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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>Z</a:mi> </a:math> rotations for any rational multiple of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>π</c:mi> </c:math> . 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 + <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>T</e:mi> </e:math> gate set. To achieve this, we present a protocol for performing controlled- 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.