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arXiv:2610.35513

Geometric aspects of fault tolerance

Jonathan Conrad

quant-phmath-phmath.DGSubmitted Oct 26, 2026• Announced Oct 27, 2026• 0 citations
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Abstract

We highlight three geometric aspects of fault tolerance. First, we construct an operator-algebraic formulation of the fiber bundle framework for fault tolerance, and show that a generalization of the Knill-Laflamme conditions implies a flat connection for this framework. Next, we develop this framework for Gottesman-Kitaev-Preskill (GKP) codes. We show that, for each polarization type, their moduli space embeds smoothly into a Grassmannian of finite-dimensional subspaces of a regularized distribution space. Finally, we explain how GKP codes associated with Jacobians give rise to magic states that can be obtained by code space projections of Gaussian states and identify a $\rm CCZ$-type magic state arising this way.

AI Summary

The paper asks how fault-tolerant logical evolution can be described geometrically beyond unitary dynamics, and how this description applies to GKP codes and magic-state preparation. It develops three related constructions using operator-algebra quantum error correction, theta functions and polarized abelian varieties. First, logical algebras are represented by completely positive idempotents with encoding/decoding frames. Generalized, pairwise Knill–Laflamme conditions compare both different channels and different encodings. These imply isometric transport and locally constant frame transitions; on smooth correctable submanifolds, the connection $\omega=\Lambda\,d\Delta$ is flat, with logical operations described by monodromy. Convex mixtures of jointly correctable channels provide smooth embedded patches of dimension $\dim_{\mathbb R}\operatorname{span}\{\mathcal E_s\Delta\}-1$. Second, for fixed GKP polarization type $D$ and every regularization parameter $\beta>0$, the code moduli space embeds smoothly into a Grassmannian of $(\det D)^2$-dimensional operator-kernel spaces. Local upper and lower Lipschitz bounds establish the parameter–subspace correspondence; global embedding uses stabilizer recovery and lattice compactness. The projectively flat theta-bundle connection induces a flat connection on logical operators. Finally, projection of Gaussian states onto Jacobian-associated GKP codes produces theta-function amplitudes. Explicit genus-three examples include a Klein-quartic state and an exact $\mathrm{CCZ}$ state from an octahedral hyperelliptic curve, at parameters satisfying $\lambda(2it)=1/4$. The flatness criteria are sufficient conditions requiring exact joint correctability. Assembling mixture patches additionally requires relative openness and common dimension. The GKP embedding uses regularized distribution spaces, and the magic-state constructions assume nonzero code-space projection.

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We highlight three geometric aspects of fault tolerance. First, we construct an operator-algebraic formulation of the fiber bundle framework for fault tolerance, and show that a generalization of the Knill-Laflamme conditions implies a flat connection for this framework. Next, we develop this framework for Gottesman-Kitaev-Preskill (GKP) codes. We show that, for each polarization type, their moduli space embeds smoothly into a Grassmannian of finite-dimensional subspaces of a regularized distribution space. Finally, we explain how GKP codes associated with Jacobians give rise to magic states that can be obtained by code space projections of Gaussian states and identify a CCZ\rm CCZCCZ-type magic state arising this way.
The paper asks how fault-tolerant logical evolution can be described geometrically beyond unitary dynamics, and how this description applies to GKP codes and magic-state preparation. It develops three related constructions using operator-algebra quantum error correction, theta functions and polarized abelian varieties.
First, logical algebras are represented by completely positive idempotents with encoding/decoding frames. Generalized, pairwise Knill–Laflamme conditions compare both different channels and different encodings. These imply isometric transport and locally constant frame transitions; on smooth correctable submanifolds, the connection ω=Λ dΔ\omega=\Lambda\,d\Deltaω=ΛdΔ is flat, with logical operations described by monodromy. Convex mixtures of jointly correctable channels provide smooth embedded patches of dimension dim⁡Rspan⁡{EsΔ}−1\dim_{\mathbb R}\operatorname{span}\{\mathcal E_s\Delta\}-1dimR​span{Es​Δ}−1.
Second, for fixed GKP polarization type DDD and every regularization parameter β>0\beta>0β>0, the code moduli space embeds smoothly into a Grassmannian of (det⁡D)2(\det D)^2(detD)2-dimensional operator-kernel spaces. Local upper and lower Lipschitz bounds establish the parameter–subspace correspondence; global embedding uses stabilizer recovery and lattice compactness. The projectively flat theta-bundle connection induces a flat connection on logical operators.
Finally, projection of Gaussian states onto Jacobian-associated GKP codes produces theta-function amplitudes. Explicit genus-three examples include a Klein-quartic state and an exact CCZ\mathrm{CCZ}CCZ state from an octahedral hyperelliptic curve, at parameters satisfying λ(2it)=1/4\lambda(2it)=1/4λ(2it)=1/4.
The flatness criteria are sufficient conditions requiring exact joint correctability. Assembling mixture patches additionally requires relative openness and common dimension. The GKP embedding uses regularized distribution spaces, and the magic-state constructions assume nonzero code-space projection.
We investigate multi-mode GKP (Gottesman–Kitaev–Preskill) quantum error-correcting codes from a geometric perspective. First, we construct their moduli space as a quotient of groups and exhibit it as a fiber bundle over the moduli space of symplectically integral lattices. We then establish the Gottesman–Zhang conjecture for logical GKP Clifford operations, showing that all such gates arise from parallel transport with respect to a flat connection on this space. Specifically, non-trivial Clifford operations correspond to topologically non-contractible paths on the space of GKP codes, while logical identity operations correspond to contractible paths.
We investigate multi-mode GKP (Gottesman--Kitaev--Preskill) quantum error-correcting codes from a geometric perspective. First, we construct their moduli space as a quotient of groups and exhibit it as a fiber bundle over the moduli space of symplectically integral lattices. We then establish the Gottesman--Zhang conjecture for logical GKP Clifford operations, showing that all such gates arise from parallel transport with respect to a flat connection on this space. Specifically, non-trivial Clifford operations correspond to topologically non-contractible paths on the space of GKP codes, while logical identity operations correspond to contractible paths.
We investigate multi-mode GKP (Gottesman–Kitaev–Preskill) quantum error-correcting codes from a geometric perspective. First, we construct their moduli space as a quotient of groups and exhibit it as a fiber bundle over the moduli space of symplectically integral lattices. We then establish the Gottesman–Zhang conjecture for logical GKP Clifford operations, showing that all such gates arise from parallel transport with respect to a flat connection on this space. Specifically, non-trivial Clifford operations correspond to topologically non-contractible paths on the space of GKP codes, while logical identity operations correspond to contractible paths.
We explain how GKP Clifford gates arise as symplectic automorphisms of the corresponding GKP lattice and show that, for scaled codes of type D=dInD=dI_nD=dIn​, their integral action agrees with the action of the mapping class group of a genus-nnn surface on first homology. This correspondence introduces a topological interpretation of fault tolerance for GKP codes and motivates the connection between GKP codes (lattices), their Clifford gates, and algebraic curves, which we explore in depth. For a single-mode GKP code, we identify the space of oriented fixed-covolume lattice realizations with S3−KS^3 - KS3−K, the three sphere S3S^3S3 with a trefoil knot KKK removed, and explain how logical degrees of freedom arise from the choice of a level structure on the corresponding curves. Specified Clifford implementations define loops in the space of lattices whose trefoil linking number is a topological invariant. Finally, we relate our observations to the idea of fiber bundle fault tolerance as proposed by Gottesman and Zhang for the GKP code, where logical Clifford and Pauli operations of the single-mode GKP code arise as the monodromy representation of a finite covering of the space of nonzero-distance GKP codes.
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.
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)H^0(X, L)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.