Efficient Magic State Cultivation for sqrt{T} Gates

Recently, experimental and theoretical quantum error correction methodology has seen remarkable breakthroughs. In particular, magic state cultivation has been shown to simplify magic-state preparation and make it feasible for near-term devices. However, recent research on magic state cultivation has focused primarily on the cultivation of Tleft| + right>_L. Only a few other magic state cultivation methods beyond Tleft| + right>_L have been investigated. Here, we generalize phase kickback checks for magic states at arbitrary Clifford hierarchy levels in specific codes. We provide an example of cultivation of sqrt{T}left| + right>_L in the doubled color code and the corresponding escape strategy using lattice surgery from the color code to large rotated surface codes. Using state vector simulation for un-grown cultivation, we observe a strong consistence between Sleft| + right>_L and sqrt{T}left| + right>_L cultivation's performance on the doubled color code. Finally, we discuss the application of the corresponding sqrt{T}left| + right>_L cultivation, incorporating the STAR architecture and T gates, for early fault-tolerant quantum computing and its potential to shorten gate synthesis in the fully fault-tolerant quantum computing era.