Deriving <i>T</i>-gates from cubic phase gates via GKP encoding

Abstract In discrete-variable quantum computation, non-Clifford T -gates play a pivotal role in achieving genuine quantum advantage. However, in the literature, T -gates are only defined for prime dimensional systems, and the absence of their high-dimensional counterparts in non-prime dimensional systems raises the issue of how to introduce certain T -gates in such systems. In continuous-variable quantum computation, universality is achieved through a combination of Gaussian gates and some non-Gaussian gates, such as the widely used cubic phase gates. In this work, we establish some connections between discrete-variable (Clifford/non-Clifford) gates and continuous-variable (Gaussian/non-Gaussian) gates via the powerful GKP (Gottesman–Kitaev–Preskill) encoding, which maps qudits to oscillators and serves as a bridge between discrete and continuous realms. By exploiting the analogies between the Clifford hierarchy and the Gaussian hierarchy, we derive (discrete-variable) T -gates in arbitrary (not necessarily prime) dimensional systems from well established (continuous-variable) cubic phase gates. We reveal some basic properties of the unified T -gates, and make a comparative study of various T -gates. As an application, we employ the T -gates to construct equidistributed n -angular frames and certain MUBs (mutually unbiased bases).