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  • Open Access

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

Minh T. P. Nguyen1,2,* and Mackenzie H. Shaw1,3

  • *Contact author: m.t.phamnguyen@tudelft.nl
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Phys. Rev. Research 8, 033107 – Published 27 July, 2026

DOI: https://doi.org/10.1103/2pg9-nsn8

Abstract

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.

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Phys. Rev. Research 8, 033107– Published 27 July, 2026
  • Received 7 December 2025
  • Revised 30 March 2026
  • Accepted 3 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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