Adaptive Purification Controller (APC)

• Adaptive Purification Controller (APC) is an automated module that optimizes entanglement distribution by dynamically adjusting purification protocols under varying quantum channel conditions.
• It employs dynamic programming with Pareto pruning to evaluate multi-hop paths, balancing fidelity, resource usage, and operational delay to maximize goodput.
• APC supports both bipartite and multipartite state strategies, adapting between BBPSSW and DEJMPS protocols to overcome fidelity cliffs in noisy quantum networks.

An Adaptive Purification Controller (APC) is an automated control-plane module designed for quantum networks to optimize entanglement distribution protocols under dynamically varying device and channel conditions. By formulating the selection of entanglement purification strategies as a resource allocation problem, the APC enables dynamic protocol-switching and depth adaptation to maximize goodput—the rate at which high-fidelity entangled pairs are delivered. It leverages a dynamic programming (DP) planner with Pareto pruning to efficiently generate purification and swapping plans that are robust to fluctuations in physical link parameters, including photon loss, memory decoherence, and gate error rates. The APC framework extends beyond bipartite purification, accommodating multipartite GHZ state distillation and continuous variable (CV) quantum systems (Kulkarni et al., 26 Jan 2026).

1. System Architecture and Workflow

The APC operates as an intermediary in the quantum network stack, interfacing between the routing layer and the quantum execution driver. Its core function is to receive a routed network path alongside per-link physical parameters and output a joint purification and swapping plan. The primary objective is to satisfy a specified end-to-end fidelity threshold $F^\star$ while maximizing metrics such as goodput or minimizing resource expenditure.

Inputs:

• Path of $H$ links $(\ell_1,\ldots,\ell_H)$
• Per-link parameters: raw generation success probability $p_i$, base fidelity $F_{0,i}$ (Werner/Bell-diagonal), effective memory coherence time $T_{2,i}^{\rm eff}$, gate error rates $(p_{1,i},p_{2,i},p_{{\rm meas},i})$, link length and associated classical round-trip $t_{\rm class}$, and BSM success $p_{{\rm BSM},i}$

Control loop and planning:

• Continuous monitoring or polling for device updates (loss, $T_2$ drift, gate calibration)
• On invocation (APCController.plan), returns plan $\Pi = \left\{ (\ell_j, r_j, P_j) \right\}_{j=1}^H$ where $r_j$ is the number of purification rounds and $P_j \in \{\text{BBPSSW},\text{DEJMPS}\}$
• Auxiliary outputs: predicted end-to-end fidelity, success probability, makespan, and expected EPR-pair cost

State and decision variables:

• Each per-link option is $u_j = (\ell_j, r, P)$ for protocol $P$ and purification depth $r$
• Partial/path prefix state after $j$ hops: $$x_j = (F_j, C_j, T^{\rm gen}_j, T^{\rm swap}_j, P_j, \Pi_j)$$ where $F_j$ is current fidelity, $C_j$ is expected EPR-pair use, $T_j$ aggregates time, $P_j$ is cumulative success probability, $\Pi_j$ is the sequence of decisions

Objectives:

• Lexicographic minimization of makespan and EPR-pair consumption
• Goodput maximization:

$$G(\Pi) = \begin{cases} 0, & F_H < F^\star,\ \dfrac{P_H}{T_{\rm makespan}}, & F_H \geq F^\star\ \end{cases} , \quad J(\Pi) = -G(\Pi)$$

2. Dynamic Programming and Pareto Frontier Planning

The APC employs a frontier-based dynamic programming recursion that propagates non-dominated prefix states across the multi-hop path, explicitly balancing trade-offs in fidelity, temporal resources, and EPR-pair consumption.

• Bellman recursion step: For each prefix $x_{j-1}$ and per-link control choice $u_j$:

$$\phi_{\ell_j}(u_j) = (F^{\rm link}, p^{\rm pur}, C^{\rm link}, t^{\rm pur})$$

• State update:

  $$\begin{aligned} F_j &= \mathrm{SwapCompose}(F_{j-1}, F^{\rm link}),\ P_j &= P_{j-1} \times p^{\rm pur} \times p^{\rm BSM},\ C_j &= C_{j-1} + C^{\rm link},\ T_j^{\rm gen} &= \mathrm{GenAgg}(T_{j-1}^{\rm gen}, t^{\rm gen}),\ T_j^{\rm swap} &= T_{j-1}^{\rm swap} + t^{\rm swap} \end{aligned}$$

  • Pareto pruning: At each stage, retain only non-dominated states: A candidate $x$ dominates $x'$ iff $F_x \ge F_{x'}$, $$T_x^{\rm gen} + T_x^{\rm swap} \leq T_{x'}^{\rm gen} + T_{x'}^{\rm swap}$$, $C_x \leq C_{x'}$ with at least one strict inequality. The frontier is capped in width to ensure tractability.
  • Completion: Iteration continues until $j=H$; select the plan $\Pi_H$ with minimal $J(\Pi_H)$.

3. Protocol Families: BBPSSW vs DEJMPS and Purification Depth

The APC supports dynamic selection between BBPSSW and DEJMPS entanglement purification protocols based on path-specific channel characteristics.

• BBPSSW (for Werner input states $\rho_W(F)$):

$$p_{\rm succ}^{\rm BBP}(F) = F^2 + \frac{2}{3}F(1-F) + \frac{5}{9}(1-F)^2$$

$$F'_{\rm BBP}(F) = \frac{F^2 + \frac{1}{9}(1-F)^2}{p_{\rm succ}^{\rm BBP}(F)}$$

• DEJMPS (for Bell-diagonal inputs $\bm{\lambda} = (\lambda_1, ..., \lambda_4)$):

$$p_{\rm succ}^{\rm DEJ}(\bm{\lambda}) = (\lambda_1 + \lambda_2)^2 + (\lambda_3 + \lambda_4)^2$$

$$\lambda_1' = \frac{\lambda_1^2 + \lambda_2^2}{p_{\rm succ}},\quad \lambda_2' = \frac{2\lambda_1\lambda_2}{p_{\rm succ}}$$

Recurrence allows protocol cycling for $r$ rounds per link:

$$P_{\rm succ}^{(r)} = \prod_{j=0}^{r-1} p_{\rm succ}^{(j)},\quad C_{\rm pairs}^{(r)} = \frac{2^r}{P_{\rm succ}^{(r)}}$$

Output fidelities and resource costs after $r$ rounds are iteratively computed through these update rules.

4. Extensions: Multipartite and Continuous Variable Distillation

The APC framework extends to more complex quantum network primitives:

• Multipartite GHZ State Distillation: For $N$-qubit GHZ states $$\rho_{\rm GHZ} = F_{\rm GHZ}|\mathrm{GHZ}_N\rangle\langle\mathrm{GHZ}_N| + (1-F_{\rm GHZ})\sigma_\perp$$, $m$ stabilizer checks (using ancilla fidelity $F_{\rm anc}$ and measurement error $p_{\rm meas}^{\rm GHZ}$) yield:

$$q_{\rm good} \approx F_{\rm anc}(1 - p_{\rm meas}^{\rm GHZ})$$

$$p_{\rm succ}^{\rm GHZ} \approx p_{\rm good} q_{\rm good}^m + p_{\rm bad} q_{\rm bad}^m$$

$$F_{\rm out} = \frac{p_{\rm good}q_{\rm good}^m}{p_{\rm succ}^{\rm GHZ}}$$

Cost includes ancillary pairs and rounds for classical and gate operations.

• Continuous-Variable (CV) Distillation via Noiseless Linear Amplification (NLA): Tracking effective squeezing $r$ ($\lambda = \tanh r$) and loss $\eta$:

$p_{\rm succ}^{\rm CV} \approx A (1/g^2)^K$

Updated triplet $(F_{\rm out}^{\rm CV}, C_{\rm CV}, t_{\rm CV})$ is computed analogously to the qubit case, allowing DP-based trade-offs for CV channels.

5. Performance Analysis

5.1 Elimination of Fidelity Cliffs and Resource Efficiency

Static protocols (fixed $r$) exhibit abrupt drops to zero goodput ("fidelity cliffs") and wasted resources when $F^\star$ exceeds achievable fidelity for the raw link or chosen purification depth. The APC avoids such cliffs by dynamically elevating the purification depth $r$ only as necessary and selecting the appropriate protocol family, ensuring that the system follows the upper-envelope of achievable goodput without over-allocating resources.

5.2 Quantitative Goodput Improvement

Simulation results show up to an order-of-magnitude higher goodput with the APC compared to any fixed-depth strategy. Non-zero delivered rates are possible even in noise regimes where all static schemes fail. Dynamically alternating between BBPSSW and DEJMPS, the APC tracks the highest goodput available within $F^\star \in [0.84, 0.91]$.

5.3 Scalability and Operational Overhead

Planning time for a chain scales approximately linearly with the number of hops: for a 1000-link chain, total planning completes in 6–7 seconds ($\sim$6-7 ms/link). Parallel planning retains linear scaling with independent requests, and per-link planning latency is always under 1% of the network makespan, confirming suitability for real-time operation.

A summary of key parameter settings used for evaluation is presented below:

Parameter	Typical Value/Range	Context
Fiber attenuation	0.2 dB/km	Channel noise
Memory coherence time	$T_2 \in [1{\rm ms},1{\rm s}]$	Storage limits
Gate error rates	$p_{2q} \in [10^{-4}, 10^{-2}]$	Gate fidelity
Max purification rounds	$R_{\max} = 6$	DP tractability
Planning time (per link)	6–7 ms (1000 links)	Real-time feasibility

6. Implementation Sketch and Algorithmic Structure

The core DP with Pareto pruning is outlined as follows:

Pseudocode:

for each link j = 1 to H: F_j := {} for each prefix state x in F_{j-1} and control u_j in U_{\ell_j}: compute extension x' via physics update add x' to F_j prune F_j to non-dominated states (cap width) return argmin_{x in F_H} J(x)

This approach ensures that only non-dominated solutions propagate through the DP recursion, balancing fidelity, resource usage, and operational delay. Maximum frontier width is enforced for computational tractability.

7. Context and Significance Within Quantum Networking

The Adaptive Purification Controller constitutes a significant advance in the automated orchestration of quantum network protocols under real-world hardware fluctuations. By mitigating "fidelity cliffs" and resource wastage endemic to static schemes, APC-mediated planning accommodates heterogeneity in devices and operational conditions. The approach is extensible to multipartite and CV quantum information tasks, demonstrating robustness and practical computational overhead compatible with real-time networked quantum applications (Kulkarni et al., 26 Jan 2026).