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Register a new userAll-Optical Long-Distance Quantum Communication with Gottesman-Kitaev-Preskill qubits
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Quantum repeaters are a promising platform for realizing long-distance quantum communication and thus could form the backbone of a secure quantum internet, a scalable quantum network, or a distributed quantum computer. Repeater protocols that encode information in single- or multi-photon states are limited by transmission losses and the cost of implementing entangling gates or Bell measurements. In this work, we consider implementing a quantum repeater protocol using Gottesman-Kitaev-Preskill (GKP) qubits. These qubits are natural elements for quantum repeater protocols, because they allow for deterministic Gaussian entangling operations and Bell measurements, which can be implemented at room temperature. The GKP encoding is also capable of correcting small displacement errors. At the cost of additional Gaussian noise, photon loss can be converted into a random displacement error channel by applying a phase-insensitive amplifier. Here we show that a similar conversion can be achieved in two-way repeater protocols by using phase-sensitive amplification applied in the post-processing of the measurement data, resulting in less overall Gaussian noise per (sufficiently short) repeater segment. We also investigate concatenating the GKP code with higher level qubit codes while leveraging analog syndrome data, post-selection, and path-selection techniques to boost the rate of communication. We compute the secure key rates and find that GKP repeaters can achieve a comparative performance relative to methods based on photonic qubits while using orders-of-magnitude fewer qubits.
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The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode is a promising bosonic code for quantum computation due to its tolerance for noise and all-Gaussian gate set. We present a toolkit for phase-space description and manipulation of GKP encodings that includes Wigner functions for ideal and approximate GKP states, for various types of mixed GKP states, and for GKP-encoded operators. One advantage of a phase-space approach is that Gaussian unitaries, required for computation with GKP codes, correspond to simple transformations on the arguments of Wigner functions. We use this fact and our toolkit to describe GKP error correction, including magic-state preparation, entirely in phase space using operations on Wigner functions. While our focus here is on the square-lattice GKP code, we provide a general framework for GKP codes defined on any lattice.
The Gottesman-Kitaev-Preskill (GKP) quantum error correcting code attracts much attention in continuous variable (CV) quantum computation and CV quantum communication due to the simplicity of error correcting routines and the high tolerance against Gaussian errors. Since the GKP code state should be regarded as a limit of physically meaningful approximate ones, various approximations have been developed until today, but explicit relations among them are still unclear. In this paper, we rigorously prove the equivalence of these approximate GKP codes with an explicit correspondence of the parameters. We also propose a standard form of the approximate code states in the position representation, which enables us to derive closed-from expressions for the Wigner functions, the inner products, and the average photon numbers in terms of the theta functions. Our results serve as fundamental tools for further analyses of fault-tolerant quantum computation and channel coding using approximate GKP codes.
Scalability of flying photonic quantum systems in generating quantum entanglement offers a potential for implementing large-scale fault-tolerant quantum computation, especially by means of measurement-based quantum computation (MBQC). However, existing protocols for MBQC inevitably impose a polynomial overhead cost in implementing quantum computation due to geometrical constraints of entanglement structures used in the protocols, and the polynomial overhead potentially cancels out useful polynomial speedups in quantum computation. To implement quantum computation without this cancellation, we construct a protocol for photonic MBQC that achieves as low as poly-logarithmic overhead, by introducing an entanglement structure for low-overhead qubit permutation. Based on this protocol, we design a fault-tolerant photonic MBQC protocol that can be performed by experimentally tractable homodyne detection and Gaussian entangling operations combined with the Gottesman-Kitaev-Preskill (GKP) quantum error-correcting code, which we concatenate with the $7$-qubit code. Our fault-tolerant protocol achieves the threshold $7.8$ dB in terms of the squeezing level of the GKP code, outperforming $8.3$ dB of the best existing protocol for fault-tolerant quantum computation with the GKP surface code. Thus, bridging a gap between theoretical progress on MBQC and photonic experiments towards implementing MBQC, our results open a new way towards realization of a large class of quantum speedups including those polynomial.
Graph states are a central resource in measurement-based quantum information processing. In the photonic qubit architecture based on Gottesman-Kitaev-Preskill (GKP) encoding, the generation of high-fidelity graph states composed of realistic, finite-energy approximate GKP-encoded qubits thus constitutes a key task. We consider the finite-energy approximation of GKP qubit states given by a coherent superposition of shifted finite-squeezed vacuum states, where the displacements are Gaussian distributed. We present an exact description of graph states composed of such approximate GKP qubits as a coherent superposition of a Gaussian ensemble of randomly displaced ideal GKP-qubit graph states. We determine the transformation rules for the covariance matrix and the mean displacement vector of the Gaussian distribution of the ensemble under tools such as GKP-Steane error correction and fusion operations that can be used to grow large, high-fidelity GKP-qubit graph states. The former captures the noise in the graph state due to the finite-energy approximation of GKP qubits, while the latter relates to the possible absolute displacement errors on the individual qubits due to the homodyne measurements that are a part of these tools. The rules thus help in pinning down an exact coherent error model for graph states generated from truly finite-energy GKP qubits, which can shed light on their error correction properties.
The Gottesman-Kitaev-Preskill (GKP) quantum error-correcting code has emerged as a key technique in achieving fault-tolerant quantum computation using photonic systems. Whereas [Baragiola et al., Phys. Rev. Lett. 123, 200502 (2019)] showed that experimentally tractable Gaussian operations combined with preparing a GKP codeword $lvert 0rangle$ suffice to implement universal quantum computation, this implementation scheme involves a distillation of a logical magic state $lvert Hrangle$ of the GKP code, which inevitably imposes a trade-off between implementation cost and fidelity. In contrast, we propose a scheme of preparing $lvert Hrangle$ directly and combining Gaussian operations only with $lvert Hrangle$ to achieve the universality without this magic state distillation. In addition, we develop an analytical method to obtain bounds of fundamental limit on transformation between $lvert Hrangle$ and $lvert 0rangle$, finding an application of quantum resource theories to cost analysis of quantum computation with the GKP code. Our results lead to an essential reduction of required non-Gaussian resources for photonic fault-tolerant quantum computation compared to the previous scheme.
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