Entanglement is the key resource for measurement-based quantum computing. It is stored in quantum states known as cluster states, which are prepared offline and enable quantum computing by means of purely local measurements. Universal quantum computing requires cluster states that are both large and possess (at least) a two-dimensional topology. Continuous-variable cluster states-based on bosonic modes rather than qubits-have previously been generated on a scale exceeding one million modes, but only in one dimension. Here, we report generation of a large-scale two-dimensional continuous-variable cluster state. Its structure consists of a 5- by 1240-site square lattice that was tailored to our highly scalable time-multiplexed experimental platform. It is compatible with Bosonic error-correcting codes that, with higher squeezing, enable fault-tolerant quantum computation.
Entanglement is the key resource for measurement-based quantum computing. It is stored in quantum states known as cluster states, which are prepared offline and enable quantum computing by means of purely local measurements. Universal quantum computing requires cluster states that are both large and possess (at least) a two-dimensional topology. Continuous-variable cluster states-based on bosonic modes rather than qubits-have previously been generated on a scale exceeding one million modes, but only in one dimension. Here, we report generation of a large-scale two-dimensional continuous-variable cluster state. Its structure consists of a 5- by 1240-site square lattice that was tailored to our highly scalable time-multiplexed experimental platform. It is compatible with Bosonic error-correcting codes that, with higher squeezing, enable fault-tolerant quantum computation.
Measurement-based quantum computation offers exponential computational speed-up through simple measurements on a large entangled cluster state. We propose and demonstrate a scalable scheme for the generation of photonic cluster states suitable for universal measurement-based quantum computation. We exploit temporal multiplexing of squeezed light modes, delay loops, and beam-splitter transformations to deterministically generate a cylindrical cluster state with a two-dimensional (2D) topological structure as required for universal quantum information processing. The generated state consists of more than 30,000 entangled modes arranged in a cylindrical lattice with 24 modes on the circumference, defining the input register, and a length of 1250 modes, defining the computation depth. Our demonstrated source of two-dimensional cluster states can be combined with quantum error correction to enable fault-tolerant quantum computation.
Continuous-variable measurement-based quantum computation, which requires deterministically generated large-scale cluster states, is a promising candidate for practical, scalable, universal, and fault-tolerant quantum computation. In this work, based on our compact and scalable scheme of generating a two-dimensional spatiotemporal cluster state, a complete architecture including cluster state preparation, gate implementations, and error correction, is proposed. First, a scheme for generating two-dimensional large-scale continuous-variable cluster state by multiplexing both the temporal and spatial domains is proposed. Then, the corresponding gate implementations by gate teleportation are discussed and the actual gate noise from the generated cluster state is considered. After that, the quantum error correction can be further achieved by utilizing the square-lattice Gottesman-Kitaev-Preskill (GKP) code. Finally, a fault-tolerant quantum computation can be realized by introducing bias into the square-lattice GKP code (to protect against phase-flip errors) and concatenating a repetition code (to handle the residual bit-flip errors), with a squeezing threshold of 12.3 dB. Our work provides a possible option for a complete fault-tolerant quantum computation architecture in the future.
One-way quantum computation is a promising scheme for realizing large-scale quantum computation, where computation is done via measurements on a universal multi-mode entangled resource called cluster state. The capability of this method is determined by the size and the dimension of the cluster state; the size is roughly equivalent to the number of input modes times number of operation steps, while the dimension determines universality and fault tolerance of the computation. To achieve universality, two-dimensional cluster state is required. Among many approaches, time-domain multiplexing using optical system [1, 2] is a promising method where quantum state is encoded in a temporal wave packet. Although universality is still lacking, one-million-mode one-dimensional cluster state has been demonstrated with this approach [3].
A large-scale continuous variable (CV) cluster state is necessary in quantum information processing based on measurement-based quantum computing (MBQC). Specially, generating a large-scale CV cluster state multiplexed in a time domain is easier to implement and has strong scalability in experiment. Here one-dimensional (1D) large-scale dual-rail CV cluster states multiplexed both in time and frequency domains are parallelly generated, which can be further extended to a three-dimensional (3D) CV cluster state by combining two time-delay nondegenerate optical parametric amplification systems with beam-splitters. It is shown that the number of parallel arrays depends on the corresponding frequency comb lines, the partite number of each array can be very large (million), and the scale of the 3D cluster state can be ultra-large. In addition, the concrete quantum computing schemes of applying the generated 1D and 3D cluster states are also demonstrated. Our schemes may pave the way for fault-tolerant and topologically protected MBQC in hybrid domains, by further combining with efficient coding and quantum error correction.
Measurement-based quantum computation with continuous variables in an optical setup shows great promise toward implementation of large-scale quantum computation, where the time-domain multiplexing approach enables us to generate the large-scale cluster state used to perform measurement-based quantum computation. To make effective use of the advantage of the time-domain multiplexing approach, in this paper, we propose the method to generate the large-scale three-dimensional cluster state which is a platform for topologically protected measurement-based quantum computation. Our method combines a time-domain multiplexing approach with a divide-and-conquer approach and has two advantages for implementing large-scale quantum computation. First, the squeezing level for verification of the entanglement of the three-dimensional cluster states is experimentally feasible. The second advantage is the robustness against analog errors derived from the finite squeezing of continuous variables during topologically protected measurement-based quantum computation. Therefore, our method is a promising approach to implement large-scale quantum computation with continuous variables.
Due to its unique scalability potential, continuous-variable quantum optics is a promising platform for large-scale quantum computing. In particular, very large cluster states with a two-dimensional topology that are suitable for universal quantum computing and quantum simulation can be readily generated in a deterministic manner, and routes towards fault tolerance via bosonic quantum error correction are known. In this article we propose a complete measurement-based quantum computing architecture for the implementation of a universal set of gates on the recently generated two-dimensional cluster states [M. V. Larsen et al., Science 366, 369 (2019); W. Asavanant et al., Science 366, 373 (2019)]. We analyze the performance of the various quantum gates that are executed in these cluster states as well as in other two-dimensional cluster states (the bilayer-square lattice and quad-rail lattice cluster states [R. N. Alexander et al., Phys. Rev. A 94, 032327 (2016); N. C. Menicucci, Phys. Rev. A 83, 062314 (2011)]) by estimating and minimizing the associated stochastic noise addition as well as the resulting gate error probability. We compare the four different states and find that, although they all allow for universal computation, the quad-rail lattice cluster state performs better than the other three states, which all exhibit similar performance.