Bosonic Pauli+: Efficient Simulation of Concatenated GKP Codes

A promising route towards fault-tolerant error correction is the concatenation of a Gottesman-Kitaev-Preskill (GKP) code with a qubit code. Development of such concatenated codes requires simulation tools which realistically model noise, while being able to simulate the dynamics of many modes. However, so far, large-scale simulation tools for concatenated GKP codes have been limited to idealized noise models and GKP code implementations. Here, we introduce the Bosonic Pauli+ model (BP+), which can be simulated efficiently for a large number of modes, while accurately capturing the rich dynamics in the bosonic multi-mode Hilbert space for a realistic finite-energy GKP code stabilized with the sBs protocol, with given physical decoherence rates. BP+ relies on a new decomposition of the GKP Hilbert space into a logical and an error subsystem, which we call the sBs basis. Confidence in the accuracy of BP+ is gained by comparing predictions of BP+ and full time evolution simulations, for several deep quantum circuits of interest. Using BP+, logical error rates of a concatenated code implementation are presented. BP+ may also be applicable to other bosonic codes.