and SU(2) subgroup constellations (2T, 2O, 2I), and with CSS concatenations widely explored for fault tolerance in bosonic systems. Prior performance analyses via channel fidelity and SDP-based optimal recovery are adopted to compare new vs. existing cat-based constructions.

- Quantum spherical code (QSC) framework: Start from a classical spherical code on an n-dimensional real or complex sphere. Partition the code constellation C into logical constellations {C_k}. Define quantum codewords as uniform superpositions |C_k> over coherent states labeled by points a in C_k lying on the complex n-sphere Ω_n with ||a||^2 = N. The minimum squared Euclidean distance d_E between points in C quantifies resolution against rotation-like (bit-flip) noise. - Coherent-state formalism: n-mode coherent states |a> with complex vector a have overlaps |<α|β>|^2 = exp(−||α−β||^2), implying approximate orthogonality with increasing energy N and larger d_E. The code resolution is d_E = min_{α≠β∈C} ||α−β||^2. - Noise model and detectability: Consider passive linear-optical rotations U_R ∈ U(n) and ladder errors L_{p,q} = ∏_j a_j^{† p_j} a_j^{q_j}. Using a_j|a> = a_j|a>, error detectability reduces to averages of monomials over each C_k. A ladder error is detectable if these averages are independent of k. - Polytope QSCs: Construct logical constellations from vertices of real or complex polytopes (and their compounds). Such vertices are well-separated (good d_E) and often form spherical designs that equalize monomial averages, suppressing phase-type errors. Define three parameters for ladder-error protection: d_⊥ (detectable losses plus one), t_⊥ (correctable symmetric gain/loss plus one), and degree distance d_+ (detectable total degree |p+q| < d_+), with floor((d_⊥+1)/2) ≤ t_⊥ ≤ d_⊥ ≤ d_+. - Code notation: ((n, K, d_+, (t_⊥, d_⊥, d_+))) with K logical codewords over n modes. The 4-component cat code is ((1, 2, 2.0, (2,2,2))). Multimode generalizations include simplex, Möbius-Kantor, Hessian, Witting, etc., chosen to trade off resolution and loss protection. - Spherical designs: If C_k is a complex spherical t-design, then averages of all monomials up to total degree t match the full-sphere averages, giving d_+ ≥ t+1. Designs are preserved under unitary rotations; thus, selecting a base design and rotating copies controls K and d_E. - CSS-based QSCs: Concatenate a CSS [[n, k, (d_x, d_z)]] code with 2-component cat code by mapping binary strings to antipodal points on the n-sphere. This yields an ((n, 2^k, d_E = 4 d_x/n, w_1 = d_z)) QSC, detecting ladder errors with Hamming weight less than d_z (phase-type) and with rotation resolution set by d_x. - Gates and stabilizers: X-type (permutation) gates are realized by passive linear optics as sphere rotations permuting constellations; stabilizers are subgroups leaving each C_k invariant. Z-type stabilizers are polynomial potentials F(a) that vanish on code constellations; logical Z-gates are monomials/polynomials evaluating to k-dependent constants on C_k. For each code, explicit stabilizer polynomials and logical operators are identified (e.g., parity for cat; two-mode phase rotations for simplex; He_3 Pauli group actions for Hessian). - Performance evaluation: Compare codes via channel fidelity under the bosonic pure-loss channel. Optimize recovery via semidefinite programming in a coherent-state basis whose size equals constellation size, avoiding Fock cutoff. Explore sweet-spot energies and performance vs. loss rate. - Autonomous protection: Engineer Lindbladians with jump operators equal to Z-type stabilizers or their generalizations using superconducting circuits with asymmetrically threaded SQUIDs (ATS). Provide a general scheme to realize required multi-monomial jumps, including examples for CSS-based QSCs (e.g., surface-cat products of lowering operators) and the Hessian code (three jump operators with specific pump tones).

- Unified framework: Many bosonic codes, including cat-like and group-orbit encodings, are instances of QSCs, enabling systematic construction via spherical codes/designs and polytope compounds. - New multimode codes with improved tradeoffs: • Simplex code: ((2, 2, 1.5, (2,3,3))) achieves higher loss detection than the p=2 cat at slightly reduced resolution, or higher resolution than the p=3 cat while correcting one fewer loss. Generalizes to ((n, 2, 2−1/n, 3)) with increasing n approaching p=2 cat resolution while detecting one more loss per mode. • Möbius-Kantor code: ((2, 3, 1.0, (3,4,4))) adds a logical state over p=3 cat and detects one additional loss; each C_k is an 8-vertex Möbius-Kantor polygon; the full code forms a 24-vertex 3{4}3 polygon. • 2T-qutrit comparison: A ((2, 3, 1.0, (2,4,4))) 2T-qutrit differs despite the same real 4D polytope mapping, highlighting complex vs real-polytope subtleties. • Hessian code: ((3, 2, 1.0, (4,5,9))) with C_k the 27-vertex Hessian polytope. It corrects as many losses as a p=4 cat but has p=3 cat resolution and detects up to 8 losses (available only for cat codes with p≥9). • Higher-complexity examples: A ((2, 2, 0.586, (5,6,12))) code from 4{3}4/2{6}4 polygons maintains p=4 cat-like resolution while correcting one more and detecting substantially more losses. The Witting code ((4, 2, ≈0.586, (6,8,12))) uses two 240-vertex Witting polytopes; it corrects as many losses as a p=6 cat, has p=4-cat-like resolution, and detects up to 11 losses. It is the first in an infinite family based on real Clifford group orbits. - Design-based lower bounds: When logical constellations are complex spherical t-designs, d_+ ≥ t+1, providing construction principles for high-degree error detectability alongside good separation. - CSS-based QSCs: The antipodal mapping of CSS codewords yields QSCs with bit-flip resolution d_E = 4 d_x/n and phase protection up to weight d_z. Asymptotically good CSS codes produce QSCs with nonvanishing d_E and weight thresholds as n grows. - Logical operations: Identified rotation-based X-type logical gate groups and stabilizers for several codes (e.g., Z_{2p} for cat codes; Z_5 and Z_5×Z_2 for simplex; He_3 and its extensions for Hessian; binary icosahedral actions for certain qudit QSCs). Z-type stabilizers and low-degree logical Z-gates are given via monomials/polynomials, including degenerate-stabilizer-like structures for Hessian. - Numerical performance: Channel fidelity comparisons show multimode QSCs (e.g., simplex and Möbius-Kantor) outperform single-mode cat codes across a range of energies and loss rates. For K>2 (quKits), multimode codes use extra dimensions efficiently, maintaining higher resolution at lower energy per mode. For K=6, the Möbius-Kantor 2{8}3 encoding outperforms various cat-code combinations at their sweet spots. - Autonomous protection feasibility: Detailed ATS-based schemes show how to implement required multi-monomial jump operators, including for concatenated surface-cat stabilizers and the Hessian code, enabling passive Z-type protection with current superconducting technology.

The QSC framework directly ties a code’s robustness to the geometry of its point constellation on a sphere. Large Euclidean separations translate to resilience against passive rotations (bit-flip-like errors), while spherical-design properties ensure monomial averages are insensitive to logical labels, suppressing phase-type errors from physical processes. By selecting polytopes whose vertices form designs and combining them in compounds, the authors systematically trade between resolution d_E and loss-detection/correction parameters (t_⊥, d_⊥, d_+). In coherent-state realizations, multimode constellations allow higher-dimensional packing: they retain good separation while distributing information across modes, enabling improved loss protection at similar or lower energy per mode compared to single-mode cats. Numerical channel-fidelity results validate these gains, particularly for quKits (K>2), with the Möbius-Kantor and simplex codes outperforming cat codes across broad parameter ranges. Interpreting CSS concatenations as QSCs reframes their protection in modal terms, providing clear metrics: d_E from X-distance and phase protection from Z-distance and Hamming weight. This perspective suggests leveraging asymptotically good CSS families to produce scalable QSCs with nonvanishing separations and phase thresholds. Operationally, QSCs admit noise-bias-preserving X-type gates via passive linear optics and Z-type stabilizers via polynomial potentials. Autonomous Z-type protection can be realized through engineered Lindbladians using ATS elements, making the proposed constructions experimentally plausible in superconducting cavity platforms. Overall, the findings address the research goal of constructing well-protected, resource-efficient bosonic encodings beyond traditional cats and unifying disparate schemes under a geometric, design-based formalism.

The paper proposes quantum spherical codes as a unifying geometric framework for bosonic, spin, and molecular encodings, showing that many known and new codes emerge from point constellations on spheres with controlled separation and design strength. Using real/complex polytopes and spherical designs, the authors construct multimode coherent-state codes that surpass standard cat codes in error-detection/correction parameters and in channel-fidelity performance at comparable overhead. They also reinterpret CSS⊗cat concatenations as QSCs, clarifying phase protection and enabling autonomous Z-type stabilization schemes. Practical implementation routes via ATS-based Lindbladian engineering are outlined. Future directions include: systematic searches over broader classes of spherical codes (group orbits, association schemes), constructing higher-strength complex designs with large separations, optimizing gates and stabilizers for qudit QSCs, extending autonomous stabilization to richer jump-operator sets, and exploring scalable families with K=O(n) that can correct losses while maintaining constant or slowly degrading resolution.

- Many distances for complex/real polytope codes are derived via numerical evaluation of monomial averages; some degree distances are lower bounds inferred from design strengths and may not be tight in all embeddings. - There is a tradeoff between resolution and loss protection that is not fully optimized; rigorous asymptotic performance analyses and thresholds are deferred to future work. - Some constructions rely on specific embeddings/mappings from real to complex spheres; protection parameters can depend on the chosen embedding. - Numerical performance results are limited to pure-loss channels and to finite code families; broader noise models and larger code sizes remain to be explored. - Autonomous stabilization schemes, while plausible with ATS technology, require nontrivial multi-tone engineering; experimental validation for multimode QSCs beyond p=2 cats is pending.