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A Time-Frequency Framework for GKP Codes

Franz Luef, Eduard Ortega

arXiv:2609.10802v3math-phcs.ITmath.FAmath.ITmath.MPmath.OAquant-ph

TL;DR

The paper asks how finite-dimensional logical information in an ideal GKP codeword can be represented by localized phase-space coefficients and stably reconstructed from finitely many of them. The paper identifies ideal codewords in M∞ with a finite logical fibre using the vector-valued Zak transform, then represents logical vectors by adjoint-lattice Gabor coefficient blocks. The normalized coefficient-block map is an isometry with an explicit recovering projection, and lattice-envelope approximations converge weak-* with asymptotically isometric logical embeddings.

  • Problem

    The paper asks how finite-dimensional logical information in an ideal GKP codeword can be represented by localized phase-space coefficients and stably reconstructed from finitely many of them.

    Source: §1 · p. 1

  • Method

    The paper identifies ideal codewords in M∞ with a finite logical fibre using the vector-valued Zak transform, then represents logical vectors by adjoint-lattice Gabor coefficient blocks.

    Source: §1 · pp. 1–2

  • Results

    The normalized coefficient-block map is an isometry with an explicit recovering projection, and lattice-envelope approximations converge weak-* with asymptotically isometric logical embeddings.

    Source: Finite coefficient blocks and logical… · p. 4, Normalizable regularizations · p. 4

  • Takeaways & Limitations

    A finite block of adjoint-lattice Gabor coefficients is sampled from the encoded codeword.

    Source: §1 · p. 2

    The results are conditional on access to complex coefficient amplitudes and do not themselves specify a physical measurement or recovery channel.

    Source: Syndrome reconstruction · p. 5

Abstract

from arXiv · show

We develop a time--frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-$*$ convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

1 Introduction

The paper asks how finite-dimensional logical information in ideal lattice GKP codewords can be represented and stably reconstructed from localized phase-space coefficients. It connects that coefficient model to normalizable approximations and displacement-syndrome recovery.

1 Introduction

  • Ideal codewords lie in M∞, where the vector-valued Zak transform identifies them with a finite logical fibre over the syndrome torus.The M∞ condition restricts the Zak transform to a lattice Dirac comb with uniformly bounded coefficients, determined by the logical vector.Source: §1 · pp. 1, 3

Finite coefficient blocks and logical reconstruction

  • The normalized finite block of adjoint-lattice coefficients identifies exactly with finite Weyl analysis of the logical vector and supports isometric, stable reconstruction by an explicit projection.The projection returns the closest admissible logical block, while the residual quantifies inconsistency with the finite-block model.Source: Finite coefficient blocks and logical… · pp. 3–4

Normalizable regularizations

  • Square-summable lattice envelopes produce normalizable states; asymptotically translation-invariant envelopes yield weak-* convergence to ideal codewords and asymptotically isometric logical embeddings.Regularized finite blocks also converge to their ideal counterparts, enabling stable reconstruction from sufficiently accurate approximations.Source: Normalizable regularizations · p. 4

Syndrome reconstruction

  • Displacement syndromes are encoded by phase relations between translated coefficient blocks, which block correlation recovers with deterministic stability under additive perturbations.Several stabilizer directions can be combined in an overdetermined, optionally weighted least-squares estimator.Source: Syndrome reconstruction · p. 5
  • The reconstruction results concern access to complex coefficient amplitudes and do not themselves specify a physical measurement or recovery channel.The block-correlation formulas also require relative complex phases.Source: Syndrome reconstruction · p. 5

2 GKP lattices, logical fibres, and syndrome space

This section sets the lattice, Weyl-operator, and stabilizer-phase conventions used to describe GKP codes and their syndrome space. It recalls the symplectic normal form and the commutation condition for stabilizers.

2.1 Weyl operators and symplectically integral lattices

  • A full lattice is symplectically integral when its symplectic pairings are integers, equivalently when its symplectic Gram matrix has integer entries; this is precisely the condition for lattice Weyl operators to commute.Changing the lattice basis leaves the lattice unchanged and transforms the Gram matrix by integral congruence.Source: §2.1 · p. 6
  • Every full symplectically integral lattice has a symplectic normal form determined by a unique divisibility tuple.The section also notes that an integral lattice is contained in its symplectic adjoint.Source: §2.1 · pp. 6–7

2.2 Stabilizer phases

  • Every symplectically integral lattice admits a stabilizer phase, which makes the associated stabilizer operators a unitary representation of the abelian lattice group.For rectangular lattices, the paper fixes a canonical stabilizer phase.Source: §2.2 · p. 7

3 Gabor frames and Heisenberg modules

This section gathers the time–frequency tools used to connect Zak descriptions of GKP codes with adjoint-lattice coefficients. It covers modulation spaces, multi-window Gabor frames, and their Heisenberg-module structure.

3 Gabor frames and Heisenberg modules

  • The framework combines modulation-space and multi-window Gabor-frame tools with Heisenberg modules and their frame-induced idempotents.Metaplectic covariance allows later coefficient constructions to be transported between symplectically equivalent lattices.Source: §3 · p. 7

3.1 Modulation spaces

  • Modulation spaces are defined by Lp integrability of the short-time Fourier transform, with M∞ identified as the weak-* dual of M1.Weyl operators act isometrically on modulation spaces, and lattice samples of STFTs of M1 functions are absolutely summable.Source: §3.1 · p. 8

3.2 Multi-window frames and super Riesz families

  • A multi-window Gabor frame has an invertible frame operator and a canonical dual family; for M1 windows, spectral invariance keeps the dual windows in M1.The multi-window duality principle relates a Gabor frame to a Riesz sequence on the adjoint lattice, and the density condition requires m ≥ vol(Λ).Source: §3.2 · pp. 8–9

3.3 Heisenberg modules and the module idempotent

  • A multi-window frame and its canonical dual yield an idempotent in the associated matrix algebra, which need not be self-adjoint.Replacing the frame by its Parseval family can produce an orthogonal projection; smooth windows under vol(Λ) < m yield smooth projections in the corresponding matrix algebras.Source: §3.3 · p. 9

4 GKP codes and the Zak transform

The section uses Zak coordinates to describe ideal GKP codes as finite logical fibres over the syndrome torus. It connects this description to stabilizer actions and extends it from rectangular normal form to general symplectically integral lattices.

4.1 The scalar Zak transform

  • The scalar Zak transform is unitary and extends to topological isomorphisms on the modulation spaces used in the analysis.Source: §4.1 · pp. 10–11

4.2 The vector-valued Zak transform and the logical fibre

  • The vector-valued Zak transform gathers the scalar transform’s finite sheets into a logical fibre and diagonalizes canonical stabilizers.Adjoint-lattice displacements act as logical Pauli operators, while displacement errors translate the syndrome variable.Source: §4 · p. 10, §4.2 · pp. 11–12

4.3 The GKP codes

  • In M∞, stabilized states have Zak transforms supported at the trivial syndrome, with bounded order-zero coefficients determined by the value at the origin.This identifies the ideal code bijectively with the finite-dimensional logical fibre Hd.Source: §4.3 · p. 13
  • The ideal code is precisely the fibre over the trivial syndrome, and the map from codewords to their logical vectors is a linear isomorphism.Source: §4.3 · p. 14

D. By Lemma 4.1, ZDψ ∈W ∞,∞

This section describes logical fibres of GKP code spaces and how Pauli operators, metaplectic symmetries, displacement syndromes, and multi-window Gabor frames act on or realize them, extending the Zak description to general symplectically integral lattices. In particular, physical displacements act on logical information, up to a phase, through finite Weyl operators.

D. By Lemma 4.1, ZDψ ∈W ∞,∞

  • Quasiperiodicity determines all logical-fibre coefficients from the coefficient at the origin, and injectivity of the Zak transform makes the logical vector unique.The converse construction uses a lattice Dirac comb with uniformly bounded coefficients, whose rapid-decaying STFT window gives membership in W ∞,∞.Source: D. By Lemma 4.1, ZDψ ∈W ∞,∞ · pp. 14–15

4.4 Pauli operators and Clifford covariance

  • The adjoint-lattice generators act on the logical Zak fibre as the generators of the logical Pauli group.This identifies phase-space displacements with logical Pauli actions in Zak coordinates.Source: §4.4 · p. 15

4.5 Metaplectic operators and code spaces

  • Metaplectic lifts transport a code space for a lattice and stabilizer phase to the code space for the transformed lattice and phase.When the lattice is preserved, the code space is preserved precisely when the stabilizer phase is invariant under the induced transformation.Source: §4.5 · p. 16

4.5.1 Clifford gates and metaplectic transformations

  • A metaplectic symmetry preserving the code induces a unitary logical operator whose conjugation preserves the logical Pauli group up to scalar phases.The induced operator is given by the adjoint of the unitary fibre map at the origin.Source: §4.5.1 · pp. 16, 18

4.5.2 Example: The square GKP qubit and the Hadamard gate

  • For the square GKP qubit, the Fourier transform realizes the lattice’s quarter-turn symmetry and acts on the encoded qubit as the logical Hadamard gate.The logical unitary exchanges X and Z.Source: §4.5.2 · p. 19

4.6 Displacement errors and the syndrome torus

  • A displacement syndrome is recorded by the Zak support, while the vector in that fibre carries logical information; the stabilizer phase selects the syndrome fibre representing the code.Displacements preserve the lattice code with a possibly changed stabilizer phase, and applying the inverse displacement returns support to the origin.Source: §4.6 · p. 19

4.7 Construction of ideal codes from multi-window Gabor frames

  • A multi-window Gabor frame constructs ideal codewords with any prescribed logical vector by choosing coefficients through the surjective Zak matrix at the origin.The Zak matrix’s continuous full-row-rank condition gives uniform upper and lower singular-value bounds; rescaling the chosen coefficients by d^1/2 yields the prescribed vector.Source: §4.7 · pp. 20–21

4.8 Zak transform for symplectically integral lattices

  • For a general symplectically integral lattice, a metaplectic change of phase-space coordinates transports ideal codewords to rectangular normal form and identifies the logical space and displacement operators.The resulting identifications are CΛ,χ ∼= Hd ∼= CdΛ.Source: §4.8 · p. 21
  • Up to a phase, the physical displacement ρ(µ) acts on ψ’s logical information through the finite Weyl operator Wqµ.Source: §4.8 · p. 22

5 Adjoint-lattice coefficients and logical reconstruction

This section builds a stable adjoint-lattice coefficient model alongside the finite-dimensional ideal code. It relates the model-space projection to a Heisenberg-module idempotent and uses lattice covariance to represent ideal codewords by finite coefficient data.

5.1 The projected model spaces

  • Multi-window Gabor expansions converge in Mp for finite p and in the weak-* topology of M∞.Source: §5.1 · p. 22

5.2 Coefficient extraction and the model-space projection

  • The adjoint-lattice model space is the range of the Heisenberg-module idempotent, and coefficient extraction gives its unique stable coordinates.The analysis and synthesis maps remain bounded across the usual modulation-space scale.Source: §5.2 · p. 23

5.3 Adjoint-lattice covariance

  • The model space is invariant under adjoint-lattice Weyl operators, whose action on coefficients is a twisted translation.Source: §5.3 · p. 24

5.4 Realization of ideal codewords in the model space

  • For ideal codewords, coefficients in each stabilizer-lattice coset are fixed by known phases, so one representative per coset determines the full sequence.Source: §5.4 · p. 25

5.5 Finite adjoint-lattice blocks recover the logical vector

The paper encodes each logical vector in a finite block of adjoint-lattice coefficients and establishes an isometric reconstruction map. It also provides a projection onto admissible blocks and results on block periodicity, probe design, minimal sampling, and Clifford action.

5.5 Finite adjoint-lattice blocks recover the logical vector

  • The normalized finite-block map is an isometry from the logical Hilbert space onto a distinguished d-dimensional coefficient subspace.Source: §5.5 · p. 27
  • An orthogonal projection maps any coefficient block to its unique nearest admissible logical block, enabling recovery of the logical vector.Source: §5.5 · p. 27
  • Phase-corrected coefficient blocks from stabilizer-lattice translates contain the same logical information.Source: §5.5 · p. 28
  • A probe can be chosen so its associated map is injective on the ideal code and realizes any prescribed nonzero vector in the logical fibre.Source: §5.5 · p. 28
  • The logical Clifford action restricts to a unitary action on admissible coefficient blocks; in the stated representation it permutes finite phase-space indices with coefficient-dependent phases.Source: §5.5 · pp. 28–29
  • For the specified Gaussian probes, a selected family of d coefficients determines every logical vector using the minimum possible number of scalar coefficients.Source: §5.5 · p. 29

5.6 General symplectically integral lattices

The rectangular-lattice construction is transported to general symplectically integral lattices by metaplectic covariance, including a correction for the stabilizer phase. The resulting Gabor frame supports the general-lattice logical coefficient block and its norm identity.

5.6 General symplectically integral lattices

  • Metaplectic transport produces a multi-window Gabor frame and dual windows on the general lattice.The transported stabilizer phase may differ from the canonical rectangular-lattice phase by a character represented by a displacement class.Source: §5.6 · p. 30
  • Corollary 5.12 extends the logical coefficient-block construction to general symplectically integral lattices.The proof uses symplectic volume preservation and metaplectic covariance.Source: §5.6 · p. 30
  • The norm identity for the transported block follows from Theorem 5.5.The passage states that the transported block map satisfies the norm identity.Source: §5.6 · p. 30

5.7 Example: the single-mode ideal comb

The single-mode ideal comb makes the finite coefficient block explicit in terms of periodized samples of the chosen Gabor windows. For the square GKP qubit, the Hadamard gate acts on the block by a simple permutation and sign change.

5.7 Example: the single-mode ideal comb

  • The ideal comb f0 has Zak transform e0 concentrated at the origin of the syndrome torus.Its coefficients separate the comb’s logical value from the frame-dependent periodized samples.Source: §5.7 · pp. 31–32
  • For fixed r, changing s changes only the Weyl phase in the explicit coefficient formula.The formula expresses the block coefficients using the periodized samples Ar(g).Source: §5.7 · p. 32
  • For the logical basis state e0, the block’s d2 coefficients are determined by d periodized samples of the chosen Gabor frame.The other adjoint-lattice coefficients follow from the stated full-lattice relation.Source: §5.7 · p. 32

5.8 Example: block action of the Hadamard gate

  • For the square GKP qubit, Hadamard exchanges coefficients (0, 1) and (1, 0), fixes (0, 0), and negates (1, 1).The example chooses a probe that is an eigenvector of Hadamard.Source: §5.8 · p. 33

6 Normalizable lattice envelopes and logical coefficient recovery

The section constructs normalizable GKP approximants by applying lattice-envelope Gabor multipliers to ideal codewords, then studies their logical geometry and coefficient recovery. The results distinguish normalizability from finite energy and give qualitative asymptotic convergence under stated envelope conditions.

6 Normalizable lattice envelopes and logical coefficient recovery

  • The general lattice-envelope construction yields L2-normalizable approximants, while finite mean oscillator energy requires additional weighted regularity in position and momentum.The section uses “normalizable” for the general result.Source: §6 · p. 33

6.2 Lattice-envelope regularization and Gabor multipliers

  • A lattice-envelope regularization is a Gabor multiplier: the localized seed carries logical information and the symbol controls decay along the stabilizer orbit.The ideal codeword corresponds formally to the constant symbol, while decaying symbols produce normalizable approximations; every ℓ2-symbol produces a normalizable state.Source: §6.2 · p. 34
  • Envelopes approaching the constant symbol converge weak-* to the ideal codeword, and approximate stabilization is controlled by symbol translation invariance.The symbol families considered are summable, uniformly bounded, and converge pointwise to 1.Source: §6.2 · p. 35

6.3 Asymptotically isometric normalizable encoding

  • After explicit normalization, asymptotically translation-invariant envelopes yield asymptotically isometric logical encodings, with qualitative operator-norm convergence.The result is uniform on bounded logical-vector sets; unlike the compared number-operator regularization, this construction provides no quantitative error estimate.Source: §6.3 · p. 36

6.4 Recovery from regularized coefficients

  • The regularized finite coefficient blocks converge to their ideal counterparts, enabling logical recovery by a least-squares left inverse and an orthogonal projection onto admissible blocks.The recovery map’s stability follows from convergence of the block maps in operator norm.Source: §6.4 · p. 37

7 Syndrome information in coefficient space

The paper extracts displacement syndromes from phase relations among translated adjoint-lattice coefficient blocks. It develops block correlation, deterministic noise stability, redundant phase estimation, and the persistence of syndrome recovery for regularized states under weak-* approximation.

7.1 Block syndrome covariance

  • A displacement changes stabilizer eigenvalues by a lattice character, which appears as a phase relation between translated finite coefficient blocks.The paper proposes recovering this phase by block correlation and combining stabilizer directions for redundant estimation.Source: §7.1 · p. 38, §7 · p. 38

7.2 Block-correlation extraction of the syndrome

  • Block correlation extracts the syndrome character without requiring any individual coefficient to be nonzero.The method uses the entire finite block rather than a single reference coefficient.Source: §7.2 · p. 39, §7.1 · p. 39

7.3 Stability under coefficient noise

  • The redundant finite Weyl block supports a deterministic coefficient-perturbation stability bound for the estimated syndrome character.The estimator is formed from observed complex coefficient blocks.Source: §7.3 · p. 40

7.4 Redundant syndrome estimation

  • A lattice basis supplies enough phases to recover the displacement class without noise; additional generating directions provide redundant measurements that can improve robustness under coefficient noise.The overdetermined phase system permits weighted or robust fitting.Source: §7.4 · pp. 41–42

7.5 Regularized states

  • For regularized states, failure of exact block covariance is controlled by envelope translation defect, and weak-* approximation makes the block-correlation estimator converge to the ideal syndrome character.Absolute block-covariance convergence alone does not ensure convergence of the correlation quotient if the normalized reference block tends to zero.Source: §7.5 · pp. 42–43

8 Conclusion and outlook

The conclusion presents a time–frequency framework for lattice GKP codes, representing ideal codewords through a finite logical fibre and adjoint-lattice coefficients. It establishes stable logical reconstruction, constructs convergent normalizable approximants, and recovers displacement syndromes from translated-block relations.

8 Conclusion and outlook

  • Ideal codewords lie in M∞, where the vector-valued Zak transform identifies them with a finite-dimensional logical fibre at the trivial point of the syndrome torus.This choice excludes derivative-type solutions of the stabilizer equations.Source: §8 · p. 44
  • The normalized fundamental block of adjoint-lattice Gabor coefficients is an isometric representation of the logical vector.Multi-window Gabor analysis supplies stable localized coordinates.Source: §8 · p. 44
  • An explicit orthogonal projection maps arbitrary coefficient data onto the space of admissible exact blocks.Source: §8 · p. 44
  • Lattice-envelope Gabor multipliers produce normalizable approximants that converge weak-* in M∞ and yield asymptotically isometric encodings; finite oscillator energy requires additional regularity.Source: §8 · p. 44
  • Block correlations recover displacement-syndrome characters from translated coefficient blocks, with deterministic stability under additive perturbations.The results concern complex time–frequency coefficients, not a physical syndrome-extraction protocol.Source: §8 · p. 44
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