GKP code

The Gottesman-Kitaev-Preskill code embeds a finite-dimensional logical system in a periodic oscillator phase-space lattice. Ideal codewords are common eigenstates of displacement operators and have infinite energy, so every physical realization must specify finite squeezing, envelope, and energy constraints.

The single-mode square GKP code is generated by two reciprocal phase-space stabilizer displacements, while logical Paulis are displacement cosets in the dual lattice but outside the stabilizer lattice. General n-mode GKP codes use a 2n-dimensional symplectic lattice and its dual quotient.

It targets small displacement noise in q and p, discretizing continuous shifts modulo the lattice into logical Pauli errors. The correctable region is a lattice Voronoi cell, while finite squeezing adds unavoidable analog noise to the syndrome itself.

Qubit-style [[n,k,d]] parameters do not fully describe a GKP code. Relevant quantities include lattice covolume, shortest dual logical vector, mean photon number, peak and envelope widths, and effective squeezing in dB; comparisons must normalize energy resources.

Gaussian symplectic transformations implement many encoded Clifford operations, and displacements support Pauli-frame and syndrome feedback. Universality needs a non-Gaussian resource, non-Clifford ancilla, or an interface to a discrete-variable code.

Maximum-likelihood decoding under displacement noise is a closest-vector or coset-likelihood problem on the phase-space lattice. Analog decoding uses the continuous syndrome remainder and generally outperforms first hard-quantizing it to a discrete bit.

Finite-energy GKP correction propagates analog displacements through ancillas, couplings, and measurements. Fault-tolerance analysis must track displacement distributions, non-Gaussian tails, loss, and preparation errors; ideal delta-comb states systematically understate risk.

Candidate carriers include microwave cavities, trapped-ion motion, and optical modes. Experimental code identity should be supported by stabilizer measurements, finite-energy codewords, repeated correction gain, and logical lifetime rather than phase-space fringes alone.

GKP is a central bosonic lattice code and is often concatenated as an inner code with repetition, surface, or multimode Gaussian codes. Square, hexagonal, and multimode lattices have different geometry at equal energy and should not be collapsed into one generic GKP point.

A statement about correcting shifts below half a lattice spacing must specify norm, lattice, and finite-energy approximation. Squeezing dB, mean photon number, and logical error rate are not interchangeable across different envelope conventions.