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Communications in Theoretical Physics

Deriving T-gates from cubic phase gates via GKP encoding

Lingxuan Feng, Shunlong Luo

Published Mar 26, 2025• 3 citations
Publisher

Abstract

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).

DOI
10.1088/1572-9494/adc54c

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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).
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.
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)
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.
Physical Gottesman-Kitaev-Preskill (GKP) states are inherently noisy as ideal ones would require infinite energy. While this is typically considered as a deficiency to be actively corrected, this work demonstrates that imperfect GKP stabilizer states can be leveraged in order to apply non-Clifford gates using only linear optical elements. In particular, Gaussian operations on normalizable GKP states, combined with homodyne measurements, permit two key primitives: clean projection onto Pauli eigenstates in the normalizable GKP codespace, thereby implementing Clifford gates with high fidelity; and probabilistic projection of unmeasured modes onto non-Pauli eigenstates. These results demonstrate that normalizable GKP stabilizer states combined with Gaussian operations provide a practical framework for computational universality within the measurement-based model of quantum computation in a realistic continuous-variable setting.
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.
Although the similarity between nonstabilizer states—also known as magic states—in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer Rényi entropy—a recent proposal for a magic measure for multiqubit states. We further provide the magic measure with an operational interpretation as a runtime of a classical simulation algorithm. In addition, we employ our results to prove that implementing a multiqubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.