Simultaneous SWAP-ASAP in Quantum Repeater Chains

• The paper presents simultaneous SWAP-ASAP as a strategy that waits for all link buffers to be ready and then performs a synchronized collapse of the swap tree, reducing intermediate buffering.
• It employs parallel elementary link generation and immediate swap triggering, leading to precise analytic expressions for fidelity, noise metrics, and secret-key rates.
• The method minimizes decoherence by avoiding stored partial chains, thus offering significant advantages in scalable quantum networks and QKD applications.

Simultaneous SWAP-ASAP is a network-layer entanglement-distribution strategy for repeater paths in which end-to-end entanglement is produced by waiting until the required elementary-link resources are available and then collapsing the swap tree without intermediate chain buffering. In one explicit formulation, the controller waits until every link buffer on an $n$-link path contains a valid pair, decoheres the selected pairs to the current time, and performs the full set of swaps in one synchronized collapse of a balanced swap tree. Closely related analytical work studies the standard homogeneous swap-ASAP repeater chain, where all elementary links are attempted in parallel every round and any repeater with two adjacent ready links swaps immediately; under that semantics, several swaps may occur in the same round, but there is no separate globally layered swap schedule. The subject therefore spans both a precise network-layer baseline and a broader ASAP semantics for local, event-driven repeater chains (Goodenough et al., 2024, Srivastava et al., 5 May 2026).

1. Definitions and protocol semantics

In the explicit network-layer usage of the term, Simultaneous SWAP-ASAP is a wait-and-swap baseline. The controller waits until every link buffer contains at least one valid pair, pops one pair from each buffer, decoheres each selected pair to the current time using the external-memory model, and then performs all entanglement swaps in one synchronized collapse of the swap tree. The recursive combination is a balanced tree: $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$ with left and right halves combined recursively. This makes the protocol connection-oriented and centrally coordinated: the path must effectively be reserved, all link buffers on the path must be known to contain valid pairs, and the swaps must be triggered in a coordinated manner (Srivastava et al., 5 May 2026).

A distinct but related usage appears in the analytical repeater literature. There, the paper does not define a separate protocol called “Simultaneous SWAP-ASAP.” Instead it studies the standard homogeneous swap-ASAP repeater chain: elementary-link attempts are made in parallel every round, and any repeater that has two adjacent ready links swaps immediately. If several repeater nodes become eligible in the same round, those swaps can occur in that same round. The model is therefore event-driven and local, not organized into deferred globally synchronized swap layers (Goodenough et al., 2024).

This terminological split is central. In the first sense, simultaneity is an explicit network-layer rule: wait for all link resources, then collapse the full swap tree. In the second, simultaneity is an emergent consequence of ASAP local semantics under round-synchronized link generation. A common misconception is to treat these as identical protocols. The literature instead supports a narrower statement: the two share an ASAP ethos, but they impose different synchronization structures.

2. Homogeneous repeater-chain model and noise observable

The exact analytical treatment considers a linear chain of $n$ elementary segments separated by $n-1$ repeater nodes, with equally spaced repeaters and homogeneous parameters. Time is discrete and synchronized across the chain. In each round, all elementary links are attempted in parallel. Segment $i\in\{1,\dots,n\}$ succeeds in round $t_i\ge 1$ with

$$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$

The chain starts a new generation cycle only after successful end-to-end delivery or after a cut-off reset; it does not keep generating fresh links while a previous generation cycle remains unresolved. Swaps are deterministic in the main analysis, and classical communication is assumed instantaneous in the main cut-off model (Goodenough et al., 2024).

The memory-noise model is based on a Bell pair mixed with white noise,

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$

with operational fidelity

$$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$

The parameter $\Lambda$ is convenient because both swapping and storage act multiplicatively: $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$0 for one round of memory decoherence, with $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$1. After $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$2 rounds of waiting, the storage factor is $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$3 (Goodenough et al., 2024).

For a realization $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$4, repeater $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$5 waits until both adjacent elementary links exist, so its local waiting time is

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$6

Ignoring endpoint storage noise, the delivered noise parameter is

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$7

The quantity

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$8

is the aggregate waiting time or roughness parameter. It makes the delivered fidelity a random variable through the independent geometric link-generation times. The main text omits the endpoint factor

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$9

justifying this by QKD use cases in which endpoint qubits can be measured immediately, or by assuming endpoints are much better protected than repeaters. Consequently,

$n$0

A key identity is that the average delivered noise parameter is exactly the probability generating function of $n$1: $n$2 All moments then follow by substitution,

$n$3

and moments of $n$4 follow by algebraic expansion from $n$5 (Goodenough et al., 2024).

3. Exact analytics, generating functions, and asymptotics

The exact framework begins with the conditional quantity

$n$6

which obeys

$n$7

with base case

$n$8

The paper shows that $n$9 lies in the span of monomials $n-1$0, $n-1$1. Writing $n-1$2, the recursion becomes a linear map $n-1$3,

$n-1$4

where

$n-1$5

The summation linear form is

$n-1$6

This yields an exact symbolic algorithm for $n-1$7, and by $n-1$8, for exact moments of arbitrary order. The paper states that this produces exact analytic formulae for all moments up to $n-1$9 segments (Goodenough et al., 2024).

The main generating-function advance is a second generating function in the segment number: $i\in\{1,\dots,n\}$0 Its $i\in\{1,\dots,n\}$1-th Maclaurin coefficient is exactly the average noise parameter for an $i\in\{1,\dots,n\}$2-segment chain. The closed form is expressed through basic $i\in\{1,\dots,n\}$3-hypergeometric series: $i\in\{1,\dots,n\}$4 with

$i\in\{1,\dots,n\}$5

$i\in\{1,\dots,n\}$6

$i\in\{1,\dots,n\}$7

$i\in\{1,\dots,n\}$8

The derivation maps the recursion to weighted walks on an unnormalized Markov chain over basis states $i\in\{1,\dots,n\}$9, decomposes those walks combinatorially, and multiplies generating functions of the resulting subwalk classes (Goodenough et al., 2024).

Singularity analysis then yields the asymptotic form

$t_i\ge 1$0

where the dominant singularity $t_i\ge 1$1 is the smallest positive real solution of

$t_i\ge 1$2

Since

$t_i\ge 1$3

the average fidelity approaches $t_i\ge 1$4 at an exponential rate governed by the same $t_i\ge 1$5. The paper emphasizes that $t_i\ge 1$6 can be interpreted as an effective per-segment decay factor, and that the approximation is “exponentially tight” in the sense that

$t_i\ge 1$7

with exponentially shrinking residual error (Goodenough et al., 2024).

Because $t_i\ge 1$8 only takes values in $t_i\ge 1$9, the exact PMF can be recovered from derivatives at $$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$0: $$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$1 The paper reports exact full delivered-noise distributions up to $$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$2 segments. This is materially stronger than moment-only characterization, because the PMF specifies exact probabilities for each delivered fidelity level.

4. Global cut-off, delivery-time trade-offs, and secret-key-rate evaluation

The global cut-off policy adds an explicit reset rule. Starting from an empty chain, a global timer increments by one every generation round. Ordinary swap-ASAP is followed, but if end-to-end entanglement is not established before the timer reaches $$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$3, all present entanglement is discarded and the protocol restarts. If delivery occurs before $$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$4, the timer resets and a new cycle begins. Analytically this corresponds to conditioning on

$$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$5

The normalized average delivered noise is

$$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$6

The corresponding recursion is

$$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$7

In the basis $$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$8, the map becomes

$$\Pr(t_i=t)=p(1-p)^{t-1}=(1-q)q^{t-1}, \qquad q\equiv 1-p.$$9

with

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$0

The finite-$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$1 linear form is

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$2

The cut-off generating function is again explicit, though cumbersome, and its smallest positive singularity yields

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$3

for the cut-off denominator singularity $$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$4 (Goodenough et al., 2024).

The practical importance is that fidelity moments and delivery-time quantities can be evaluated without Monte Carlo simulation. The single-attempt success probability under cut-off is

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$5

Without cut-off,

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$6

Under global cut-off, the expected delivery time is

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$7

where

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$8

For QKD implications, the paper uses fully asymmetric BB84. The secret-key fraction is

$$\Lambda \ket{\psi}\!\bra{\psi} + (1-\Lambda)\frac{\mathbb I}{4}, \qquad \sqrt{2}\ket{\psi}=\ket{00}+\ket{11},$$9

with

$$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$0

The secret-key rate is secret-key fraction divided by expected delivery time. Without binning, one uses $$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$1. With binning by roughness $$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$2, one uses the exact PMF,

$$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$3

By convexity,

$$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$4

The paper’s practical conclusion is that both cut-off and binning can materially improve repeater performance, and in low-success-probability regimes a cut-off can be essential to obtaining nonzero key rate at all (Goodenough et al., 2024).

5. Network-layer comparison with sequential swapping

A separate study fixes the link layer and varies only the network-layer protocol on a chain of length $$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$5. Each elementary link is controlled by an independent reinforcement-learning policy from the WN2M2 setting of Yau et al., trained for a six-state-QKD secret-key-rate objective; the network-layer rule is therefore the sole independent variable. Simultaneous SWAP-ASAP is compared against sequential entanglement swapping, where partial chains are grown hop by hop and stored in explicit chain buffers $$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$6 (Srivastava et al., 5 May 2026).

The system distinguishes internal communication memory with coherence time $$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$7 from external storage memory with coherence time $$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$8. Link buffers are per-link deques of capacity $$F=\Lambda+\frac{1-\Lambda}{4}=\frac{3}{4}\Lambda+\frac{1}{4}.$$9, while sequential additionally maintains external-memory chain buffers. The key timing parameter is the per-link heralding latency

$\Lambda$0

for homogeneous links, or

$\Lambda$1

more generally. The central organizing variable is the dimensionless ratio $\Lambda$2. The stored-state model is

$\Lambda$3

and swapping two Werner states uses

$\Lambda$4

The mechanistic difference is explicit. Sequential swapping “holds partial chains in intermediate buffers between successive swaps.” Simultaneous SWAP-ASAP “holds no intermediate chain storage”: it waits only at the level of independent link buffers, then consumes one valid pair from each buffer and executes the full swap tree. The paper states that all $\Lambda$5 swaps are performed in

$\Lambda$6

rounds rather than $\Lambda$7. For $\Lambda$8, the four link-level pairs are combined through a balanced binary tree within the same controller action. The decoherence exposure is therefore limited to the ages of elementary-link pairs in the link buffers, not to the age of already-swapped partial chains (Srivastava et al., 5 May 2026).

The empirical regime structure is stark. Simultaneous SWAP-ASAP delivers a constant rate across the stressed sweep of $\Lambda$9. For symmetric topologies,

• on $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$00 km it delivers $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$01 bps across the full sweep;
• on $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$02 km it delivers $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$03 bps across the full sweep.

For bottleneck topologies it delivers $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$04–$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$05 bps, again invariant in $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$06. In the off-diagonal $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$07 sweep at $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$08 km, the simultaneous heatmap is flat to four decimals and gives $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$09 bps at every cell.

Sequential swapping exhibits a collapse regime, a recovery onset, and a saturation regime:

• no end-to-end deliveries for

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$10

• recovery begins at

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$11

• the simultaneous rate is reached only in the relaxed regime

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$12

Quantitatively, for symmetric $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$13 km, sequential gives $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$14 bps for

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$15

then $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$16 bps at $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$17, and $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$18 bps at $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$19, versus simultaneous’s constant $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$20 bps. For symmetric $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$21 km, sequential gives zero at

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$22

while simultaneous remains at $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$23 bps. At $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$24 and $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$25, sequential reaches only $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$26–$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$27 of the simultaneous rate depending on $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$28. In the relaxed reference regime $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$29, the two protocols agree to within $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$30; for the symmetric $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$31 s case, the reported values are $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$32 bps versus $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$33 bps at $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$34, and $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$35 bps for both at $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$36 (Srivastava et al., 5 May 2026).

The paper’s interpretation is careful. Sequential swapping is not claimed to be fundamentally inferior. Rather, the penalty is presented as a near-term, hardware-limited effect caused by finite external-memory coherence and the need to buffer partial chains between swaps. This suggests that the decisive issue is not the abstract possibility of local-growth protocols, but the current ratio of external-memory coherence time to heralding latency.

6. Performance metric, assumptions, and terminological boundaries

The comparative network-layer metric is a secret-key-rate-like utility,

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$37

where $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$38 is the six-state-protocol entropy expression

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$39

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$40 is the mean end-to-end fidelity over deliveries in a trial, and $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$41. This makes the reported bps values comparable across simultaneous and sequential protocols because fidelity and mean inter-delivery interval are computed symmetrically (Srivastava et al., 5 May 2026).

Both analytical and simulation results are subject to strong scope conditions. In the homogeneous exact analysis, the chain is equally spaced and homogeneous, time is discrete, swaps are deterministic and immediate, endpoint decoherence is neglected in the main formulas, link attempts are synchronized and parallel, and the chain does not keep generating fresh links while a previous cycle remains unresolved. Classical communication delay is neglected in the main cut-off analysis. Exact PMF reconstruction stops at $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$42, and exact moment expressions are reported up to $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$43, for symbolic-computation rather than conceptual reasons (Goodenough et al., 2024).

In the network-layer comparison, the study is a proof of principle at fixed $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$44, on a single pre-selected path, in a single-flow setting. Ideal local operations are assumed, the external-memory decoherence model is Werner-state depolarizing decay, and classical communication overhead is idealized. The paper notes that including signaling budgets would sharpen the comparison, especially because sequential would require $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$45 signaling rounds versus

$$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$46

for simultaneous SWAP-ASAP. It also notes that sequential’s chain-assembly window scales linearly with $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$47, whereas simultaneous remains a single-tick collapse, suggesting that the near-term penalty may worsen for longer chains, though it is not quantified beyond $$\text{Sim.\ SWAP-ASAP}([F_1,\ldots,F_m]): \quad \text{if }m=1\text{ return }F_1,\quad F_{\mathrm{swap}}(F_L,F_R)$$48 (Srivastava et al., 5 May 2026).

A further source of confusion is the phrase “SWAP-ASAP” outside quantum networking. In quantum-control literature, “ASAP” can mean shortcuts to adiabatic passage. A cavity-QED paper on a deterministic SWAP gate uses Lewis–Riesenfeld invariant-based inverse engineering and quantum Zeno dynamics, but its gate is realized as three sequential shortcut-assisted population-transfer steps rather than as a simultaneous entanglement-swapping schedule. That usage is therefore terminologically related but operationally distinct (Liang et al., 2015).

The most precise synthesis is therefore narrow. Simultaneous SWAP-ASAP, in the quantum-network sense, denotes an ASAP entanglement-swapping policy with strong coordination: either an explicitly centralized wait-until-all-links-ready collapse of the full swap tree, or the closely related canonical repeater semantics in which links are generated in parallel and eligible swaps fire immediately. The central technical questions are then the stochastic accumulation of memory noise, the exact fidelity distribution and its moments, the cut-off trade-off between fidelity and delivery time, and the architectural advantage gained by eliminating intermediate chain storage under finite coherence.