Fault-tolerant non-Clifford Gottesman-Kitaev-Preskill gates using polynomial phase gates and on-demand noise biasing

The Gottesman-Kitaev-Preskill (GKP) error correcting code uses a bosonic mode to encode a logical qubit and has the attractive property that its logical Clifford gates can be implemented using Gaussian unitary gates. In contrast, a direct unitary implementation of the T gate using the cubic phase gate has been shown to have a logical error floor unless the GKP codestate has a biased-noise profile. In this work, we propose a method for on-demand noise biasing based on a standard GKP error-correction circuit. This on-demand biasing circuit can be used to bias the GKP codestate before a T gate and return it to a nonbiased state afterward. With the on-demand biasing circuit, we prove that the logical error rate of the T gate can be made arbitrarily small as the quality of the GKP codestates increases. We complement our proof with a numerical investigation of the cubic phase gate subject to a phenomenological noise model, showing that the T gate can achieve average gate fidelities above 99 % , with 12 dB of GKP squeezing without the use of postselection. Moreover, we develop a formalism for finding optimal unitary representations of logical diagonal gates in higher levels of the Clifford hierarchy that is based on a framework of “polynomial phase stabilizers” whose exponents are polynomial functions of one of the quadrature operators. This formalism naturally extends to multiqubit logical gates and even to number-phase bosonic codes, providing a powerful algebraic tool for analyzing non-Clifford gates in bosonic quantum codes.