The Octo-Rail Lattice: a four-dimensional cluster state design

TL;DR

The paper introduces the Octo-Rail Lattice (ORL), a four-dimensional, time-multiplexed continuous-variable cluster state built from GKP Bell pairs that enables surface-code-based fault tolerance with a passive optical hardware footprint. By extending the Quad-Rail Lattice into higher dimensions and reusing time-domain multiplexing, the ORL implements a Raussendorf-Harrington-Goyal-like lattice suitable for fault-tolerant quantum computation, with gating achieved via measurement-based teleportation on an eight-mode eightsplitter network. Universality is achieved using GKP qunaught inputs and heterodyne-based magic-state generation, with the architecture supporting arbitrary GKP encodings and higher-dimensional topological codes. A squeezing threshold of 9.75 dB demonstrates the viability of fault-tolerant operation, while the design scales linearly with dimension and supports state multiplexing and injection, marking a practical path toward large-scale photonic quantum computing.

Abstract

Macronode cluster states are promising for fault-tolerant continuous-variable quantum computation, combining gate teleportation via homodyne detection with the Gottesman-Kitaev-Preskill code for universality and error correction. While the two-dimensional Quad-Rail Lattice offers flexibility and low noise, it lacks the dimensionality required for topological error correction codes essential for fault tolerance. This work presents a four-dimensional cluster state, termed the Octo-Rail Lattice, generated using time-domain multiplexing. This new macronode design combines the noise properties and flexibility of the Quad-Rail Lattice with the possibility to run various topological error correction codes including surface and color codes. Besides, the presented experimental setup is easily scalable and includes only static optical components allowing for a straight-forward implementation. Analysis demonstrates that the Octo-Rail Lattice, when combined with GKP qunaught states and the surface code, exhibits noise performance compatible with a fault-tolerant threshold of 9.75 dB squeezing. This ensures universality and fault-tolerance without requiring additional resources such as other non-Gaussian states or feed-forward operations. This finding implies that the primary challenge in constructing an optical quantum computer lies in the experimental generation of these highly non-classical states. Finally, a generalisation of the design to arbitrary dimensions is introduced, where the setup size scales linearly with the number of dimensions. This general framework holds promise for applications such as state multiplexing and state injection.

Figures (17)

• Figure 1: Setup of the Octo-Rail Lattice cluster state. Each time step, four Bell pairs are generated and partially delayed by one, $n$, $mn$ and $kmn$ clock cycles, respectively. The non-delayed halves from the current together with the delayed halves from previous time steps are then entangled by a beamsplitter network known as eightsplitter and measured by eight homodyne detectors (HD).
• Figure 2: Depiction of a macronode of the Octo-Rail Lattice cluster state within the generated macronode lattice. The four-dimensional layout is created by the Bell pairs linking each macronode to macronodes at eight different time steps.
• Figure 3: Reduction of an ORL macronode into two separate QRL macronodes by applying the same measurement bases across modes. Equally colored detectors measure in the identical basis.
• Figure 4: Setup of the Octo-Rail Lattice configuration used to efficiently run the surface code. The GKP Bell pairs are equipped with an additional $\tfrac{\pi}{2}$-rotation, while the longest delay line is removed by setting $k=0$.
• Figure 5: Depiction of a macronode of the Octo-Rail Lattice cluster state adapted to the surface code. Removing the longest delay line by setting $k=0$ reduces the macronode lattice to three dimensions and creates a Bell pair link within each macronode. The shown links are given by the adapted Bell pairs of Eq. \ref{['eq:GKPBellPairH']}.