Quantum computation with light, compared with other platforms, offers the unique benefit of natural high-speed operations at room temperature and large clock rate, but a big obstacle of photonics is the lack of strong nonlinearities which also makes loss-tolerant or generally fault-tolerant quantum computation (FTQC) complicated in an all-optical setup. Typical current approaches to optical FTQC that aim at building suitable large multi-qubit cluster states by linearly fusing small elementary resource states would still demand either fairly expensive initial resources or rather low loss and error rates. Here we propose reintroducing weakly nonlinear operations, such as a weak cross-Kerr interaction, to achieve small initial resource cost and high error thresholds at the same time. More specifically, we propose an approach to generate a large-scale cluster state by hybridizing Gottesman-Kitaev-Preskill (GKP) and single-photon qubits. Our approach enables us to implement FTQC based on GKP squeezing of 7.4 and 8.4 dB and a photon loss rate of 1.0 and 5.0 %, respectively. In addition, our scheme has a reduced resource cost, i.e., number of physical qubits/photons per logical qubit or initial entanglement, compared to high-threshold FTQC with optical GKP qubits or fusion-based quantum computation with encoded single-photon-qubit states, respectively. Furthermore, our approach, when assuming very low photon loss, allows to employ GKP squeezing as little as 3.8 dB, which cannot be achieved by using GKP qubits alone.
Quantum computation with light, compared with other platforms, offers the unique benefit of natural high-speed operations at room temperature and large clock rate, but a big obstacle of photonics is the lack of strong nonlinearities which also makes loss-tolerant or generally fault-tolerant quantum computation (FTQC) complicated in an all-optical setup. Typical current approaches to optical FTQC that aim at building suitable large multi-qubit cluster states by linearly fusing small elementary resource states would still demand either fairly expensive initial resources or rather low loss and error rates. Here we propose reintroducing weakly nonlinear operations, such as a weak cross-Kerr interaction, to achieve small initial resource cost and high error thresholds at the same time. More specifically, we propose an approach to generate a large-scale cluster state by hybridizing Gottesman-Kitaev-Preskill (GKP) and single-photon qubits. Our approach enables us to implement FTQC based on GKP squeezing of 7.4 and 8.4 dB and a photon loss rate of 1.0 and 5.0 %, respectively. In addition, our scheme has a reduced resource cost, i.e., number of physical qubits/photons per logical qubit or initial entanglement, compared to high-threshold FTQC with optical GKP qubits or fusion-based quantum computation with encoded single-photon-qubit states, respectively. Furthermore, our approach, when assuming very low photon loss, allows to employ GKP squeezing as little as 3.8 dB, which cannot be achieved by using GKP qubits alone.
To implement fault-tolerant quantum computation (FTQC) with continuous variables, continuous variables need to be digitized using an appropriate code such as the Gottesman-Kitaev-Preskill (GKP) qubit. The scheme introduced in [Fukui et al. Phys. Rev. X 8, 021054 (2018)] has reduced the threshold of the squeezing level required for continuous-variable FTQC to less than 10 dB, assuming noise derived from the GKP qubit itself. In this paper, we propose a scheme to improve noise tolerance during the construction of a large-scale cluster state used for FTQC with the GKP qubits. In our scheme, a small-scale cluster state is prepared by employing maximum-likelihood estimation, the entanglement generation via the Bell measurement, and probabilistic reliable measurement. Then, a large-scale cluster state is constructed from the small-scale cluster states via the reliable encoded Bell measurement. In the numerical calculations, we assume errors derived from the two-mode gate and loss in the homodyne measurement in addition to noise from the GKP qubit itself. The results show that the thresholds of a squeezing level are around 8.1, 9.6, and 12.4 dB for loss in the homodyne measurement 0, 5, and 10%, respectively. Hence, this paper provides a way toward continuous-variable FTQC with a feasible squeezing level.
Hybridizing different physical systems or degrees of freedom offers significant advantages for realizing practical, universal, scalable, and fault-tolerant quantum computation (FTQC). Here, we propose optical FTQC schemes with low squeezing thresholds by leveraging the strengths of both discrete-variable (DV) and continuous-variable (CV) systems while utilizing frequency, time, and orbital angular momentum degrees of freedom. First, we design an optical entanglement generator (OEG) capable of producing various types of entangled pairs, including cluster pairs, hybrid entangled pairs, and GKP Bell pairs, which can be flexibly chosen by adjusting the measurement basis. Additionally, the OEG features extra ports for directly inputting (outputting) data (result) states via quantum teleportation, eliminating the need for optical switches. Secondly, large-scale one-dimensional, two-dimensional, and three-dimensional (3D) hybrid cluster states, composed of DV Gottesman-Kitaev-Preskill (GKP) qubits and CV squeezed states, are deterministically generated using the entangled pairs passed through a time-delay system. Moreover, we optimize the surface-GKP code to further reduce logical errors during the stabilizer measurements in the surface code. By combining the 3D cubic hybrid cluster state with the modified surface-GKP code and accounting for full circuit-level noise, FTQC is achieved with a squeezing threshold of 10 dB. Moreover, our method can also generate a 3D macronode Raussendorf-Harrington-Goyal (RHG) cluster state, facilitating an alternative FTQC scheme via the RHG-GKP code. Our work provides a viable pathway toward future optical FTQC architectures.
The Gottesman-Kitaev-Preskill (GKP) qubit is a promising ingredient for fault-tolerant quantum computation (FTQC) in optical continuous variables due to its advantage of noise tolerance and scalability. However, one of the main problems in the preparation of the optical GKP qubit is the difficulty in obtaining the nonlinearity. Cross-Kerr interaction is one of the promising candidates for this nonlinearity. There is no existing scheme to use the cross-Kerr interaction to generate the optical GKP qubit for FTQC. In this work, we propose a generation method of the GKP qubit by using a cross-Kerr interaction between a squeezed light and a superposition of Fock states. We numerically show that the GKP qubit with the 10 dB can be generated with mean fidelities of 99.99 and 99.9% at the success probabilities of 2.7 and 4.8%, respectively. Therefore, our method has the potential to generate the optical GKP qubit with a quality required for FTQC when we obtain the sufficient technologies for the preparation of ancillary Fock states and a cross-Kerr interaction.
The scalability of photonic implementations of fault-tolerant quantum computing based on Gottesman-Kitaev-Preskill (GKP) qubits is injured by the requirements of inline squeezing and reconfigurability of the linear optical network.In this work we propose a topologically error-corrected architecture that does away with these elements at no cost-in fact, at an advantage-to state preparation overheads.Our computer consists of three modules: a two-dimensional (2D) array of probabilistic sources of GKP states; a depthfour circuit of static beam splitters, phase shifters, and short delay lines; and a 2D array of homodyne detectors.The symmetry of our proposed circuit allows us to combine the effects of finite squeezing and uniform photon loss within the noise model, resulting in more comprehensive threshold estimates.These jumps over both architectural and analytical hurdles considerably expedite the construction of a photonic quantum computer.
The integration of diverse quantum resources and the exploitation of more degrees of freedom provide key operational flexibility for universal fault-tolerant quantum computation. In this work, we propose a flexible Gottesman-Kitaev-Preskill-state-embedded fault-tolerant quantum computation architecture based on a three-dimensional cluster state constructed in polarization, frequency, and orbital angular momentum domains. Specifically, we design optical entanglement generators to produce three diverse entangled pairs, and subsequently construct a three-dimensional cluster state via a beam-splitter network with several time delays. Furthermore, we present a partially squeezed surface-GKP code to achieve fault-tolerant quantum computation and ultimately find the optimal choice of implementing the squeezing gate to give the best fault-tolerant performance (the fault-tolerant squeezing threshold is 11.5 dB). Our scheme is flexible, scalable, and experimentally feasible, providing versatile options for future optical fault-tolerant quantum computation architecture.
Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction.However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction.In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation.To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources.Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation.To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state.We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.