Bosonic algebraically-restricted constellation (BARC) codes encode quantum information in finite superpositions of coherent states whose constellation points are solution sets of multivariate complex polynomial systems, with the polynomial constraints chosen to reflect photon-gain and photon-loss error symmetries [1],[2]. This paper develops a self-contained geometric analysis of the single-mode BARC construction. We restate the polynomial-to-symmetry pipeline, derive explicitly the upper bound on minimum pairwise separation for N equally weighted points on a circle of fixed mean photon number, and evaluate it for the hexagonal BARC family: for N = 6 points at mean photon number n̄, the maximal minimum chord is d = √n̄, giving worst-case coherent-state overlap |⟨α|β⟩|² = exp(−n̄/2). We then compute, with full arithmetic, the exact pairwise overlaps and the loss-error Gram matrix entries for a concrete hexagonal instance at n̄ = 8, obtaining |⟨αᵢ|αⱼ⟩|² = e⁻⁴ ≈ 0.0183 and post-loss overlaps e^(−3.8) ≈ 0.0224 at transmissivity η = 0.95, and we verify the approximate Knill–Laflamme diagonal condition, finding that the off-diagonal-to-diagonal deviation for the loss error is ~13.5% (ratio 0.1353353, O(10⁻¹)), while the next-nearest and opposite-pair contributions are individually bounded at O(10⁻²) or below. We compare these numbers against the qualitative expectations for four-point and cubature-style constellations, and situate BARC codes within the broader resource-commensurable landscape of bosonic encodings [10]. We argue that the polynomial-symmetry viewpoint is valuable primarily as a *generator* of candidate constellations, and that its ultimate worth must be decided by entanglement-fidelity benchmarks rather than by algebraic elegance. ## 1. Introduction Continuous-variable quantum information requires encodings that live in infinite-dimensional Hilbert space yet remain correctable under physically dominant errors. For optical and microwave cavities, those err Full text and updates: papers.qnfo.org/papers/barc-codes-as-algebraically-restricted-coherent-state-constellations-geometry-se/
Bosonic algebraically-restricted constellation (BARC) codes encode quantum information in finite superpositions of coherent states whose constellation points are solution sets of multivariate complex polynomial systems, with the polynomial constraints chosen to reflect photon-gain and photon-loss error symmetries [1],[2]. This paper develops a self-contained geometric analysis of the single-mode BARC construction. We restate the polynomial-to-symmetry pipeline, derive explicitly the upper bound on minimum pairwise separation for N equally weighted points on a circle of fixed mean photon number, and evaluate it for the hexagonal BARC family: for N = 6 points at mean photon number n̄, the maximal minimum chord is d = √n̄, giving worst-case coherent-state overlap |⟨α|β⟩|² = exp(−n̄/2). We then compute, with full arithmetic, the exact pairwise overlaps and the loss-error Gram matrix entries for a concrete hexagonal instance at n̄ = 8, obtaining |⟨αᵢ|αⱼ⟩|² = e⁻⁴ ≈ 0.0183 and post-loss overlaps e^(−3.8) ≈ 0.0224 at transmissivity η = 0.95, and we verify the approximate Knill–Laflamme diagonal condition, finding that the off-diagonal-to-diagonal deviation for the loss error is ~13.5% (ratio 0.1353353, O(10⁻¹)), while the next-nearest and opposite-pair contributions are individually bounded at O(10⁻²) or below. We compare these numbers against the qualitative expectations for four-point and cubature-style constellations, and situate BARC codes within the broader resource-commensurable landscape of bosonic encodings [10]. We argue that the polynomial-symmetry viewpoint is valuable primarily as a generator of candidate constellations, and that its ultimate worth must be decided by entanglement-fidelity benchmarks rather than by algebraic elegance. ## 1. Introduction Continuous-variable quantum information requires encodings that live in infinite-dimensional Hilbert space yet remain correctable under physically dominant errors. For optical and microwave cavities, those err Full text and updates: papers.qnfo.org/papers/barc-codes-as-algebraically-restricted-coherent-state-constellations-geometry-se/
We introduce an algebraic framework for constructing a new class of quantum error-correcting codes, bosonic algebraically-restricted constellation (BARC) codes. The code states are finite superpositions of coherent-state constellations constrained by symmetries of solution sets to multivariate complex polynomial systems. The constraints are associated with photon-gain and photon-loss errors, thereby imposing approximate Knill--Laflamme conditions for such errors. For equally-weighted constellation points, we derive upper bounds on the minimum geometric separation between points, indicating how noise-resilient the code is. In the single-mode case, we characterize degree-two polynomial solution sets using an orthogonal group symmetry to obtain explicit ellipsoidal and hexagonal BARC codes. We benchmark representative instances of these families against spherical and cubature codes using the entanglement fidelity with optimal recovery under pure-loss noise.
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.
Photon loss is a dominant noise process in bosonic quantum hardware, including superconducting cavities. Fixed-total-photon-number qubit encodings in two bosonic modes retain the loss-detection advantage of dual-rail qubits while supporting photon loss correction. Optimizing entanglement fidelity in this setting for 4≤n≤25 over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of SU(2) and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number nd=⌈(3d2+1)/4⌉ and construct each corresponding code through distance d=10. To our knowledge, the constructed (n,d)=(28,6),(49,8),(76,10) codes give the smallest reported n for their respective distances; all members through d=9 attain the fixed-n LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.
Tiger codes form a family of multimode bosonic quantum codes that unify several previously known constructions, including cat, paircat, and the two-mode binomial code. In this work, we give a rigorous algebraic treatment of these codes. Starting from a kernel definition of the codespace, we prove that the annihilation-type constraints admit a finite generating set, construct an explicit orthonormal basis, and show that the logical structure of the code is governed by the homology of an underlying chain complex, as expected in the original work on Tiger codes of Xu et al. We then develop a Fourier transform over the codespace to prove that the span of phase-rotated projected coherent states is dense therein, and to yield dual X- and Z-type descriptions of the code. We further extend the framework to non-linear number constraints, encompassing codes such as the four-legged cat or the repetition cat code. Finally, we investigate the implementation of logical operations. We first generalise the construction of logical Pauli operators proposed by Xu et al. to arbitrary logical spaces, and then construct non-Clifford gates using physical polynomial phase rotations of the form eiP(n^). We derive criteria on the real polynomial P which, for positive single-logical-qubit Tiger codes satisfying an additional sign assumption, such as the paircat code, characterise the polynomials P that preserve the codespace by decomposing them into a family of univariate polynomials. Through this decomposition, we relate the degrees of the resulting components to the induced logical action in the Clifford hierarchy. These results establish Tiger codes as a mathematically robust framework for describing a broad class of bosonic encodings.
The approximate coherent state rank is the minimal number of (classical) coherent states required to approximate a continuous-variable bosonic quantum state and directly relates to the classical complexity of simulating bosonic computations. Despite its importance, little is known about lower bounds on this quantity, even for basic families of states. In this work, we initiate a systematic study of lower bounds on the approximate coherent state rank. Our contributions are as follows.
(i) We introduce a technique based on low-rank approximation theory yielding generic lower bounds on the approximate coherent state rank of arbitrary single-mode states.
(ii) Using this technique, we find a complete characterization of all single-mode states of finite approximate coherent state rank, and we obtain in particular analytical expressions for the approximate coherent state rank of squeezed states and of finite superpositions of Fock states.
(iii) We further show that our single-mode lower bounds can be lifted to multimode lower bounds for finite superpositions of multimode Fock states.
(iv) Finally, we prove a super-polynomial lower bound on the approximate coherent state rank of the n-mode Fock state ∣1⟩⊗n, by exploiting a connection to the permanent. To do so, we show that the algebraic complexity of approximate multi-linear formulas for the permanent is super-polynomial, building upon the proof of a lower bound for exact formulas due to [Raz, JACM 2009].
Our results establish an unconditional barrier to efficient classical simulation of Boson Sampling via coherent state decompositions and connect non-classicality of bosonic quantum systems to central questions in algebraic complexity.
Continuous-variable cat codes are encodings into a single photonic or phononic mode that offer a promising avenue for hardware-efficient fault-tolerant quantum computation. Protecting information in a cat code requires measuring the mode’s occupation number modulo two, but this can be relaxed to a linear occupation-number constraint using the alternative two-mode pair-cat encoding. We construct multimode codes with similar linear constraints using any two integer matrices satisfying a Calderbank-Shor-Steane-like homological condition of a quantum rotor code. Just like the pair-cat code, syndrome extraction can be performed in tandem with stabilizing dissipation using current superconducting-circuit designs. The framework includes codes with various finite- or infinite-dimensional code spaces and codes with finite or infinite Fock-state support. It encompasses two-component cat, pair-cat, dual-rail, two-mode binomial, various bosonic repetition codes, and aspects of χ -squared encodings, while also yielding codes from homological products, lattices, generalized coherent states, and algebraic varieties. Among our examples are analogs of repetition codes, the Shor code, and a surface-code-like construction that is not a concatenation of a known cat code with the qubit surface code. Code words are coherent states projected into a Fock-state subspace defined by an integer matrix, and their overlaps are governed by Gelfand-Kapranov-Zelevinsky hypergeometric functions.