Measurement-free code-switching protocol for low-overhead quantum computation using permutation-invariant codes

Transversal gates on quantum error correction codes have been a promising approach for fault-tolerant quantum computing, but are limited by the Eastin-Knill no-go theorem. Existing solutions such as gate teleportation and magic state distillation are resource-intensive. We present a measurement-free code-switching protocol for universal quantum computation, switching between a stabilizer code for transversal Clifford gates and a permutation-invariant (PI) code for transversal non-Clifford gates that are logical Z rotations for any rational multiple of π. The novel non-Clifford gates enabled by this code-switching protocol provide for a lower gate count implementation of a universal gate set relative to the Clifford + T gate set. To achieve this, we present a protocol for performing controlled-not gates between the codes using near-term quantum control operations that employ a catalytic bosonic mode. We also present a new class of PI codes with tunable code distance, supporting transversal non-Clifford gates, and demonstrate their reduced gate count overhead relative to a comparable stabilizer code to stabilizer code-switching scheme.