Large-Scale Tree-Type Photonic Cluster State Generation with Recurrent Quantum Photonic Neural Networks

TL;DR

This paper tackles the challenge of scalable generation of large-scale photonic cluster states for quantum networks and measurement-based computing. It introduces a recurrent quantum photonic neural network (QPNN) that learns to generate unit-cell entanglement and recursively build tree-type cluster states using a combination of linear MZI networks and photon-number dependent nonlinearities. The authors demonstrate loss-tolerant, high-fidelity operation across three platform models and analyze the scalability, showing feasible generation of clusters from tens to hundreds of photons today and potentially much larger clusters with modest loss reductions, along with a one-way repeater performance assessment for global-scale networks. The work suggests that QPNN-based generators can overcome many fundamental limitations of current approaches, offering a path toward practical, loss-limited, large-scale quantum networks and related technologies.

Abstract

Large, multi-dimensional clusters of entangled photons are among the most powerful resources for emerging quantum technologies, as they are predicted to enable global quantum networks or universal quantum computation. Here, we propose an entirely new architecture and protocol for their generation based on recurrent quantum photonic neural networks (QPNNs) and focusing on tree-type cluster states. Unlike other approaches, QPNN-based generators are not limited by the the coherence of quantum emitters or by probabilistic multi-photon operations, enabling arbitrary scaling only limited by loss (which, unavoidably, also affects all other methods). We show that a single QPNN can learn to perform all of the many different operations needed to create a cluster state, from photon routing to entanglement generation, all with near-perfect fidelity and at loss-limited rates, even when it is created from imperfect photonic components. Although these losses ultimately place a limit on the size of the cluster states, we show that state-of-the-art photonics should already allow for clusters of 60 photons, which can grow into the 100s with modest improvements to losses. Finally, we present an analysis of a one-way quantum repeater based on these states, determining the requisite platform quality for a global quantum network and highlighting the potential of the QPNN to play a vital role in high-impact quantum technologies.

Figures (5)

• Figure 1: a Architecture of the tree state generator. One single-photon source (that can emit each $\Delta t_s$) and Hadamard gate (H) prepare and inject each photon of the tree into the generator, where they are subsequently routed through the QPNN, switch, and delay lines according to a timing protocol for the desired tree shape. Each wire corresponds to a single photonic qubit, thus two spatial modes in the dual-rail encoding scheme. b Example tree-type photonic cluster state, with branching vector $\vec{b} = \left[2, 2, 2, 2\right]$, generated in a. Faded photons are yet to be generated in a, while those present can be identified by the colors of the lines connecting them, all of which represent maximal entanglement. One unit cell of the tree is outlined by a dashed box. c Inset of the Hadamard gate in a, realized by a Mach-Zehnder interferometer (MZI) with two phase shifters $(\phi, \theta)$, where each emitted photon in state $\left|0\right\rangle$ is transformed to state $\left|+\right\rangle$ by selecting phase shifts $(\phi, \theta) = (0, \pi / 4)$. d 2-layer, 6-mode QPNN that operates on three photonic qubits as shown in a. Each linear layer, $\mathbf{U}(\boldsymbol{\phi}_i, \boldsymbol{\theta}_i)$, is formed by a mesh of MZIs and separated by single-site nonlinearities $\boldsymbol{\Sigma}(\varphi_1, \varphi_2)$. e, f Comparison of tree generation protocols based on QPNNs (this work), active control of different quantum emitters (qd, SiV, atom), and linear optics (lo), assuming that each photon in each tree has a $10\%$ chance of being lost in the generator, regardless of the tree size (i.e. number of photons), then traverses a 5 km fiber channel, accruing $\sim 18\%$ additional loss. In e, f respectively, the effective loss of the logical qubit and repetition rate of the generator are shown as the tree scales. Further details on this comparison are given in Sec. S1 of the Supplementary Information.
• Figure 2: a-d Circuit diagrams for each QPNN operation required by the generation protocol. e Minimization of the network cost (i.e. average error) during 200 optimization trials of 1000 epochs each for the single-, multi- and future-platform (see main text and Sec. \ref{['sec:methods']} for platform details). Dashed lines denote the loss limit (i.e. minimum achievable cost due to loss). f-i Hinton diagrams resolved in the X-basis for each operation of the multi-platform QPNN outlined in black in e, where the uppermost photonic qubit is $\left|+\right\rangle$ at the input (vertical axis), yet can belong to a superposition of $\left|+\right\rangle$ and $\left|-\right\rangle$ at the output (horizontal axis). Each box is colored according to its argument, which is always within $\pi/100$ of either $0$ or $\pi$ up to an insignificant global phase. When a photonic qubit is missing at any input or output port of the network, $\emptyset$ is written in its place. The fidelity of each operation is given above its Hinton diagram, never falling below 0.999929 for this network.
• Figure 3: Procedure for generating a tree state with branching vector $\vec{b} = \left[2, 2\right]$. a-b Individual photons from row 2 of the tree are emitted in subsequent timesteps $t$, separated by $\Delta t_s$, the time between source triggers, then traverse through the QPNN which acts as an identity operation. The switch is adjusted at each timestep, directing each pair of consecutive photons to each of the two longest delay lines ($4\Delta t_s$ and $3\Delta t_s$), such that the latter photon catches up to the former. c-d Delayed photons arrive at the QPNN input with a newly emitted photon such that all three are subsequently entangled. The top photon is routed to a delay line by the switch while the other photons reach the output of the generator. These operations are separated by 2$\Delta t_s$, and the shortest delay line ($2\Delta t_s$) is selected at $t=6\Delta t_s$ to accommodate this change. g The root photon of the tree, $(0, 0)$, is emitted such that it arrives at the QPNN simultaneously with the delayed photons. The switch ensures the root is also routed to the output after it becomes entangled with the others. f Each dark line specifies the timesteps that a photon, as labeled on the vertical axis, is traversing the generator. Photon markers denote timesteps where the QPNN acts on a given photon, with connecting lines added to indicate entangling operations.
• Figure 4: a Fidelity of tree states of increasing depth, with the mean (95% confidence intervals) of fit beta distributions shown by markers (shaded regions), when generated using a single-, multi- and future-platform QPNN, respectively (cf. Fig. \ref{['fig:training']}). b The corresponding rate at which entire trees (i.e. not missing any photons) can be generated for the three platforms as a function of the tree depth, assuming that the QPNN is trained to loss-limited operation. The shaded region is unattainable, even with lossless components, due to the time needed to generate all of the individual photons in the protocol. c Generation rate for different depth trees as a function of the percent loss reduction starting from the single-platform QPNN (i.e. at 0 reduction), allowing us to understand the network performance if all elements are improved together. The solid black line separates the regime where MZI losses dominate from the one where fiber losses dominate.
• Figure 5: a The number of photons in the optimal tree-type cluster state as a function of the total channel length (keeping a constant 5 km node separation; number of nodes shown on top axis) for the three different QPNN platforms (cf. Fig. \ref{['fig:training']}). Dashed curves represent trees with a constant branching ratio $b=2$, while solid curves are arbitrarily-shaped trees with $\max\{\vec{b}\}\leq 4$. A change in the depth (branching) of the tree shape is denoted by vertical (horizontal) arrowheads. Two exemplary trees are shown in insets for the future QPNN, comprised of 15 and 268 photons, demonstrating that tree growth is primarily accomplished through increased branching. b The corresponding communication rates for the different clusters in a, benchmarked to the rate using single photons (black line). In this model, we assume that information is transferred between logical qubits perfectly at each node, such that all rate reduction is due to losses. Further details can be found in Supplementary Information Sec. S5.