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arXiv:2009.05309 · Physical Review A

Unsuitability of cubic phase gates for non-Clifford operations on Gottesman-Kitaev-Preskill states

Jacob Hastrup, Mikkel V. Larsen, Jonas Schou Neergaard-Nielsen, Nicolas C. Menicucci, Ulrik L. Andersen

Published Mar 10, 2021• 27 citations
arXivHTMLPDF

Abstract

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 $T$ 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.

DOI
10.1103/physreva.103.032409

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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.
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.
The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.
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.