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arXiv:2511.20355 · Physical Review Research

Fault-Tolerant Non-Clifford GKP Gates using Polynomial Phase Gates and On-Demand Noise Biasing

Minh T. P. Nguyen, Mackenzie H. Shaw

quant-phSubmitted Nov 25, 2025• Announced Nov 26, 2025• 0 citations• Revised Aug 18, 2026 (v2)
v1 · Nov 25, 2025v2 · Aug 18, 2026
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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 logical error floor unless the GKP codestate has a biased noise profile [1]. 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 non-biased state afterwards. 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 multi-qubit logical gates and even to number-phase bosonic codes, providing a powerful algebraic tool for analyzing non-Clifford gates in bosonic quantum codes. [1] J. Hastrup, M. V. Larsen, J. S. Neergaard-Nielsen, N. C. Menicucci, and U. L. Andersen, Phys. Rev. A 103, 032409 (2021)

DOI
10.1103/2pg9-nsn8
Comments
44 pages, 8 figures. Add feedback from Reviewers. Update DOI to the published version

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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{T}T gate using the cubic phase gate has been shown to have logical error floor unless the GKP codestate has a biased noise profile [1]. 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 TTT gate and return it to a non-biased state afterwards. With the on-demand biasing circuit, we prove that the logical error rate of the TTT 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{T}T gate can achieve average gate fidelities above 99%99\%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 multi-qubit logical gates and even to number-phase bosonic codes, providing a powerful algebraic tool for analyzing non-Clifford gates in bosonic quantum codes.
[1] J. Hastrup, M. V. Larsen, J. S. Neergaard-Nielsen, N. C. Menicucci, and U. L. Andersen, Phys. Rev. A 103, 032409 (2021)
With the Gottesman-Kitaev-Preskill (GKP) encoding, Clifford gates and error correction can be carried out using simple Gaussian operations. Still, non-Clifford gates, required for universality, require non-Gaussian elements. In their original proposal, GKP suggested a particularly simple method of using a single application of the cubic phase gate to perform the logical non-Clifford TTT gate. Here we show that this cubic phase gate approach performs extraordinarily poorly, even for arbitrarily large amounts of squeezing in the GKP state. Thus, contrary to common belief, the cubic phase gate is not suitable for achieving universal fault-tolerant quantum computation with GKP states.
Achieving a fault-tolerant universal quantum computer with an optical system is one of the goals of quantum optics. For that purpose, the most promising candidate is the Gottesman-Kitaev-Preskill (GKP) encoding of qubits in the continuous-variable system [1] . In order to perform universal computation with GKP qubits, the current open problem is the realization of non-Clifford operations, which require non-Gaussianity. In the original GKP’s paper where [1] , two different approaches are suggested to implement a non-Clifford T gate, i.e. T^=∣0L⟩⟨0L∣+eiπ4∣1L⟩⟨1L∣\hat T = \left| {{0_{\text{L}}}} \right\rangle \left\langle {{0_{\text{L}}}\left| { + {{\text{e}}^{i\frac{\pi }{4}}}} \right|{1_{\text{L}}}} \right\rangle \left\langle {{1_{\text{L}}}} \right|T^=∣0L​⟩⟨0L​​+ei4π​​1L​⟩⟨1L​∣ , where |0 L 〉 and |1 L 〉 are the logical bases of the GKP qubits. The first approach is using a cubic phase gate, a non-Gaussian operation, combined with Gaussian operations. In this context, the cubic phase gate has been widely developed with nonlinear feedforward as a key technology [2] , [3] . In recent research, however, it is shown that this approach is not optimal for GKP qubits [4] . Hence, the second approach, which is based on the magic state injection method [5] , is more promising.
In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.
The Gottesman-Kitaev-Preskill (GKP) code is an exciting route to fault-tolerant quantum computing since Gaussian resources and GKP Pauli-eigenstate preparation are sufficient to achieve universal quantum computing. However, there is a disconnect between the noise model that GKP qubits are in theory designed to correct—uniform random displacement errors—and the conditions that affect GKP qubits in superconducting devices in practice: realistic noise channels, logical gates, and inefficient measurements. In this work we bridge this gap in three ways. First, we approximate the effect loss and dephasing on approximate GKP codestates using a random displacement channel, and show that this approximation matches well with numerics. Second, we analyze the error-spreading properties of GKP Clifford gates and describe how a modification in the decoder following the implementation of each gate can reduce the gate infidelity by multiple orders of magnitude. Finally, we consider the effect of homodyne measurement inefficiencies on logical state readout and analyze a scheme to improve the measurement efficiency using the theory of quantum trajectories.
In a continuous-variable optical system, the Gottesman-Kitaev-Preskill (GKP) qubit is a promising candidate for fault-tolerant quantum computation. To implement non-Clifford operations on GKP qubits, non-Gaussian operations are required. In this context, the implementation of a cubic phase gate by combining nonlinear feedforward with ancillary states has been widely researched. Recently, however, it was pointed out that the cubic phase gate is not the most suitable for non-Clifford operations on GKP qubits. In this paper, we show that we can achieve linear optical implementation of non-Clifford operations on GKP qubits with high fidelity by applying the nonlinear feedforward originally developed for the cubic phase gate and using a GKP-encoded ancillary state. Our paper shows the versatility of the nonlinear feedforward technique---important for optical implementation of the fault-tolerant continuous-variable quantum computation.
Concatenation of a bosonic code with a qubit code is one of the promising ways to achieve fault-tolerant quantum computation. As one of the most important bosonic codes, the Gottesman-Kitaev-Preskill (GKP) code is proposed to correct small displacement errors in phase space. If the noise in phase space is biased, the square-lattice GKP code can be concatenated with the repetition code that promises a high fault-tolerant threshold to suppress the logical error. In this work, we study the performance of GKP repetition codes with finite-energy ancillary GKP qubits in correcting biased noise. We find that there exists a critical value of noise variance for the ancillary GKP qubit such that the logical Pauli error rate decreases when increasing the code size. Furthermore, one round of GKP error correction has to be performed before concatenating with the repetition code. Our study paves the way for practical implementation of error correction by concatenating the GKP code with low-level qubit codes.