Lattices, Gates, and Curves: GKP codes as a Rosetta stone

arXiv:2407.03270v3 Announce Type: replace Abstract: We explain how GKP Clifford gates arise as symplectic automorphisms of the corresponding GKP lattice and show that, for scaled codes of type $D=dI_n$, their integral action agrees with the action of the mapping class group of a genus-$n$ 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 $S^3 - K$, the three sphere $S^3$ with a trefoil knot $K$ 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.

Abstract:We explain how GKP Clifford gates arise as symplectic automorphisms of the corresponding GKP lattice and show that, for scaled codes of type $D=dI_n$, their integral action agrees with the action of the mapping class group of a genus-$n$ 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 $S^3 - K$, the three sphere $S^3$ with a trefoil knot $K$ 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.

31 pages, 13 figures. v2 is significantly revised with a clearer presentation and general corrections

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Differential Geometry (math.DG); Geometric Topology (math.GT)

arXiv:2407.03270 [quant-ph]

(or arXiv:2407.03270v3 [quant-ph] for this version)

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Lattices, Gates, and Curves: GKP codes as a Rosetta stone

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