Encoding qubits in multimode grid states - Part 1

Bosonic error-correction codes leverage the continuous variables of the Hilbert space to design low-error quantum memories. The Gottesman-Kitaev-Preskill (GKP) code is an example of a bosonic code that provides logical encoding of qubit in a phase space lattice of an oscillator mode. Recent experiments in microwave cavities and trapped ions show promising realization of these codes. In order to further improve over single mode codes, one can concatenate such codes with a qubit code. In this talk, we take a different approach, studying codes based on lattices in the 2n-dimensional phase space of n oscillators. We highlight the advantages of this approach with the specific example of a two-mode code based on a hypercubic lattice. Compared to single mode codes, this code consists of shorter stabilizers and longer logical operators which consequently provides increased protection against errors while reducing ancilla-induced faults during stabilizer measurements. We also discuss the gate set for this code, as well as more general relations between lattice symmetries and logical gates.