Papers › Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays
Computation with quantum Reed-Muller codes and their mapping onto 2D atom arrays
Anqi Gong, Joseph M. Renes
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We give a fault tolerant construction for error correction and computation using two punctured quantum Reed-Muller (PQRM) codes. In particular, we consider the [[127,1,15]] self-dual doubly-even code that has transversal Clifford gates (CNOT, H, S) and the triply-even [[127,1,7]] code that has transversal T and CNOT gates. We show that code switching between these codes can be accomplished using Steane error correction. For fault-tolerant ancilla preparation we utilize the low-depth hypercube encoding circuit along with different code automorphism permutations in different ancilla blocks, while decoding is handled by the high-performance classical successive cancellation list decoder. In this way, every logical operation in this universal gate set is amenable to extended rectangle analysis. The CNOT exRec has a failure rate approaching 10⁻⁹ at 10⁻³ circuit-level depolarizing noise. Furthermore, we map the PQRM codes to a 2D layout suitable for implementation in arrays of trapped atoms and try to reduce the circuit depth of parallel atom movements in state preparation. The resulting protocol is strictly fault-tolerant for the [[127,1,7]] code and practically fault-tolerant for the [[127,1,15]] code. Moreover, each patch requires a permutation consisting of $7$ sub-hypercube swaps only. These are swaps of rectangular grids in our 2D hypercube layout and can be naturally created with acousto-optic deflectors (AODs). Lastly, we show for the family of [[2²ʳ,2rr,2ʳ]] QRM codes that the entire logical Clifford group can be achieved using only permutations, transversal gates, and fold-transversal gates.
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