Integrating Entanglement Purification into All-Photonic Quantum Repeaters

TL;DR

This work tackles the challenge of incorporating entanglement purification into all-photonic repeater networks based on repeater graph states (RGS). By introducing optimistic purification implemented directly on half-RGS primitives and delaying the join of half-RGSs, the authors enable flexible purification scheduling along the connection path with modest overhead, while keeping end-to-end latency dominated by transmission time $L_{\text{total}}/c$. The key contributions include a purification-enhanced RGS framework, detailed overhead analysis comparing raw, end-node purification, and optimistic schemes, and a numerical study showing fidelities above $0.9$ and order-of-magnitude rate improvements ($\sim 45$–$65\times$) over baselines. This approach brings memory-based purification scheduling concepts into all-photonic implementations, potentially enabling near-deterministic high-fidelity entanglement across long distances with reduced memory requirements at end nodes. The work also provides open-source code for reproducing the numerical results and highlights avenues for further optimization and network-level validation.

Abstract

We propose a purification-enhanced all-photonic quantum repeater scheme based on repeater graph states (RGS) framework that leverages the recently proposed half-RGS building block. This framework addresses a longstanding open question--how to naturally integrate entanglement purification with an all-photonic scheme--by enabling long-distance purification without disrupting the core design. Our framework utilizes optimistic purification performed directly on the half-RGS primitives across long distances without waiting for heralding outcomes. The overhead is modest: the RGS generation slows down proportionally with the number of purification rounds, and each round requires only one additional quantum emitter per half-RGS source. However, since the generation time is negligible compared to the end-to-end communication delay, the total latency remains effectively dominated by communication time, similar to frameworks without purification. Our framework enables flexible purification scheduling along the connection path, making it compatible with memory-based strategies, a rich body of research on purification scheduling and optimization that was previously thought inapplicable to the RGS scheme. Through numerical evaluation, we compare the performance of our framework with purification between memories at end nodes.

Figures (6)

• Figure 1: An overview of the RGS scheme, illustrating the node types and qubits involved. (a) A half-RGS with three arms, where purple, orange, and red circles represent anchor, inner, and outer qubits, respectively. Two half-RGSs can be combined into an RGS by applying a CZ gate and performing X-basis measurements on the anchor qubits. (b) A complete RGS formed by joining two half-RGSs at their anchor qubits. (c) An example of a tree-encoded inner qubit with branching parameters (2,3,1). (d) A schematic of Bell pair distribution in the RGS-based repeater chain. End nodes (Alice and Bob) and all RGS sources (RGSS) along the path generate and transmit their half-RGSs to adjacent advanced Bell state analyzer (ABSA) nodes. For clarity, the conversion of half-RGSs into full RGSs at RGSS is omitted. Bell state measurements (BSMs) are performed on pairs of outer qubits arriving from opposite sides of each ABSA. Each ABSA selects one successful BSM arm to retain and measures the connected inner qubits in the X basis (shown), while all other inner qubits are measured in the Z basis. Once the measurement outcomes are communicated to Alice and Bob, a Bell pair is effectively shared between them.
• Figure 2: (a) Schematic of two-way heralded entanglement purification. The process begins with the probabilistic distribution of Bell pairs, followed by heralding signals from intermediate nodes confirming successful pair creation. A purification circuit is then applied, during which some Bell pairs are measured and consumed. The measurement results are exchanged between the two parties; if the outcomes agree (i.e., correlate or anti-correlate as expected), the remaining Bell pair is kept, resulting in improved fidelity. (b) Schematic of optimistic purification. In this approach, distribution and purification operations are performed immediately in sequence, without waiting for heralding confirmation. Classical outcomes are compared only once at the end, combining both the heralding of Bell pair generation and the results of purification in a single round of communication.
• Figure 3: (a) Integration of entanglement purification into the RGS generation process. Instead of joining left and right half-RGSs immediately after their creation as in the original protocol naphan-rgs-protocol, multiple half-RGSs are generated on each side. Their anchor emitters serve as inputs to a purification circuit. After purification, the retained high-fidelity half-RGSs from the left and right are paired and joined to form complete RGSs. (Note that dark red lines represent the circuits or their evolution in time and do not represent entangled properties of graph states.) (b) Graph state purification circuits that measures the parity of stabilizer $Z_a X_b$ where $a$ represents the Alice's side and $b$ for Bob dur-graph-state-purificationkruszynska-graph-state-purification Top (bottom) qubits of Alice and Bob are Bell pairs in the two-qubit graph state. The top qubits for both side are retained after the purification circuit. The purification circuit for $X_a Z_b$ are similar, by exchanging the circuits that Alice and Bob locally performed. (c) Graph state purification circuit that measures the parity of $Y_a Y_b$ stabilizers.
• Figure 4: Illustration of flexible purification scheduling enabled by our purification-enhanced RGS framework. Using five half-RGSs per side at each hop (bottom line represents a copy), three end-to-end Bell pairs are created via different purification schedules, resulting in varying fidelities. These pairs then undergo further purification at the end nodes to yield a single high-fidelity Bell pair. The first is constructed by purifying ($YY$) each link-level connection and stitching the purified links across the full path (blue border). The second is formed by stitching link-level pairs into two copies of two 5-hop segments, purifying ($YY$) within each segment, and then connecting them (pink border). The third is created by directly stitching raw link-level pairs without intermediate purification (black thick border). These three pairs are then used in a pumping-like purification sequence: the first and second are purified (with the second as the sacrificial pair with $ZX$), followed by purification between the result and the third (with the third as the sacrificial pair with $YY$). The final output is a high-fidelity end-to-end Bell pair, significantly improved over the unpurified case upon success.
• Figure 5: End-to-end fidelity comparison between unpurified Bell pairs, baseline purification using direct end-to-end Bell pairs, and our purification-enhanced scheme. Both of the purified schemes consume five half-RGSs per instead of one in the unpurified setting.