We present the algebraic structure of bosonic quantum error-correcting codes on genus-two Riemann surfaces. We explicitly construct the code words as automorphic forms and analytically generate the full tower of code spaces at all weights. We prove a fundamental no-go theorem: for any genus greater than one, the stabilizer group is non-amenable, forcing a strictly positive spectral gap in the stabilizer Hamiltonian. Consequently, no normalizable quantum state can satisfy all stabilizer conditions. This sharply contrasts with standard Gottesman-Kitaev-Preskill (GKP) codes, where the amenability of the stabilizer group $\mathbb{Z}^2$ permits approximate code words with arbitrary precision.
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7 pages, 3 figures + Supplemental Material (30 pages, 4 figures). Code, notebooks, and data at https://doi.org/10.5281/zenodo.20585488
We present the algebraic structure of bosonic quantum error-correcting codes on genus-two Riemann surfaces. We explicitly construct the code words as automorphic forms and analytically generate the full tower of code spaces at all weights. We prove a fundamental no-go theorem: for any genus greater than one, the stabilizer group is non-amenable, forcing a strictly positive spectral gap in the stabilizer Hamiltonian. Consequently, no normalizable quantum state can satisfy all stabilizer conditions. This sharply contrasts with standard Gottesman-Kitaev-Preskill (GKP) codes, where the amenability of the stabilizer group Z2 permits approximate code words with arbitrary precision.
We develop a theory of stabilizer codes whose stabilizer groups consist of commuting qudit Paulis, oscillator displacements, and/or planar-rotor displacements. We consider codes whose stabilizer groups form generalized lattices in quantum phase space, a condition that guarantees a finite logical dimension. We construct oscillator-rotor, rotor-qudit, and oscillator-rotor-qubit codes that cannot be decomposed into separate subsystems by generalized Clifford transformations. We also introduce an oscillator-qubit code arising from a partially lifted Golay code, where Construction A is applied to half of the Golay code coordinates. We derive analytical forms of logical operators from the symplectic dual lattice, determine Clifford-Gaussian gates from lattice symmetries, show that the code dimension equals the covolume of the stabilizer lattice, and organize the syndrome subspaces into a vector bundle. Our technique builds on earlier non-commutative geometric results by Rieffel, but it can often be interpreted simply as concatenating a given stabilizer code with a Gottesman-Kitaev-Preskill (GKP) code, applying conventional lattice-theoretic results, and unconcatenating. We formulate distances, optimal decoding, and quantum weight enumerators, recovering the Pauli, qudit, and GKP versions together with known MacWilliams identities. We construct finite-energy codewords whose encoding is an isometry up to an error exponentially small in the inverse damping parameter.
Replication package — code, Jupyter notebooks, and precomputed data — for "Bosonic codes from compact phase spaces" (D. Roberts, A. Slipper, A. Parhizkar, V. V. Albert, and M. Hafezi). The paper develops the correspondence between Gottesman–Kitaev–Preskill (GKP) codes and elliptic curves — established concurrently by Conrad, Burchards & Flammia [1] and the present authors [2] — and extends it, via geometric quantization, to bosonic quantum codes associated with algebraic curves of arbitrary genus. Following geometric quantization, a quantized phase space is a family of coherent states holomorphically parametrized by a complex manifold; this work realizes the phase spaces parametrized by a compact Riemann surface. Code states are then holomorphic sections of a line bundle over that surface (automorphic forms), and the code dimension is fixed by Riemann–Roch — recovering GKP codes when the surface is an elliptic curve (genus 1) and yielding a new class of hyperbolic CV quantum codes at genus greater than or equal to 2. The package also numerically illustrates the central no-go result: an amenability dichotomy under which arbitrarily accurate approximate code words exist at genus one but not at any higher genus. The genus-two Bolza surface — the most symmetric genus-two curve — serves as the explicit worked example throughout, and is what the numerics compute. For that example the package: constructs the k=2 code space, verifies the Veronese factorization Y^2 = XZ that generates the full tower of code spaces, and renders the Husimi Q-functions of the code basis (main-text Fig. 2); builds the Poincaré theta series spanning the code space and studies their convergence; verifies the weight-2 canonical gate representation — V_R, V_U in the main text, ρ(R), ρ(U) in the notebooks — for the automorphism group Aut(C) = GL(2, F_3), symbolically and numerically; derives the circuit-QED conditional two-mode-squeezing Hamiltonian via a lab-frame dispersive Schrieffer–Wolff reduction; simulates the conditional-kick stabilization dynamics (main-text Fig. 3) and diagonalizes the stabilizer Hamiltonian to exhibit a spectral gap, a genus-two instance of the amenability dichotomy. Every computational figure in the main text and Supplemental Material is regenerated; the two non-computational schematics (main-text Fig. 1 and the SM circuit diagram) are illustrations, not code output. See README.md for the full file-to-figure map. [1] https://arxiv.org/abs/2407.03270 (2024)[2] https://github.com/droberts01/higher-genus-gkp (2024)
We connect the structure theorem for quasiprobability representations to quantum error correction. For a code space invariant under a subgroup of an extended Weyl-Heisenberg group, the Brif-Mann construction gives a semi-functorial representation whenever its kernel satisfies the Stratonovich-Weyl axioms, and composes on channels covering full error-correction cycles. With phase-space parity available, a kernel other than the ordinary Wigner one exists iff the code space is invariant under a lattice of displacements: among compact-phase-space bosonic codes, this is only the ideal Gottesman-Kitaev-Preskill family; in finite dimension, this is every qudit stabiliser code, whose negative volume for odd prime dimension is the syndrome-averaged logical magic. For rotation-symmetric, one-photon, non-Pauli qudit and spin codes only the ordinary kernel remains, its negativity reflecting the physical carrier, not logical content; a sector-graded representation built from recovery isometries restores a resource reading. We add a channel atlas, a closed-form magic lifetime under displacement noise, a Kirkwood-Dirac layer, and finite-energy corrections.
We study a class of quantum error-correcting codes through the geometry of complex abelian varieties. These codes, introduced by Gottesman--Kitaev--Preskill, are built from symplectically integral lattices and therefore naturally define polarized complex abelian varieties. We give a precise mathematical formulation of this relationship and extend it to a dictionary between the main structures of GKP code theory and classical objects in the theory of abelian varieties. For instance, under this dictionary, the finite-dimensional code space becomes the space of theta functions H0(X,L), logical Pauli gates arise from the theta group, passive logical Clifford gates correspond to automorphisms of the polarized abelian variety, and concatenation with stabilizer codes corresponds to isogeny. We also prove several key results that give precise mathematical formulations of statements about these codes that often appear in heuristic form in the physics literature. In particular, we prove that the encoding is asymptotically isometric, that every logical Clifford gate is realized by a Gaussian unitary, and that, for noise of small variance, the failure probability is governed to first order by the shortest nontrivial displacement in the kernel of the polarization isogeny, a systolic invariant of the underlying polarization. This leads naturally to optimization problems on the moduli space of polarized abelian varieties.
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode is a promising bosonic code for quantum computation due to its tolerance for noise and all-Gaussian gate set. We present a toolkit for phase-space description and manipulation of GKP encodings that includes Wigner functions for ideal and approximate GKP states, for various types of mixed GKP states, and for GKP-encoded operators. One advantage of a phase-space approach is that Gaussian unitaries, required for computation with GKP codes, correspond to simple transformations on the arguments of Wigner functions. We use this fact and our toolkit to describe GKP error correction, including magic-state preparation, entirely in phase space using operations on Wigner functions. While our focus here is on the square-lattice GKP code, we provide a general framework for GKP codes defined on any lattice.
We present a general approach to error detection of bosonic quantum error-correction codes via an adaptive quantum phase estimation algorithm assisted by a single ancilla qubit. The approach is applicable to a broad class of bosonic codes whose error syndromes are described by symmetry or stabilizer operators, including the rotation-symmetric codes and Gottesman-Kitaev-Preskill (GKP) codes. The detection precision scales inversely with the total evolution time and thus reaches the Heisenberg limit. We numerically demonstrate the approach for several examples, such as detecting bosonic excitation loss errors in high-order cat or binomial codes and displacement errors in finite-energy GKP codes. We also extend the approach to efficiently generate arbitrary Fock states. Our schemes are feasible in present-day experiments.