Repeater Graphs in Quantum Networks

• Repeater graphs are versatile graph constructions that model entanglement resources, network topologies, and dynamic self-replication in quantum and classical systems.
• In quantum communication, repeater graph states use complete or complete-bipartite cores with attached arms (often tree-encoded) to improve loss tolerance and boost secret-key rates.
• Beyond quantum repeaters, repeater graphs inform network performance analysis and graph dynamics studies, showcasing unique iterative behavior and self-similarity properties.

Searching arXiv for papers on “repeater graphs” and related usages to ground the synthesis. Repeater graphs are graph-structured objects that arise in several technically distinct literatures. In quantum communication, the term is most closely associated with repeater graph states: multipartite photonic resource states used to replace long-lived matter memories in all-photonic repeaters, often built from a complete or complete-bipartite core with attached arms and, in practical schemes, augmented by tree-encoded logical qubits for loss tolerance (Chan, 2018, Li et al., 2019). In quantum-network modeling, repeater graphs also denote communication topologies such as star graphs, chains, and layered multi-hop networks whose vertices are repeater stations and whose edges are elementary links or effective channels (Kunzelmann et al., 23 May 2025, Jones et al., 2015, Siwerson et al., 10 Dec 2025). In a different graph-dynamical sense, “repeater” behavior refers to periodicity under graph iteration, and for iterated jump graphs this behavior is almost completely absent: only $C_5$ and the net graph $N$ are fixed points, and there are no non-trivial periodic sequences (Herr et al., 2022).

1. Scope of the term

In the cited literature, “repeater graphs” is not a single standardized object. It names different graph constructions according to the role played by repetition, routing, or self-replication.

Context	Graph object	Defining role
All-photonic quantum repeaters	Repeater graph state (RGS)	Memory-free entanglement-routing resource
Quantum repeater networks	Star, chain, or layered repeater topology	Rate, latency, or SNR analysis
Iterated graph dynamics	Jump-graph orbit under $J$	Periodicity or its absence
Structural self-similarity	Copies or recursive products of graphs	Internal or hierarchical repetition

The quantum-information usage is the most operationally developed. There, the graph is not merely a descriptive topology: it is the entangled resource itself, and local measurements alter connectivity in a controlled way. The graph-dynamical and self-similarity usages are different in emphasis. They study whether a graph reproduces itself under an operator, embeds copies of itself, or reappears through recursive graph products (Chan, 2018, Kunzelmann et al., 23 May 2025, Herr et al., 2022, Kurilić et al., 2014, Bondarenko, 2014).

2. Repeater graph states

The canonical repeater graph state in the all-photonic architecture of Azuma, Tamaki, and Lo consists of a complete subgraph of $K$ core photons, each connected to an additional photon to form $K$ external arms (Li et al., 2019). Operationally, outer photons are transmitted to neighboring nodes or end users, while inner photons remain local and mediate graph connectivity. In the full protocol, Bell-state measurements (BSMs) are performed between incoming signal photons and designated leaf photons; if a local BSM succeeds, an associated qubit is measured in the $X$ basis to connect the successful branch, while failure triggers a $Z$-basis measurement that removes the failed branch (Li et al., 2019).

Later work sharpened the internal graph requirement. For repeater functionality, the inner photons do not need to form a fully connected clique; the minimum useful condition is that they form a complete bipartite graph. Missing entanglement edges between inner photons on the same side do not affect repeater operation (Chan, 2018). This reduction is significant because it removes unnecessary entangling overhead from deterministic generation schemes.

A central distinction is between a bare RGS and an encoded RGS. A bare RGS is a repeater graph state whose inner photons are not encoded. An encoded RGS replaces each inner photonic qubit by a tree-encoded logical qubit, following the loss-tolerant encoding used in all-photonic repeaters (Chan, 2018). For a tree with branching vector

$\vec b=\{b_0,b_1,\dots,b_l\},$

the total number of photons is

$Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$

The success probabilities of logical $X$- and $N$0-measurements are

$N$1

with recursion

$N$2

and boundary conditions

$N$3

These relations encode the Varnava-style indirect-measurement mechanism that turns local tree structure into loss tolerance (Chan, 2018).

A continuous-variable variant uses small CV graph states as a memory-free multiplexing resource. In that architecture, each elementary source station $N$4 locally prepares a small graph state whose nodes are cat-code photonic modes, sends those nodes through $N$5 parallel channels, and prunes unsuccessful branches by logical $N$6-measurements after syndrome detection at $N$7 stations. If at least one desired branch remains on each side, the surviving graph reduces to a useful bipartite entangled state. The probability that at least one of the $N$8 channels per elementary link yields the desired syndrome outcome across all $N$9 elementary links is

$J$0

and, because no quantum memory waiting is required, the repetition time is

$J$1

The paper reports that graph-state or few-memory multiplexing can improve the secret-key rate by more than 10 orders of magnitude for cat codes (Li et al., 2024).

3. Generation, local equivalence, and experiment

The most developed deterministic generation scheme for encoded repeater graph states is based on coupled quantum emitters and four graph-construction rules: pumping plus emitter Hadamards, pumping plus photon Hadamards, CZ gates between emitters, and the graph identity that local complementation followed by $J$2-measurement is equivalent to direct $J$3-measurement (Chan, 2018). The protocol is symmetric and parallelized: two encoded arms are generated in parallel, joined by an emitter-emitter CZ, and then extended to an arbitrary even number of arms.

For a $J$4-arm encoded RGS with $J$5, the total CZ-gate count is

$J$6

The paper’s repeater-level success probability is written as

$J$7

and the secret-key rate per mode is

$J$8

In the benchmark with $J$9, branching vector $K$0, and $K$1 source nodes, the maximum repeater key rate for the deterministic encoded-RGS protocol is

$K$2

compared with

$K$3

for the probabilistic scheme, an improvement factor of about $K$4. For the same benchmark, the deterministic scheme uses about

$K$5

photons versus order

$K$6

for the probabilistic approach (Chan, 2018).

The locality structure of repeater graph states has also been studied through LU–LC equivalence. Complete graph states, biclique states, and certain imperfect repeater graph states belong to families for which local-unitary equivalence implies local-Clifford equivalence. In particular, biclique states satisfy $K$7, and so do complete-graph-based repeater graph states with one leaf missing and biclique-based repeater graph states with two leaves missing. For perfect repeater graph states with one leaf attached to every core vertex, the same question remains open (Tzitrin, 2018).

An experimental demonstration of the all-photonic repeater switching primitive was carried out with a 12-photon interferometer implementing a $K$8 parallel all-photonic quantum repeater (Li et al., 2019). The experiment used a four-photon GHZ state,

$K$9

as a local-unitary equivalent surrogate of a complete graph state. In the simplified protocol, passive-choice measurement devices automatically acted either as Bell-state analyzers or as $K$0-basis projectors. The measured enhancement in entanglement-generation rate over standard parallel entanglement swapping was $K$1, corresponding to

$K$2

at down-conversion probability $K$3, consistent with the derived relation

$K$4

This was a proof-of-principle rather than a full realization of a loss-tolerant encoded RGS, but it directly implemented the graph-state switching logic (Li et al., 2019).

4. Repeater networks as graphs

A second major usage treats the repeater system itself as a graph. In the stationary-rate analysis of multipartite quantum repeaters, the network is a star graph with one central quantum router and $K$5 leaves, each leaf connected to the center by $K$6 parallel elementary links or memories (Kunzelmann et al., 23 May 2025). If $K$7 is the number of filled memories of party $K$8 after storage in a round, then the number of GHZ states generated in that round is

$K$9

The stationary router rate is defined as

$X$0

For $X$1, the exact stationary yield is

$X$2

and the same abstract formula applies to a bipartite repeater chain with $X$3 elementary segments (Kunzelmann et al., 23 May 2025).

Two asymptotic conclusions are especially important. First, without multiplexing the stationary rate decreases only logarithmically with network size: $X$4 Second, with multiplexing the per-memory rate saturates: $X$5 The general stationary balance relation

$X$6

connects yield to the total post-measurement occupancy $X$7 (Kunzelmann et al., 23 May 2025).

In communication-protocol analyses, the graph edges themselves are protocol-dependent stochastic objects rather than static capacities. The link-level study of MeetInTheMiddle, SenderReceiver, and MidpointSource treats a repeater network as a graph whose vertices are repeater nodes and whose edges are optical links with protocol-specific success laws, round durations, and memory lock-up times (Jones et al., 2015). For MeetInTheMiddle,

$X$8

For MidpointSource, under the fast-clock assumption,

$X$9

so throughput scales linearly rather than quadratically in $Z$0, with

$Z$1

This is why MidpointSource can outperform the other link protocols by orders of magnitude in low-success-probability regimes (Jones et al., 2015).

A layered interpretation appears in POLARNet, which studies a directed acyclic repeater graph with a single base station at layer $Z$2, a single user equipment at layer $Z$3, and $Z$4 repeater layers in between (Siwerson et al., 10 Dec 2025). If $Z$5 is the diagonal gain matrix of layer $Z$6, the end-to-end effective channel is

$Z$7

POLARNet solves

$Z$8

by layerwise forward-backward updates over compact activation sets. The method is gradient-free, step-size-free, and monotonic in the objective. In the reported Rician-fading scenario, distributing power across multiple repeaters within a layer under the nonnegative $Z$9-ball constraint is approximately $\vec b=\{b_0,b_1,\dots,b_l\},$0 better than the optimal select-one policy; in the IID Gaussian scenario, the reported gains relative to random-allocation baselines are approximately $\vec b=\{b_0,b_1,\dots,b_l\},$1 and $\vec b=\{b_0,b_1,\dots,b_l\},$2 (Siwerson et al., 10 Dec 2025).

A related all-photonic architecture treats the network as a chain of encoded Bell-pair edges rather than as a large graph state. In that 1000 km design, repeater stations are spaced by about 9 km, and each elementary edge is a concatenated-code-protected encoded Bell pair built from GKP qubits and the $\vec b=\{b_0,b_1,\dots,b_l\},$3 Steane code (Shiina et al., 24 Jun 2026). This suggests a broader notion of repeater graph in which the elementary graph edge is itself a fault-tolerant photonic resource.

5. Repetition as graph dynamics and self-embedding

In graph dynamics, repeating behavior can mean periodicity under an operator. For the jump graph $\vec b=\{b_0,b_1,\dots,b_l\},$4, whose vertices are the edges of $\vec b=\{b_0,b_1,\dots,b_l\},$5 and whose adjacencies correspond to non-incident edge pairs, the iterated sequence

$\vec b=\{b_0,b_1,\dots,b_l\},$6

has a complete qualitative classification (Herr et al., 2022). The decisive theorem states that for any non-empty graph $\vec b=\{b_0,b_1,\dots,b_l\},$7, the following are equivalent: $\vec b=\{b_0,b_1,\dots,b_l\},$8, $\vec b=\{b_0,b_1,\dots,b_l\},$9 for some $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$0, and $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$1 is $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$2 or the net graph $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$3. Hence there are no cycles of period $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$4, and no graph other than $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$5 or $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$6 satisfies $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$7 for any $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$8. Every starting graph therefore either terminates, becomes one of these two fixed points, or grows without bound in edge count (Herr et al., 2022).

A different structural meaning of repetition appears in the study of the Rado graph. There, the relevant objects are the monoid of self-embeddings $Q_l=\sum_{j=0}^{l}\prod_{i=0}^{j} b_i.$9, the family of copies

$X$0

and the ideal

$X$1

The poset of copies, the quotient algebra $X$2, and the inverse right Green preorder on self-embeddings are forcing equivalent to a two-step iteration $X$3, where $X$4 is Sacks-like and $X$5 is forced to be $X$6-distributive (Kurilić et al., 2014). In this sense, the Rado graph is a paradigmatic self-repeating graph: it contains abundant induced copies of itself, and these copies have a highly structured order-theoretic organization.

A stochastic growth interpretation appears in randomised reproducing graphs. Starting from $X$7, every vertex reproduces simultaneously at each step, creating parent and child descendants with edge channels controlled by probabilities $X$8, $X$9, and $N$00 (Jordan, 2011). The degree process of a random lineage exhibits a phase transition at

$N$01

while total edge growth changes regime at

$N$02

If $N$03, the empirical degree distribution converges almost surely to a stationary law; if $N$04, the proportion of vertices with any fixed degree tends to zero. Separately, if $N$05, the graph densifies with exponent

$N$06

This is not repeater-graph terminology in the quantum sense, but it is a rigorous model of simultaneous graph reproduction (Jordan, 2011).

6. Recursive graph families and related graph-based notions of repetition

Recursive repetition also appears in explicit graph constructions. Bondarenko showed that finitely generated self-similar groups can produce graph sequences $N$07 that satisfy

$N$08

that is, iterated zig-zag or replacement products combined with graph powering (Bondarenko, 2014). These constructions yield bounded-degree recursive families; in the zig-zag automaton case, the action graphs have linear level diameter,

$N$09

and bounded girth. This provides explicit examples of small-diameter, self-similar interconnection networks generated by repeated reuse of a fixed local wiring rule (Bondarenko, 2014).

In quantum-network state distribution, graph growth can be organized without relying primarily on repeater-assisted central distribution. A protocol for distributing graph states over noisy quantum networks constructs and purifies small GHZ subgraphs locally and then merges them to mimic the target topology (Cuquet et al., 2012). The paper benchmarks this against two repeater-based bipartite protocols and shows that the resulting graph-state fidelity can be written as the partition function of a classical Ising-type Hamiltonian: $N$10 For the linear cluster, the leading first-order decay rates are $N$11, $N$12, $N$13, and $N$14 for Bipartite A, Bipartite B, Subgraphs S1, and Subgraphs S2 respectively, showing that local subgraph growth can match or outperform repeater-heavy bipartite distribution (Cuquet et al., 2012). This suggests that, within graph-state networking, “repeater graph” design is inseparable from the choice between centralized edge distribution and distributed graph assembly.

A related but terminologically distinct use of graph structure concerns repeat detection in metagenomic assembly. GraSSRep formulates the problem as node classification on a unitig graph, uses a high-precision low-recall pseudo-label heuristic based on unitig length and coverage,

$N$15

and then trains a two-layer GraphSAGE model plus a random forest on handcrafted and learned graph features (Azizpour et al., 2024). On the Shakya 1 benchmark, the pseudo-label heuristic attains $N$16 precision but only $N$17 recall, while the graph-based propagation raises final $N$18 to $N$19. This is not a repeater-graph-state construction, but it exemplifies another rigorous use of graph topology to identify repeated structure (Azizpour et al., 2024).

Taken together, these literatures show that repeater graphs are best understood as a family of graph concepts rather than a single definition. In quantum communication they are concrete entanglement resources or network topologies; in graph dynamics they are periodic, self-embedding, or recursively generated structures; and in adjacent graph-learning applications they are the topological signatures of repetition itself.