Qubit CSS codes with Universal Transversal Gates

Abstract:The Eastin-Knill theorem rules out a universal set of transversal logical gates on any code that detects single-qubit errors. The theorem constrains one partition of the qubits at a time. We show that depth-one layers of few-qubit gates, each respecting its own partition, can together generate a universal logical gate set on an error-correcting code. Each logical gate generator is then a single depth-one layer followed by error correction of the same stabilizer code, with no magic-state distillation, code switching, gauge fixing, lattice surgery, or other gadgets. We show that such codes can be found by applying an entangling circuit to a lower-distance "seed" code admitting a depth-one non-Clifford gate and an appropriate "kernel" code. We construct 47 codes with depth-one generating sets that tolerate at least one faulty gate per layer, including a [[37,1,7]] and a [[59,1,9]] code, both of check weight at most eight.

25 pages, 5 figures, 6 tables. The ancillary file this http URL lists all 110 codes with their stabilizers, logical operators, depth-one generators, seed and kernel codes, and coupling circuits

arXiv:2610.06730 [quant-ph]

(or arXiv:2610.06730v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.06730 arXiv-issued DOI via DataCite (pending registration)