Rate-Fidelity Control for Wide-Area Quantum Links

Rate-Fidelity Control for Wide-Area Quantum Links

arXiv:2608.07163v1 [quant-ph] 7 Aug 2026

CONNOR CLAYTON, University of Maryland CORY NUNN, National Institute of Standards and Technology; University of Maryland QUINN CARMACK, University of Maryland WAYNE MCKENZIE, Laboratory for Telecommunication Sciences ANNE MARIE RICHARDS, Laboratory for Telecommunication Sciences XIAODI WU, University of Maryland BOBBY BHATTACHARJEE, University of Maryland Quantum network links must distribute entanglement at high rates while satisfying application-specified fidelity demands. However, wide-area deployed fiber links suffer from polarization drift which destabilizes end-to-end fidelity and forces periodic compensation. Current deployments often use active stabilization with fixed control policies, and improvements generally stem from advances in quantum hardware. Meanwhile, software control remains relatively underexplored. Here, we formulate quantum link operation as a joint control problem over tunable rate-fidelity tradeoffs and uncontrollable link drift. From this framework, we construct a link control protocol that dynamically adapts source pump power and polarization compensation to maximize entanglement distribution rate subject to a minimum fidelity constraint. We evaluate the protocol through trace-driven simulations driven by data from a 64 km deployed optical fiber. Compared with optimized static policies, our adaptive controller improves mean entanglement distribution rate by 14% over a 24 hour trace, without requiring any offline policy optimization. Our results show that software-based physical layer control can provide a practical mechanism for improving near-term quantum link performance without requiring additional quantum hardware.

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INTRODUCTION

Quantum networks promise to provide revolutionary capabilities in areas such as secure communications [5, 16], distributed quantum sensing [17, 19], and modular quantum computing [1, 32]. To realize the utility of these technologies, quantum network architectures and protocols must scale both in terms of geographic distance and quality of information transfer [47]. The scaling of these metrics are at odds with each other. Information in a quantum network exists in the form of quantum bits (qubits) whose signals are necessarily weak, with each transmitted qubit generally consisting of a single photon. Quantum communication is therefore highly susceptible to loss and noise, both of which scale unfavorably with increasing distance. Most applications of quantum networks require the distribution of high-quality entanglement between remote parties. Entanglement is a property of two or more qubits that allows for correlations even across large distances. The quality of an entangled quantum system can be measured by its fidelity 𝐹 , where 0 ≤ 𝐹 ≤ 1 and 𝐹 = 1 indicates that the system is in the “ideal” target state. In practice, system imperfections and environmental noise cause 𝐹 to be less than 1; however, such noisy states are often sufficient and a higher-fidelity state may be attained by consuming many noisy copies through a process known as entanglement distillation [21]. Significant efforts are underway to design, develop, test, and deploy hardware components which will enable high-fidelity transmission of entanglement [6, 24, 29]. Concurrently, considerable effort has been put into developments at higher layers of the quantum protocol stack [22] which manage Authors’ Contact Information: Connor Clayton, cbclayto@cs.umd.edu, University of Maryland; Cory Nunn, cory.nunn@nist. gov, National Institute of Standards and Technology; University of Maryland; Quinn Carmack, qcarmack@terpmail.umd.edu, University of Maryland; Wayne McKenzie, wmckenzie@ltsnet.net, Laboratory for Telecommunication Sciences; Anne Marie Richards, arichards@ltsnet.net, Laboratory for Telecommunication Sciences; Xiaodi Wu, xwu@cs.umd.edu, University of Maryland; Bobby Bhattacharjee, bobby@cs.umd.edu, University of Maryland.

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resources to deliver entanglement in complex networks [9, 12, 13, 42, 46]. However, there is a gap in the current research landscape: most protocols for wide-area quantum networks generally assume high-volume, consistent-fidelity entanglement generation — conditions that the physical layer is not expected to meet in the near- to medium-term [29]. We propose a framework for considering optimization problems relevant to the performance of present-day quantum networks. The family of optimization problems we define here accounts for both controllable parameters of quantum link hardware, which admit tradeoffs between fidelity and performance metrics like entanglement distribution rate, and uncontrollable parameters that diminish link fidelity, such as polarization drift. Neither set of parameters has been systematically analyzed for wide-area links, especially in conjunction. We introduce an optimization problem that considers both sets of parameters simultaneously, and present a link control protocol that tunes the controllable parameters and adapts to the uncontrollable to achieve high entanglement distribution rates over a wide-area link. Our adaptive control protocol takes as input an end-to-end entanglement fidelity target, and aims to maximize the entanglement distribution rate over the link. Absent link degradation, the entanglement rate can be easily maximized with a proper selection of controllable parameters. However, in a realistic wide-area setting, physical properties and environmental factors (e.g., temperature, cloud cover, wind) degrade entanglement in volatile ways. Our protocol queries a pre-built model of link degradation to try to predict how the link will behave, and then sets the controllable parameters to compensate for the expected degradation. Our main result is a classical control protocol that boosts the practical utility of wide-area quantum links by maximizing the rate of entanglement distribution at a consistent, user-defined fidelity. The protocol requires no additional quantum hardware and introduces minimal classical overhead. To make things concrete, we analyze an instance of our protocol based around a 64-km deployed fiber in a major metropolitan area of the US (specific location omitted for anonymity during double-blind review). Our link layer protocol directly benefits wide-area quantum networks based on quantum repeaters, as these networks operate by swapping elementary entanglements and therefore the rate and fidelity of each physical hop directly constrains end-to-end performance. The rest of this paper is structured as follows: we introduce relevant hardware properties and discuss the specific quantum hardware used in our experiments in Section 2. In Section 3, we present modeling of the components used by our adaptive protocol. Section 4 introduces a sequence of optimization problems that decompose different aspects of entanglement distribution, while Section 5 describes the control protocol itself. Section 6 evaluates the protocol against existing techniques, Section 7 discusses generalization to other hardware and higher network layers, and Section 8 concludes. 2

ENTANGLEMENT DISTRIBUTION AT THE PHYSICAL LAYER

The elementary unit of a quantum network is a physical link that generates and transmits entanglement. The hardware involved in this point-to-point entanglement distribution consists of a source, link, and receiver, and is directly relevant for applications like entanglement-based quantum key distribution [40], sensing, and metrology [50], wherein the receiver immediately characterizes entangled states. This framework can be extended to more sophisticated link models, like those in future quantum repeater networks, which rely on additional technologies like quantum memories [48]. 2.1

Entanglement Sources

An entanglement source is a device capable of generating pairs of entangled qubits. For longdistance quantum communication, information is encoded into one of many physical degrees

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of freedom within individual photons. These “flying qubits” must be generated with dedicated quantum hardware. Entanglement can be generated via clever manipulation of individual qubits from single-photon sources (as in [14]), or from sources that natively produce entangled photon pairs. In practice, entanglement generation is typically a probabilistic process, though technology for on-demand, deterministic sources is improving [15, 29]. Sources can be characterized in part by their photon generation rate, single-photon purity (quantifying the degree of errant multi-photon emission), and the resulting quantum state fidelity [7]. Many source properties are fixed during the engineering process; however, these systems still typically allow some degree of tunability. In this work, we consider a simple example of modulating the pump laser power for a source based on spontaneous parametric down-conversion (SPDC); details are covered in Section 3.1. Sources based on four-wave mixing or emission from solid state quantum memories also often exhibit tunability via pump laser power. Additional examples of controllable parameters include central wavelength and bandwidth temperature tuning for both SPDC and quantum dot sources. 2.2

Quantum Receivers

Once an entangled qubit reaches its destination, the information may be stored, measured, or processed through some logical operation. Ubiquitous across all receiver operations is the singlephoton detector, a device capable of identifying the presence of individual photons [7]. Single-photon detectors are characterized by their efficiency and dark count rate (among other measures) which indicate the true-positive and false-positive rate of photon detection, respectively. State-of-the-art performance is achieved using superconducting nanowire single-photon detectors (SNSPDs), reaching upwards of 98% efficiency at standard telecom wavelengths [30, 39]. Similar to sources, detectors can offer some degree of tunability. For example, SNSPD bias current can be adjusted to achieve (limited) increased detector efficiency. This comes at the expense of increased dark counts, but may still yield an overall advantage in some applications [34]. Additional tradeoffs can be introduced in the processing of photodetection signals, such as tuning the “coincidence window” to qualify the simultaneity of detection events [37], or the duration of a measurement for state characterization, trading integration time for reduced statistical uncertainty. 2.3

Polarization Drift Across a Link

Ideal quantum channels would perfectly preserve quantum states between source and receiver, but real-world links invariably introduce some degree loss and decoherence of quantum information as photons interact with the environment. A central characteristic of optical communication, classical or quantum, is polarization. Physically, polarization describes the direction of oscillating electromagnetic fields that comprise light. For most quantum networking purposes, maintaining consistent polarization between all parties is strictly required, especially when the polarization itself encodes the quantum information. This poses a great challenge for transmission over realistic quantum links, whereby polarization state fidelities inevitably decay without intervention. The objective of polarization stabilization is to ensure a photon’s state of polarization (SOP) remains unchanged from source to receiver. For our purposes, the SOP can be described by three parameters (𝑆 1, 𝑆 2, 𝑆 3 ) which define a unit vector known as a Stokes vector. The received vector 𝑣®′ = (𝑆 1′ , 𝑆 2′ , 𝑆 3′ ) will be altered by optical fiber imperfections and environmental conditions. These can also change over time — a phenomenon known as polarization drift. While the magnitude of 𝑣®′

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Fig. 1. (a) Schematic of our quantum link. Entanglement generation and automated polarization compensation (APC) signals are time multiplexed. Measurements at the APC receiver inform corrections at the polarization controller. (b) Feedback loop used for link control. Polarization fidelity (𝐹 pol ) measured by the APC acts as input to the link control protocol and its associated models, which in turn determine hardware parameters at the source site.

may change1 , the most significant and dynamic changes are usually in the vector direction. The angle between the desired and received vectors is the drift angle 𝜃 , from which the polarization fidelity 𝐹 pol is calculated [23]: 1 𝐹 pol = (1 + cos(𝜃 )) (1) 2 The exact drift conditions are unique to each link, and can be highly variable over time depending on the environment. Significant effort goes into maintaining polarization over a quantum link. This process commonly works by sending classical reference signals, measuring the change in polarization of those signals, then applying a correction to offset this change [26, 28]. In the following subsection, we discuss a commercial solution to polarization stabilization that operates on this principle. 2.4

Concrete Hardware Instance

To make the ideas of this paper concrete, we analyze a protocol instance based around hardware available to us, described here and sketched in Figure 1. However, the same ideas extend naturally to many other experimental setups, see Sections 2.5 and 7.2. Our simulations are driven by polarization data gathered over a 64km quantum link in the the Washington DC metropolitan quantum network (DC-QNet) [33]. The link consists of a 2km buried optical fiber connecting the Laboratory for Telecommunication Sciences (LTS) to a switch at the University of Maryland, which routes photons through a 62km partially-aerial fiber to the National Institute of Standards and Technology (NIST). Figure 2 sketches the route of this link. Transmission loss over this fiber link was measured to be approximately 18dB. Over 70% of the fiber is aerial, which leads to greater polarization instability compared to buried fiber due to its more direct interaction with environmental changes. A schematic of our entanglement source and detector setup is shown in Figure 3. Our source of polarization-entangled photon pairs is based on SPDC, the most mature (and therefore widelydeployed) class of sources used for quantum communications [15]. The power of the input pump laser driving the source can be easily modulated to determine the rate and fidelity of entanglement generation (see Section 3.1 and Figure 4a). A more detailed description and diagram can be found in Appendix A. Our receiver is comprised of a pair of polarization analyzers and SNSPDs (single-photon detectors). The analyzers allow us to perform full quantum state characterization (tomography) on the 1 via polarization-dependent loss or polarization mode dispersion [18].

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Fig. 2. Metropolitan context of the 64km deployed fiber link, part of the Washington DC metropolitan quantum network (DC-QNet). The source is located at the Laboratory for Telecommunication Sciences (LTS) in College Park, MD, USA. The route passes through a switch at the University of Maryland, then continues through a 62km partially-aerial fiber to the National Institute of Standards and Technology (NIST) in Gaithersburg, MD, USA. The exact routes of these two fibers are not known and therefore sketched as straight dashed lines.

polarization-encoded photons emitted from the source [2]. From this, we compute the entanglement fidelity to an ideal Bell state. Note that the procedure in Figure 3 depicts local characterization of the source-receiver system. For entanglement distribution, one qubit is detected locally near the source, while the other is instead measured at remote location across a fiber link (see Figure 1a). 2.4.1 Polarization Drift. In our experiments, polarization drift causes the fidelity 𝐹 pol of our polarization-encoded qubits to degrade over time. Our protocol relies on a sufficient understanding of the rate of change in polarization (the “drift rate”) in order to make informed parameter adjustment decisions. Figure 5 displays measured polarization drift over a 48 hour period over the link. The plot shows the raw polarization Stokes parameters during this period with the distilled mean drift rate | 𝑑𝜃 𝑑𝑡 | overlaid. Drift rates are strongly correlated with time of day, with significantly higher drift over daytime hours.

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Detectors

Fig. 3. Schematic of experimental setup used to characterize SPDC entanglement source and receiver with controllable pump laser power. A complete diagram and its description can be found in Appendix A.

2.4.2 Polarization Compensator. During operation, the polarization drift on our link is measured and corrected using a commercial2 Automated Polarization Compensation (APC) device [10]. The APC measures polarization using in-band classical signals, meaning that no quantum data can be transmitted while it is in operation. Figure 1a illustrates how the path must be fully switched away from the entanglement source and detector while the APC is in use. The APC may be probed at any time. On each probing, three parameters must be passed to the device: (1) a triggering fidelity threshold 𝐹 trigger , (2) a target fidelity threshold 𝐹 target , and (3) a timeout duration 𝑇timeout . When the APC is probed, it initiates a “fidelity check”. A fidelity check takes a constant amount of time, 44ms for our link, and measures the current polarization fidelity 𝐹 pol . If 𝐹 pol > 𝐹 trigger , the APC concludes operation and entanglement distribution resumes. However, if 𝐹 pol ≤ 𝐹 trigger , the device initiates a compensation routine, during which it adjusts the polarization controller in an attempt to raise the polarization fidelity until 𝐹 pol ≥ 𝐹 target . This routine takes an unspecified amount of time and is not guaranteed to succeed. If the target fidelity has not been met after 𝑇timeout seconds, the routine aborts and entanglement distribution resumes. 2.5

Generality of the Control Abstraction

While the results in this paper are evaluated using a particular wide-area link with a particular source, detector, and polarization controller, the control logic is defined over a hardware-agnostic link interface. This interface consists of three replaceable components: (1) a set of controllable hardware settings that define a tradeoff between entanglement generation rate and fidelity, (2) a model for time-varying link degradation, and (3) a compensation mechanism and corresponding model mapping corrective actions to expected fidelity improvement and overhead. For the source-detector setup, the relevant requirement is a controllable rate-fidelity operating frontier. In our evaluated system, pump power provides this control. Figure 4 shows this behavior for our SPDC source and for a second commercial source based on four-wave mixing [11]. Four-wave mixing probabilistically generates entangled photon pairs by a fundamentally different physical mechanism than SPDC [15], yet importantly, both systems exemplify the same control abstraction: increasing pump power increases the entanglement generation rate while reducing entanglement fidelity over the operating regime. Note, in addition, that our approach extends well beyond these examples. The same abstraction encompasses many quantum memory-based links as well, with controllable rate-fidelity tradeoffs common in links that entangle pairs of trapped ions [25], neutral atoms [38], and color centers in diamond [12, 45]. In all these examples, the tradeoffs are modulated by source laser power; however, 2 Commercial equipment and software referred to in this work is identified for informational purposes only, and does not

imply recommendation of or endorsement by the National Institute of Standards and Technology, nor does it imply that the products so identified are necessarily the best available for the purpose.

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Fig. 4. Controllable rate-fidelity tradeoffs across two entanglement source technologies. In both sources, increasing pump power increases entanglement generation rate but reduces entanglement fidelity. The simulations in this paper use the SPDC characterization in (a).

this need not be the case, as many other control knobs exist depending on source and detector configurations. We list several such parameters in Sections 2.1 and 2.2. The protocol also does not require the rate-fidelity relationship to be linear or one-dimensional, as the same role can be played by any calibrated set of operating points, including a multidimensional Pareto front over rate and fidelity. On the link side, the relevant requirement is time-varying degradation that can be measured or conservatively bounded over the control horizon, together with a corrective action that can improve the link state at some cost. Polarization drift is a ubiquitous example of a link-degrading phenomenon in optical quantum communication, and its magnitude depends on link-specific factors such as length and deployment environment. These factors enter the protocol through the learned drift model rather than through hardware-specific assumptions. A different deployment would replace or update the training data and prediction model, but the structure of the controller remains unchanged. Similarly, our use of a commercial APC device is one instance of a more general compensation primitive. The protocol’s only requirement is a way to estimate when compensation is worth its cost through a sufficiently validated model. Other compensation hardware may expose different parameters or incur different overheads, but these differences can be incorporated by reworking the compensation response model. The APC model used in our simulator is therefore an instantiation of this interface, not a protocol requirement. 3

MODELING THE EVALUATED QUANTUM LINK

We now instantiate the three required interfaces described in Section 2.5 for the deployed link used in our evaluation by describing the modeling of relevant hardware components. These models

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abstract the performance characteristics of each component and are used as input to our control protocol described in Section 5. 3.1

Source-Detector Model

For the evaluated deployment, we instantiate the source-detector model using empirical data from an SPDC source designed to produce polarization-entangled photon pairs. In Figure 4a, our source is characterized for a range of input pump powers, from 25 to 300 mW. The upper subplot shows the quantum state fidelity to an ideal Bell state, computed from the tomographic measurement shown in Figure 3. The lower subplot shows the rate of coincident detection events at each of the two SNSPDs, normalized to the maximum observed value. This rate depends on the efficiency of the SPDC process, coupling into optical fiber, detector efficiency, and other link losses between source and receiver. Details of the data in Figure 4a and its processing are given in Appendix B. As pump power increases, the photon pair generation rate increases linearly in this regime. However, due to the probabilistic nature of SPDC, increasing the probability of emission per each laser pulse also increases multipair emission– the likelihood that more than one photon pair is generated from a single pulse. This effectively adds noise and reduces the entanglement fidelity 𝐹 sd , as the measured polarization correlations are masked by independently generated pairs that are not correlated to one another. The maximum fidelity in the source-detector setup of Figure 3 (average 𝐹 sd ≈ 0.95) is observed at low pump powers where multipair probability is negligible, but the rate is also reduced. Hence, the pump power serves as a control parameter to appropriately balance rate and source fidelity. 3.2

Polarization Drift Prediction Model

Polarization drift is a complex physical process that we do not attempt to model precisely in this paper, although related models have been constructed [4]. In principle, one could model polarization

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Algorithm 1 Simulation of Polarization Compensation Input: the initial fidelity 𝐹 0,pol , the target fidelity 𝐹 target , the start time 𝑡 0 Output: total compensation time, final fidelity 1: 𝜃, 𝜃 target ← angles_from_fids(𝐹 0,pol, 𝐹 target ) ⊲ Convert fidelities to drift angles via Eq. 1 2: 𝑡 ← 𝑡 0 3: while 𝜃 > 𝜃 target do 𝜂 ⊲ Simulate a noisy gradient descent step 4: 𝜃 = 𝜃 − 2 sin 𝜃 + Δ · 𝑑 (𝑡) 5: 𝑡 ←𝑡 +Δ 6: end while 7: return 𝑡 − 𝑡 0, fid_from_angle(𝜃 )

drift by monitoring the relevant environmental factors (wind, incident sunlight, vibrations from vehicles, etc.); however, in practice, it is impractical over a link with 10s of kilometers of aerial fiber. Since we are only interested in the aggregate change in polarization over time, a simpler empirical method suffices for our purposes. To develop our model, we instead turn to a more readily accessible variable: measured drift rates. We find that polarization drift over a short period of time is strongly correlated with drift over a subsequent short period of time. We build our model empirically from a dedicated experiment which tracked the state of polarization over a period of 48 hours3 . This data is shown in Figure 5. The training of our model is simple: for a pair of durations (Δ𝑡 1, Δ𝑡 2 ), we sample a time 𝑡 and find the corresponding three stokes vectors 𝑣𝑡 −Δ𝑡1 , 𝑣𝑡 , 𝑣𝑡 +Δ𝑡2 . Then we compute the two consecutive drift angles as 𝜃 1 = angle_between(𝑣𝑡 −Δ𝑡1 , 𝑣𝑡 ) and 𝜃 2 = angle_between(𝑣𝑡 , 𝑣𝑡 +Δ𝑡2 ). Repeating this for 𝑁 = 106 randomly chosen values of 𝑡 results in a distribution of drift angles 𝜃 2 given parameters (Δ𝑡 1, Δ𝑡 2, 𝜃 1 ). We repeat this for 64 representative values of (Δ𝑡 1, Δ𝑡 2 ) to create a general and practical model. 3.3

Modeling the Polarization Compensator

Simulating the effects of the APC requires a model to predict its run time and resulting fidelity upon termination, given the current link state. These aspects of the device have not been studied, so we developed a model based on conversations with the device manufacturer and 70 hours of experimental data from an APC test over a similar link (62km aerial fiber). The details of the APC’s compensation algorithm are proprietary, so we make several assumptions in the construction of our model. We know that the routine is gradient-descent-based, repeatedly measuring the polarization state and applying corrections until the fidelity target is met or the routine times out. We assume vanilla Riemannian gradient descent, which is designed for optimization on smooth manifolds such the surface of the 3-dimensional unit sphere of Stokes vectors. The update step of this algorithm is 𝜂 𝜃 𝑘+1 = 𝜃 𝑘 − sin 𝜃 𝑘 (2) 2 where 𝜃 𝑘 is the angle between the ideal and realized vectors at iteration 𝑘 of gradient descent and 𝜂 is the step size. Second, we assume that the effect of polarization drift on compensation can be modeled by a noise term within each gradient descent step. We assume this term is equal to the product of the current drift rate 𝑑 (𝑡) and the duration of each step Δ = 27.7ms. 3 Given a sufficiently long operation time, such a model could be built without a dedicated experiment by gathering drift

statistics during operation.

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The modeled compensation routine is shown in Algorithm 1; more details on the fit to experimental data can be found in Appendix C. We note that the results in this paper assume this model, and deviations from this model may affect the results upon deployment. 4

OPTIMIZING ENTANGLEMENT DISTRIBUTION

Given an understanding of the available hardware and modeling techniques, described in the previous section for one deployed link, we now introduce a more general framework that decomposes entanglement generation into a sequence of optimization problems. In this section, we focus on optimizing the performance of a single link. We extend the framework to consider entanglement across multiple links in Section 7.3. 4.1

Opt-0: Optimizing Controllable Parameters

We first consider a stable experimental setting in which link drift is negligible or ignored, such as a short, closed laboratory link or a brief experiment with pre- and post-calibration checks. Even in this regime, source- and receiver-side parameters must be tuned to maximize entanglement rates under a minimum (and possibly time-varying) fidelity constraint. We define Opt-0 as the problem of selecting controllable parameters (e.g., pump power at the source, integration time at the receiver) to maximize entanglement rate subject to a target fidelity. An Opt-0-type optimization has been integrated into a quantum link layer protocol [12, 36], where a parameterized microwave pulse at the source is used to control the rate-fidelity tradeoff. The class of problems posed by Opt-0 is more general, as it encompasses multiple controllable parameters at both the source and receiver and potentially more complex tradeoffs. Nevertheless, these remain the primary studies to consider link-level control in this context. 4.2

Opt-1: Managing Uncontrollable Parameters

For wide-area entanglement distribution, links are inevitably subject to time-varying and unpredictable noise, particularly for above-ground deployments exposed to environmental factors such as temperature, wind, and human activity. Some effects exhibit predictable patterns (e.g., diurnal temperature cycles), while others do not, complicating long-duration experiments. We define Opt-1 as the problem of optimizing the mitigation of these uncontrollable, performancedegrading link parameters. Prior work has addressed physical-layer aspects of Opt-1 over longdistance links, including clock synchronization [33] through the IEEE-1588 Standard (PTP [41] and related White Rabbit [31] protocols) and polarization stabilization using commercial compensation devices [10, 43]. However, prior to this work, these techniques have not been systematically optimized, particularly with respect to their impact on end-to-end system performance. Solutions to Opt-1 must maintain acceptable operating conditions while balancing downtime, compensation accuracy, and interference with the quantum signal. Compensation often requires pausing quantum data transmission, making its frequency and duration critical design parameters. Thus, Opt-1 involves trading off improved link quality against compensation overhead to maximize overall link utility. 4.3

Opt-2: Joint Optimization

Finally, consider again a general wide-area quantum link. Practical entanglement distribution can only be achieved if the link is able to consistently produce high-fidelity entanglement at a sufficient rate. Achieving this requires tuning of controllable parameters according to applicationspecified fidelity requirements and the current state of the link. Further, invocation of any link compensation algorithm must take into account the overhead of drift correction in conjunction with the current link state. This is problem Opt-2: joint optimization of controllable parameters

Rate-Fidelity Control for Wide-Area Quantum Links Variable 𝐹 𝑟 𝐹 min 𝐹 sd 𝐹 pol 𝛿 𝐹 pol 𝐹 trigger 𝐹 target 𝑇timeout 𝑇eff 𝑟¯

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Definition end-to-end fidelity end-to-end entanglement distribution rate minimum fidelity requirement source-detector fidelity polarization fidelity 𝛿-conservative prediction of 𝐹 pol APC triggering fidelity threshold APC target fidelity threshold APC timeout duration expected time until the next fidelity check average rate within a period

Table 1. Summary of variables used in this paper.

while compensating for the uncontrollable. A salient feature of Opt-2 solutions is the feedback loop between compensation overhead and controllable parameter settings, as both must be considered in concert to achieve high entanglement rates. The focus of this paper is a protocol for Opt-2. Part of our work concerned with source control is similar to [12], though the hardware involved, as we have described, is markedly different. The primary focus of our protocol is to explore the feedback loop intrinsic in solutions for Opt-2: scheduling compensations based on link performance, which depends on the parameter settings and compensation quality. We describe this protocol next. 5

DYNAMIC CONTROL PROTOCOL

The experimental setup described above defines a control problem. Our link protocol therefore has access to five control variables: • Source-detector parameter: (1) pump laser power • Drift parameters: (2) timing of fidelity checks, (3) APC triggering fidelity threshold, (4) APC target fidelity threshold, (5) APC timeout duration At time 𝑡, the link distributes entanglement at a rate of 𝑟 (𝑡) entangled pairs per second with approximate4 fidelity 𝐹 (𝑡) ≈ 𝐹 sd (𝑡)𝐹 pol (𝑡). The objective of a protocol here is to maximize the expected long-run average entanglement distribution rate subject to a hard minimum fidelity constraint 𝐹 min . Entangled pairs are discarded when the fidelity requirement is not met; i.e., 𝑟 (𝑡) = 0 whenever 𝐹 (𝑡) < 𝐹 min . The choice of this strict fidelity requirement is to ensure robust entanglement generation at the link level, as applications of quantum networks require entanglement of sufficient fidelity [12] and protocols involving multiple links typically assume a constant fidelity for each link. Static Protocols. Existing entanglement distribution protocols of this type make straightforward use of available resources and APIs by setting constant values for the five control variables. A recent experiment demonstrating state-of-the-art entanglement distribution across a metropolitan scale link uses constant values for all five control variables [43]. 4 The true end-to-end fidelity would be estimated experimentally by a prolonged measurement of many copies of the

transmitted quantum states, destroying the entanglement [3, 49]. In contrast, this 𝐹 (𝑡 ) model is efficiently computable for our real-time protocol, and should well approximate and scale monotonically with the true fidelity.

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5.1

Intuition

It is useful to think of the system as two independent components: (1) The source and detector form a single logical unit responsible for the creation and measurement of Bell states. The rate 𝑟 (𝑡) and fidelity 𝐹 sd of entangled pair generation are fully and continuously controllable by the pump laser power (see Section 3.1, Figure 4a). (2) The transmission medium responsible for distributing entangled photons between the source and detector is not directly controllable and only adds noise (and loss) to the entanglement generated through subsystem (1). Much of the effect of transmission is determined by the environment, can only be measured through active probing, and can only be controlled through active compensation (see Figure 5). From this decomposition, it becomes clear that the role of the pump power should be to counteract the effect of polarization drift. Since 𝐹 pol tends to decrease over time, we want to lower the pump power over time in order to raise 𝐹 sd and therefore sustain 𝐹 (𝑡) = 𝐹 sd (𝑡)𝐹 pol (𝑡) ≥ 𝐹 min . This approach will decrease our end-to-end rate over time, as a consequence of the decreasing pump laser power. Ideally, we would like keep our pump power as high as possible to maximize the end-to-end rate. However, this is likely sub-optimal as the polarization drift is unknown. Instead, we use our polarization drift prediction model to estimate the extent of polarization drift, and select pump power conservatively to ensure that we maintain 𝐹 (𝑡) ≥ 𝐹 min with high probability. Our methodology for meeting this criterion is explained in Section 5.2. This conservative power adjustment, while necessary to avoid high-penalty 𝐹 min violations, tends to reduce our rate more than is strictly necessary. In addition, the uncertainty in our polarization estimate increases with time. In order to correct possible sub-optimality, it is desirable to invoke a fidelity check to learn the current value of 𝐹 pol (𝑡). Doing so allows us to reset the pump power to the optimal value and gives us updated information on the current polarization drift levels. However, we do not want to invoke this capability too often, as we want to balance the information gain of a fidelity check with the downtime it incurs. The full policy is explained in Section 5.3. After some amount of time, we should perform polarization compensation in order to mitigate the effects of polarization drift, which forces our rate to generally decrease over time. To do so, we set the APC triggering fidelity such that we trigger a compensation when the fidelity has drifted low enough that the increase in rate from compensating outweighs the opportunity cost of downtime during compensation. We also set the value of the target fidelity threshold and timeout duration to balance the increase in 𝐹 pol with downtime. The details of our policy for setting the APC device parameters are given in Sections 5.4 and 5.5. 5.2

Pump Power Control Policy

Given perfect information of 𝐹 pol (𝑡), the optimal pump power control policy is to choose the pump power such that 𝐹 (𝑡) = 𝐹 min , i.e., 𝐹 sd = 𝐹 min /𝐹 pol .

(3)

This choice is guaranteed to maximize rate subject to the minimum fidelity constraint. During execution, however, the exact value of 𝐹 pol is only known immediately after a fidelity check. Given this uncertainty, our polarization drift model allows us to maintain a belief distribution over 𝐹 pol (𝑡). Using this, we control the pump power conservatively such that Pr[𝐹 (𝑡) < 𝐹 min ] ≤ 𝛿

(4)

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for a fixed confidence parameter 𝛿 ∈ (0, 1). Equivalently, power is chosen as if the transmission fidelity were equal to the 𝛿-quantile of the belief distribution. In our simulations, we select 𝛿 = 0.10, meaning that we assume a polarization drift rate in the 90th percentile according to our model. 5.3

Fidelity Check Initiation Policy

A fidelity check returns the exact current value of 𝐹 pol after a fixed downtime duration 𝑡 check , during which the rate 𝑟 (𝑡) = 0. Fidelity checks may be initiated at any time. The protocol schedules fidelity checks adaptively, based on a value-of-information criterion. Let 𝛿 ) denote the rate achieved using our current predicted 𝛿-conservative polarization fidelity. Let 𝑟 (𝐹 pol 𝑇eff denote the expected time until the next check, and let Λ1 and Λ2 be the expected throughput over the next 𝑇eff seconds if we do and do not perform a check, respectively. A fidelity check is initiated when 𝛿 Λ1 − Λ2 ≥ 𝑟 (𝐹 pol )𝑡 check .

(5)

This condition compares the expected rate gain from resolving uncertainty to the opportunity cost of downtime. Fidelity checks are therefore performed only when uncertainty induces a sufficiently large rate penalty. Λ1, Λ2 may be calculated through estimates of future rates using the polarization drift predictor — see Appendix D for details on efficient computation of Λ𝑖 . 5.4

Fixing Compensation Duration

Our choice of polarization compensation target fidelity 𝐹 target and timeout duration 𝑇timeout is based on a simple observation: that setting 𝐹 target = 1.0 ; 𝑇timeout = 𝑐

(6)

for some constant 𝑐 ensures that the compensation routine will take time 𝑐. To see this, note that the APC’s gradient descent approach combined with continuous drift effectively ensures 𝐹 pol (𝑡) < 1.0 at all times 𝑡. In our simulations, we choose 𝑐 := 1 second. This choice is unconventional but carries several benefits. First, assuming a constant compensation duration greatly simplifies the analysis of optimal compensation initiation, allowing for a much cleaner triggering policy (see Section 5.5). It also removes the need for potentially complex control policies governing 𝐹 target and 𝑇timeout . Second, limiting compensation time makes efficient use of the diminishing returns seen during the compensation routine. From Section 3.3, we expect that the increase in fidelity decays exponentially with each gradient descent iteration, while the loss of throughput during compensation is linear in time. As a result, the net benefit of additional compensation time is strictly decreasing; fixing the compensation time truncates this low-value tail in a principled manner. Finally, this policy is robust against unknown, time-varying polarization drift. From Section 3.3, we find that a high drift rate can prevent the compensator from reaching the target fidelity; this phenomenon commonly results in expensive compensation routines trying to reach an unachievable 𝐹 target (see [43]). In contrast, a fixed-duration policy avoids implicit assumptions about the achievable polarization fidelity and guarantees bounded downtime regardless of drift conditions. 5.5

Polarization Compensation Initiation

Polarization compensation routines naturally define “periods” of protocol execution, which we define to be the time between the end of one compensation routine and the end of the next. As an approximation to our objective of maximizing the long-run average entanglement distribution rate, we aim to maximize the average rate of each individual period.

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For a given time 𝑡, let 𝑟¯(𝑡) be the average rate of the current period if we were to initiate compensation at time 𝑡. Then we can show that the average rate of the period is maximized when the instantaneous rate equals the period’s average rate, and therefore we should initiate a compensation whenever 𝑟 (𝑡) ≤ 𝑟¯(𝑡). (7) See Appendix E for a derivation of this policy. Since the rate is determined by our chosen pump power and the compensation duration is set to a constant, we can easily track the quantities 𝑟 (𝑡) and 𝑟¯(𝑡). We initiate a fidelity check whenever the condition in Equation 7 is met to ensure that we do not operate in this suboptimal regime5 . Upon initiation, it is possible that 𝑟 (𝑡) is simply suboptimal due to our conservative choice of pump power. Therefore, we set 𝐹 trigger to be the value of 𝐹 pol that, if measured, would result in a rate that violates the condition in Equation 7. This is computed directly from our pump power policy. 5.6

Putting It All Together

We explain how the components of our link control protocol work together through a representative run over a 29-second period between two polarization compensations, shown in Figure 6. The following descriptions correspond to the circled numbers in Figure 6: (0) The solid green curve in the topmost subplot shows the end-to-end entanglement distribution rate 𝑟 (𝑡) at each time 𝑡. Notice that 𝑟 (𝑡) decreases in the time between each fidelity check (thin vertical lines), and that 𝑟 (𝑡) trends downward throughout the period overall. (1) The second subplot shows pump power and 𝐹 sd over time. Pump power is chosen to satisfy the condition in Equation 4. Notice that the trends in 𝐹 sd mirror that of the pump power, while 5 Note that this is a second criterion to initiate a fidelity check, supplemental to the condition given in Section 5.3

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trends in 𝑟 (𝑡) mimic the pump power. This stems directly from the experimental relationships between these quantities shown in Figure 4a. (2) The third subplot shows the trends in actual and predicted polarization fidelity. After each 𝛿=0.1 changes based on the observed drift since the previous check. fidelity check, our estimate 𝐹 pol 𝛿=0.1 < 𝐹 𝐹 pol pol throughout this example; in general, we expect this to be true at least 90% of the time based on our choice of 𝛿 = 0.10. (3) The bottom-most subplot shows the end-to-end fidelity as a result of the parameters from the plots above. At all times, the algorithm attempts to maintain an end-to-end fidelity above the minimum requirement 𝐹 min . (4) Most fidelity checks occur due to the condition in Equation 5. In this instance, the protocol determines that the expected rate gain from performing a fidelity check outweighs the associated loss. (5) To ensure 𝑟¯ is monotonically increasing, the protocol initiates a fidelity check whenever the instantaneous rate drops below the running average for the period (Equation 7). In this particular 𝛿 was overly pessimistic, so 𝑟 (𝑡) was unnecessarily low instance, the polarization prediction 𝐹 pol and a full compensation was not triggered. (6) Once again 𝑟 (𝑡) dips below 𝑟¯(𝑡), but this time the rate cannot be recovered above 𝑟 (𝑡) and a polarization compensation is initiated. Notice that this event coincides with a substantial drop in 𝐹 pol . (7) Each polarization compensation routine lasts a fixed duration of 1 second, as described in Section 5.4.

6

RESULTS

Here we present evaluations of our link control protocol, starting with a discussion of how the results were derived. Next, we compare our protocol to currently adopted practices, and present results under varying fidelity requirements as determined by higher level applications. Finally, we benchmark against a theoretical upper bound which assumes no polarization drift. 6.1

Methodology

All results are derived from a custom trace-driven simulator. Source code for this simulator and the underlying trace data are publicly available [8]. The underlying trace data were derived over the 64 km partially-aerial LTS-NIST fiber link and span three days. Each data point contains the Stokes parameters measured every 100 milliseconds, enabling fine-grained reconstruction of polarization drift. In our simulations, we use the drift data to compute the polarization fidelity 𝐹 pol over time, which is then used to evaluate the decisions made by the control algorithms. Now consider a “static” protocol that issues an APC probe with some fixed input parameters (𝐹 trigger, 𝐹 target,𝑇timeout ) every 5 seconds and maintains constant pump power. After each probe, 𝐹 pol ≥ 𝐹 trigger by construction; polarization fidelity then degrades over the next five seconds until the next probe. The end-to-end fidelity is given by the product of the fixed source–receiver fidelity 𝐹 sd and 𝐹 pol (𝑡), and the entanglement rate is derived directly from the pump power. While evaluating static protocols, we optimize over fidelity check frequency and 𝐹 sd (a linear function of pump power, see Figure 4a). The values of the APC parameters are fixed at 𝐹 trigger = 0.98, 𝐹 target = 0.99,𝑇timeout = 55𝑠, consistent with a recent demonstration using the same device [43]. We apply the same method to evaluate our adaptive protocol, accounting for dynamically adjusted parameters. Pump power is updated every 100ms. attenuator that sets the pump power to the desired value. APC probes are scheduled as needed based on the protocol state (Sections 5.3-5.5).

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6.2 Static Application Demands Figure 7 compares our adaptive protocol to different static protocols over different time intervals: 24 hours (Figure 7a), 8 hours of night time (Figure 7b), and 8 hours of day time (Figure 7c). Here, we set the application fidelity demand (𝐹 min ) as a constant (0.85). For the static protocols, we vary both 𝐹 sd and fidelity check intervals. The result shows that the adaptive protocol performs well across all time intervals, whereas static protocol parameters require careful selection for reasonable performance. Over a 24 hour period, the adaptive protocol outperforms the best static protocol by 14%, and this increases to 16% at night. The gap is smaller during the day (6%), but it is important to understand that the “best” static protocol for one day could be very different for another. Static protocol performance shows an expected trend: if the source-detector fidelity 𝐹 sd is set too close to 𝐹 min , then these protocols are not able to provide high entanglement throughput. Conversely, setting a more conservative 𝐹 sd does not allow the static protocols to maximize entanglement rate. Even with a well-chosen 𝐹 sd , the best fidelity check interval varies over time (e.g., ~1s during the day versus ~10s at night). Further, these parameter values are sensitive to application demands (𝐹 min ) and the polarization drift rate, both of which may vary unpredictably. No parameter choices are required for the adaptive algorithm. 6.3

Varying Application Demands

The previous result fixed the application requirement of fidelity (𝐹 min ) to 0.85. However, we expect a real-world deployment of quantum networks will have to support multiple applications, each of which may have different requirements. One of the major advantages of our adaptive protocol is that it can adapt to changing 𝐹 min “on-the-fly”. Figure 8 shows entanglement distribution rates achieved (𝑦-axis) by the adaptive protocol and different static protocols with changing 𝐹 min (𝑥-axis). The results (obtained over a 24-hour run) again show that the adaptive protocol is able to outperform each static protocol, but more importantly, each static protocol optimizes for a single 𝐹 min . The result also shows the limits of the adaptive protocol, as entanglement rate steadily dips (and reaches 0 for 𝐹 min ≈ 0.9475). This is expected, as the higher values of 𝐹 min cannot be sustained by

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the adaptive (or indeed, any protocol), as even the most conservative source setting (25mW pump power) provides 𝐹 sd ≈ 0.9475 (see Figure 4a). The previous result shows that the adaptive protocol is able to adapt to different 𝐹 min requirements (over 24 hours). However, the protocol can also adapt to instantaneously changing 𝐹 min , as 𝐹 min is simply used as an input to Equation 3. We show the results from an artificial scenario where the required 𝐹 min changes continuously in the shape of a sine curve. While we do not expect application demands to necessarily change with every entanglement requested, this experiment provides an “extreme” scenario to understand and visualize the dynamics of the adaptive protocol. It’s important to note that the adaptive protocol is not only adapting to the changing 𝐹 min , but also to the unknown link drift simultaneously. Finally, it is perhaps not a stretch to emphasize that no static protocol would perform well in such a scenario. Figure 9 shows the performance of the adaptive protocol with continuously changing 𝐹 min (red curve in Figure 9a). The figure shows that the adaptive protocol is able to maintain requisite fidelity (blue line in Figure 9a) with changing 𝐹 min . Figures 9b and 9c shows the entanglement rate achieved and pump powers set by the protocol over time. As expected, the achievable entanglement rate mirrors 𝐹 min , as does the pump power set. It is interesting to note that the protocol is compensating for two changing variables at each step here: polarization drift over the link, and 𝐹 min . We also see that the protocol invokes the APC probes more frequently as we approach the maximum feasible 𝐹 min , and less frequently as 𝐹 min recedes. This behavior can be explained as follows: as 𝐹 min approaches the maximum attainable 𝐹 min with this experimental setup, the protocol is frequently forced to set pump power to the minimum available, triggering an APC probe. On the other hand, as the required 𝐹 min decreases, the instantaneous rate and average rate curves tend to not converge (See Equation 7 in Section 5.5), allowing for longer runs without re-calibration. 6.4

Evaluation Against Theoretical Upper Bound

The adaptive protocol presented here provides no guarantees of optimality, and indeed the process of defining an optimal protocol is unclear. Even if we consider an “optimal” that has perfect future knowledge of drift, it is unclear how often to initiate APC probing, and with what parameters. If we further assume polarization compensations take zero time, then the best policy would be to continuously compensate.

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In this “optimal” scenario, the system is equivalent to a link with no drift, and pump power should be chosen such that 𝐹 sd = 𝐹 min . The throughput in the zero-drift scenario therefore constitutes an (unachievable) upper bound on the performance of any protocol. We compare the performance of the adaptive protocol to this theoretical upper bound. For the experiments in Figure 7, this bound is a rate of 85.6 entanglements/second. Over a 24-hour period, the adaptive protocol performs within 21% of the upper-bound (with ~9% of that time being devoted to probes and compensation). The comparison at night is more interesting, as drift rate is lower and more closely resembles the upper bound scenario. At night, the gap is < 8%, of which 3.4% is due to probes and compensation. These results indicate that the scope for improvement for more sophisticated prediction and compensation is relatively modest (<5%). Moreover, the adaptive protocol is relatively simple and lightweight, and can easily be integrated into existing deployments without major investment in computing hardware. 7 7.1

DISCUSSION Deployment Considerations

A practical advantage of the proposed protocol is that it requires only software access to control interfaces already common in deployed wide-area quantum links. In the system studied here, the controller adjusts the source pump power via an electronically-tunable attenuator and adjusts the APC probe parameters through an API provided by the manufacturer. Thus, deployment requires no additional quantum hardware. It requires only a classical control process with access to existing control knobs The online control logic is lightweight. Runtime decisions consist of table lookups and 𝑂 (1) time parameter calculations. The APC simulation used in our evaluation (Algorithm 1) is not part of deployment, since compensation is executed directly by the APC device. Similarly, the controller requires only low-bandwidth communication for distributing parameters and fidelity check results. The relevant timescale in our simulations is 100ms; we expect that the lightweight computation, control, and communication requirements will fit well within this window.

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The main deployment cost is characterization. The entanglement source must have a wellcharacterized rate-fidelity curve and there must be sufficient polarization drift statistics to initialize the prediction model. These measurements are standard characterization procedures that can be performed during commissioning and refined during normal operation. This makes the protocol suitable for near-term deployment as a software control layer over existing wide-area quantum link infrastructure. 7.2

Extension to General Quantum Links

While the control protocol presented here was developed around specific quantum hardware, we believe that the fundamental concepts apply broadly. The problem classes Opt-0 (optimization of controllable parameters) and Opt-1 (compensation of uncontrollable parameters) are universal to all quantum links and therefore Opt-2 (the joint optimization of Opt-0 and Opt-1 over the same link) represents a general and practical class of problems for which the methods presented here provide a blueprint. Opt-0 formalizes the idea of managing tunable hardware parameters to improve end-to-end system performance, rather than simply maximizing the performance of individual components. Quantum hardware often exposes many parameters that can be tuned. Many intrinsic hardware tradeoffs beyond pump power can be exploited for optimization; we discussed several examples in Sections 2.1 and 2.2. As a further example, consider the following binary tradeoff: two successful measurement outcomes occur with equal probability, but only one corresponds to higher-fidelity entanglement [24]. This forces a choice between discarding low-fidelity events to maximize fidelity or retaining all events to maximize rate. Opt-1 concerns compensation of uncontrollable noise parameters, which is again universal. Polarization stabilization is required for all photonic quantum links regardless of encoding, and notably is relevant for optical entanglement swapping, where increasing polarization mismatch reduces success probability. The magnitude and timescale of polarization drift depend on link characteristics (e.g., length, aerial versus buried fiber), but every compensation strategy induces tradeoffs. Coexistence schemes that multiplex reference light with quantum signals avoid system downtime but yield imperfect compensation due to wavelength-dependent drift. Conversely, using same-wavelength reference signals enables high-precision compensation but necessaitates downtime as the reference signals overwhelm the quantum signals. Taken together, these considerations make Opt-2 a general problem that applies to all wide-area quantum settings. While the specific control and response variables will vary by link, a common general structure emerges: controllable parameters induce tunable tradeoffs between entanglement rate and fidelity, while uncontrollable drift introduces noise that must be mitigated at the cost of rate and/or fidelity. The core control-theoretic principle is to adjust controllable parameters to offset the effects of drift while compensating for uncontrollable parameters in a way that balances the cost and benefit of compensation. We expect these concepts to extend naturally to a wide range of quantum links beyond the specific system studied here. 7.3

Extension to Higher Layers of the Quantum Networking Stack

A natural extension of the above framework is the joint optimization of two or more links working in tandem to distribute entanglement over multiple hops. Each link has its own sets of controllable parameters (Opt-0) and time-varying noise parameters (Opt-1); we refer to the resulting multi-link problem as Opt-3. Unlike earlier cases, Opt-3 must account for network topology while jointly configuring parameters and compensation strategies on a per-link basis.

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Even for two links, Opt-3 is a broad and largely open problem when realistic hardware constraints are considered. Although out of scope for this paper, we illustrate the key considerations with a twohop example. Consider three nodes 𝐴, 𝐵, and 𝐶, connected by links 𝐴-𝐵 (Link 0) and 𝐵-𝐶 (Link 1), with entanglement swapping at 𝐵 [35]. Suppose Link 0 is stable (e.g., short underground fiber), while Link 1 exhibits significant drift. An application specifies a minimum end-to-end entanglement quality requirement, which constrains the allowable operating points of each link. Under an Opt-2 protocol, each link exposes a set of achievable rate–fidelity operating points. Because Link 1 is more unstable, its feasible region is more restricted. An Opt-3 controller must then choose how Link 0 operates relative to Link 1: matching its rate, increasing fidelity to improve the end-to-end fidelity, increasing its rate to enable entanglement distillation [27], or adopting another strategy. The optimal choice depends on hardware capabilities and how entanglement is ultimately consumed by higher layers. This scenario demonstrates that Opt-2 (and our protocol) provides a useful abstraction for link behavior. By exposing a feasible set of rate–fidelity operating points, Opt-2 grants higher-layer Opt-3 protocols flexibility in optimizing end-to-end performance. 8

CONCLUSIONS

In this work, we presented a control protocol for quantum links that explicitly accounts for physical-layer impediments while exploiting controllable rate-fidelity tradeoffs in modern quantum networking hardware. Improvements in the performance of present-day quantum networks are typically driven by advances in hardware, yet our results demonstrate that carefully designed software control can yield substantial gains without increasing hardware requirements. Importantly, the proposed protocol has a clear path to deployment within existing experimental infrastructures. More broadly, the optimization framework introduced here extends beyond the specific link studied and highlights the potential for software-based control to play a significant role in improving the performance of near-term quantum networks. 9

ACKNOWLEDGMENTS

We thank Mael Flament, Gabriel Bello Portmann, Rourke Sekelsky, and Aditya Varma of the Qunnect team for helpful discussions about the mechanics of the APC and for characterization data of the Qu-SRC. We also thank Ivan Burenkov, Alan Migdall, and Thomas Gerrits for helpful conversations and comments during the preparation of the manuscript. This work was supported by a National Science Foundation Graduate Research Fellowship (Grant No. DGE 2236417). REFERENCES [1] H Aghaee Rad, T Ainsworth, RN Alexander, B Altieri, MF Askarani, R Baby, L Banchi, BQ Baragiola, JE Bourassa, RS Chadwick, et al. 2025. Scaling and networking a modular photonic quantum computer. Nature (2025), 1–8. [2] J. B. Altepeter, E. R. Jeffrey, and P. G. Kwiat. 2005. Photonic State Tomography. Advances In Atomic, Molecular, and Optical Physics, Vol. 52. Academic Press, 105–159. doi:10.1016/S1049-250X(05)52003-2 [3] Joseph B Altepeter, Evan R Jeffrey, and Paul G Kwiat. 2005. Photonic state tomography. Advances in atomic, molecular, and optical physics 52 (2005), 105–159. [4] Patrick R. Banner, Steven L. Rolston, and Joseph W. Britton. 2025. BIFROST: A First-Principles Model of Polarization Mode Dispersion in Optical Fiber. arXiv:2510.01212 [physics.optics] https://arxiv.org/abs/2510.01212 [5] Charles H Bennett and Gilles Brassard. 1984. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of the international conference on computers, systems and signal processing. 175–179. [6] Eric Bersin, Matthew Grein, Madison Sutula, Ryan Murphy, Yan Qi Huan, Mark Stevens, Aziza Suleymanzade, Catherine Lee, Ralf Riedinger, David J. Starling, Pieter-Jan Stas, Can M. Knaut, Neil Sinclair, Daniel R. Assumpcao, Yan-Cheng Wei, Erik N. Knall, Bartholomeus Machielse, Denis D. Sukachev, David S. Levonian, Mihir K. Bhaskar, Marko Lončar, Scott Hamilton, Mikhail Lukin, Dirk Englund, and P. Benjamin Dixon. 2024. Development of a Boston-area 50-km fiber quantum network testbed. Phys. Rev. Appl. 21 (Jan 2024), 014024. Issue 1. doi:10.1103/PhysRevApplied.21.014024

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[7] Joshua Bienfang, Thomas Gerrits, Paulina Kuo, Alan Migdall, Sergey Polyakov, and Oliver Slattery. 2025. Single-photon Sources and Detectors Dictionary. NIST (Aug. 2025). https://www.nist.gov/publications/ single-photon-sources-and-detectors-dictionary-0 Last Modified: 2025-08-04T15:08-04:00. [8] Connor Clayton and Quinn Carmack. 2026. Source Code for “Rate-Fidelity Control for Wide-Area Quantum Links”. https://github.com/cbclayton/dynamic-quantum-link-control [9] Connor Clayton, Xiaodi Wu, and Bobby Bhattacharjee. 2024. Efficient Routing on Quantum Networks using Adaptive Clustering. In 2024 IEEE 32nd International Conference on Network Protocols (ICNP). IEEE, 1–12. [10] Alexander N Craddock, Anne Lazenby, Gabriel Bello Portmann, Rourke Sekelsky, Mael Flament, and Mehdi Namazi. 2024. Automated distribution of polarization-entangled photons using deployed New York City fibers. PRX Quantum 5, 3 (2024), 030330. [11] Alexander N Craddock, Yang Wang, Felipe Giraldo, Rourke Sekelsky, Mael Flament, and Mehdi Namazi. 2024. High-rate subgigahertz-linewidth bichromatic entanglement source for quantum networking. Physical Review Applied 21, 3 (2024), 034012. [12] Axel Dahlberg, Matthew Skrzypczyk, Tim Coopmans, Leon Wubben, Filip Rozpędek, Matteo Pompili, Arian Stolk, Przemysław Pawełczak, Robert Knegjens, Julio de Oliveira Filho, et al. 2019. A link layer protocol for quantum networks. In Proceedings of the ACM special interest group on data communication. 159–173. [13] C Delle Donne, M Iuliano, B Van Der Vecht, GM Ferreira, H Jirovská, TJW Van Der Steenhoven, A Dahlberg, M Skrzypczyk, D Fioretto, M Teller, et al. 2025. An operating system for executing applications on quantum network nodes. Nature 639, 8054 (2025), 321–328. [14] L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller. 2001. Long-distance quantum communication with atomic ensembles and linear optics. 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Ou, and L. Mandel. 1987. Measurement of subpicosecond time intervals between two photons by interference. Phys. Rev. Lett. 59 (Nov 1987), 2044–2046. Issue 18. doi:10.1103/PhysRevLett.59.2044 [21] Michał Horodecki, Paweł Horodecki, and Ryszard Horodecki. 1997. Inseparable Two Spin- 12 Density Matrices Can Be Distilled to a Singlet Form. Phys. Rev. Lett. 78 (Jan 1997), 574–577. Issue 4. doi:10.1103/PhysRevLett.78.574 [22] Jessica Illiano, Marcello Caleffi, Antonio Manzalini, and Angela Sara Cacciapuoti. 2022. Quantum internet protocol stack: A comprehensive survey. Computer Networks 213 (2022), 109092. [23] Richard Jozsa. 1994. Fidelity for Mixed Quantum States. Journal of Modern Optics 41, 12 (1994), 2315–2323. arXiv:https://doi.org/10.1080/09500349414552171 doi:10.1080/09500349414552171 [24] Can M Knaut, Aziza Suleymanzade, Y-C Wei, Daniel R Assumpcao, P-J Stas, Yan Qi Huan, Bartholomeus Machielse, Erik N Knall, Madison Sutula, Gefen Baranes, et al. 2024. Entanglement of nanophotonic quantum memory nodes in a telecom network. Nature 629, 8012 (2024), 573–578. [25] Viktor Krutyanskiy, Maria Galli, Vojtech Krcmarsky, Simon Baier, DA Fioretto, Yunfei Pu, Azadeh Mazloom, Pavel Sekatski, Marco Canteri, Markus Teller, et al. 2023. Entanglement of trapped-ion qubits separated by 230 meters. Physical Review Letters 130, 5 (2023), 050803. [26] Stephan Kucera, Christian Haen, Elena Arenskötter, Tobias Bauer, Jonas Meiers, Marlon Schäfer, Ross Boland, Milad Yahyapour, Maurice Lessing, Ronald Holzwarth, Christoph Becher, and Jürgen Eschner. 2024. Demonstration of quantum network protocols over a 14-km urban fiber link. npj Quantum Information 10, 1 (Sept. 2024), 88. doi:10.1038/ s41534-024-00886-x [27] Paul G Kwiat, Salvador Barraza-Lopez, Andre Stefanov, and Nicolas Gisin. 2001. Experimental entanglement distillation and ‘hidden’non-locality. Nature 409, 6823 (2001), 1014–1017. 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[33] Wayne McKenzie, Anne Marie Richards, Shirali Patel, Thomas Gerrits, T. G. Akin, Steven Peil, Adam T. Black, David Tulchinsky, Alexander Hastings, Ya-Shian Li-Baboud, Anouar Rahmouni, Ivan A. Burenkov, Alan Mink, Matthew Diaz, Nijil Lal, Yicheng Shi, Paulina Kuo, Pranish Shrestha, Mheni Merzouki, Alejandro Rodriguez Perez, Eleanya Onuma, Daniel E. Jones, Atiyya A. Davis, Thomas A. Searles, J. D. Whalen, Qudsia Sara Quraishi, Kate S. Collins, La Vida Cooper, Harry Shaw, Bruce Crabill, Oliver Slattery, and Abdella Battou. 2024. Clock synchronization characterization of the Washington DC metropolitan quantum network (DC-QNet). Applied Physics Letters 125, 16 (Oct. 2024), 164004. doi:10.1063/5.0225082 [34] C. M. Nunn, J. D. Franson, and T. B. Pittman. 2021. Heralding on the detection of zero photons. Phys. Rev. A 104 (Sep 2021), 033717. Issue 3. doi:10.1103/PhysRevA.104.033717 [35] Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter, and Anton Zeilinger. 1998. Experimental entanglement swapping: entangling photons that never interacted. Physical review letters 80, 18 (1998), 3891. [36] Matteo Pompili, Carlo Delle Donne, Ingmar te Raa, Bart van der Vecht, Matthew Skrzypczyk, Guilherme Ferreira, Lisa de Kluijver, Arian J Stolk, Sophie LN Hermans, Przemysław Pawełczak, et al. 2022. Experimental demonstration of entanglement delivery using a quantum network stack. npj Quantum Information 8, 1 (2022), 121. [37] Anouar Rahmouni, PS Kuo, Ya-Shian Li-Baboud, IA Burenkov, Yicheng Shi, MV Jabir, N Lal, Dileep Reddy, Mheni Merzouki, Lijun Ma, et al. 2024. 100-km entanglement distribution with coexisting quantum and classical signals in a single fiber. Journal of Optical Communications and Networking 16, 8 (2024), 781–787. [38] Mohsen Razavi and Jeffrey H Shapiro. 2006. Long-distance quantum communication with neutral atoms. Physical Review A—Atomic, Molecular, and Optical Physics 73, 4 (2006), 042303. [39] Dileep V. Reddy, Robert R. Nerem, Sae Woo Nam, Richard P. Mirin, and Varun B. Verma. 2020. Superconducting nanowire single-photon detectors with 98% system detection efficiency at 1550nm. Optica 7, 12 (Dec 2020), 1649–1653. doi:10.1364/OPTICA.400751 [40] Davide Rusca and Nicolas Gisin. 2024. Quantum Cryptography: an overview of Quantum Key Distribution. arXiv:2411.04044 [quant-ph] https://arxiv.org/abs/2411.04044 [41] Vinay Shankarkumar, Laurent Montini, Tim Frost, and Greg Dowd. 2017. Precision Time Protocol Version 2 (PTPv2) Management Information Base. RFC 8173. doi:10.17487/RFC8173 [42] Shouqian Shi and Chen Qian. 2020. Concurrent Entanglement Routing for Quantum Networks: Model and Designs. In Proceedings of ACM SIGCOMM (Virtual Event, USA). 62–75. doi:10.1145/3387514.3405853 [43] Yicheng Shi, Jing Su, Anouar Rahmouni, Pranish Shrestha, Mheni Merzouki, Gabriel Bello Portmann, Anne Lazenby, Mael Flament, Mehdi Namazi, Abdella Battou, et al. 2026. Entanglement distribution over a polarization-stabilized aerial fiber. Journal of Optical Communications and Networking 18, 8 (2026), 813–819. [44] Y. H. Shih and C. O. Alley. 1988. New Type of Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta Produced by Optical Parametric Down Conversion. Phys. Rev. Lett. 61 (Dec 1988), 2921–2924. Issue 26. doi:10.1103/PhysRevLett.61.2921 [45] P-J Stas, Y-C Wei, M Sirotin, YQ Huan, U Yazlar, F Abdo Arias, E Knyazev, G Baranes, B Machielse, S Grandi, et al. 2026. Entanglement-assisted non-local optical interferometry in a quantum network. Nature 651, 8105 (2026), 326–332. [46] E. Sutcliffe and A. Beghelli. 2023. Multiuser Entanglement Distribution in Quantum Networks Using Multipath Routing. IEEE Transactions on Quantum Engineering 4, 01 (jan 2023), 1–15. doi:10.1109/TQE.2023.3329714 [47] Stephanie Wehner, David Elkouss, and Ronald Hanson. 2018. Quantum internet: A vision for the road ahead. Science 362, 6412 (2018), eaam9288. [48] Shi-Hai Wei, Bo Jing, Xue-Ying Zhang, Jin-Yu Liao, Chen-Zhi Yuan, Bo-Yu Fan, Chen Lyu, Dian-Li Zhou, You Wang, Guang-Wei Deng, Hai-Zhi Song, Daniel Oblak, Guang-Can Guo, and Qiang Zhou. 2022. Towards Real-World Quantum Networks: A Review. Laser & Photonics Reviews 16, 3 (March 2022), 2100219. doi:10.1002/lpor.202100219 [49] William K Wootters and Wojciech H Zurek. 1982. A single quantum cannot be cloned. Nature 299, 5886 (1982), 802–803. [50] Zheshen Zhang and Quntao Zhuang. 2021. Distributed quantum sensing. Quantum Science and Technology 6, 4 (jul 2021), 043001. doi:10.1088/2058-9565/abd4c3

Rate-Fidelity Control for Wide-Area Quantum Links

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Fig. 10. A more detailed diagram of the entanglement source sketched in Figure 3.

A DETAILS OF ENTANGLEMENT SOURCES This section provides additional detail on the SPDC setup beyond the simplified diagram introduced in Figure 3. As a general overview: the SPDC process occurs within some nonlinear media, whereby high-energy pump photons are converted into lower-energy, entangled photon pairs. Additional components are then used to harness entanglement in the desired degree of freedom. Each of these aspects is detailed below for our source. The full experimental setup is shown in Figure 10. A mode-locked pump laser generates pulses at a repetition rate of 76.142 MHz. Pump power control is executed via a variable beam splitter (VBS) consisting of an adjustable half wave plate (HWP) followed by a polarization beam splitter (PBS). Adjusting the angle of the HWP changes the amount of pump light that used for SPDC. The pump pulses then enter a periodically poled potassium titanyl phosphate (PPKTP) non-linear crystal which serves as a medium for the SPDC process. The crystal is engineered to convert pump photons with central wavelength 775 nm into photon pairs centered at 1550 nm. With some probability (~10−4 per pulse), a pair of photons with orthogonal polarization states are generated, separated from the pump with spectral filters, and then split at a PBS. One of the photons passes through a HWP to (optionally) put both photons into the same polarization state. The photons are coupled from free space into single-mode fiber and interfered at a 50:50 fiberbased beam splitter (BS) [44]. Interference is maximized through tuning of a variable delay (Δ𝑡) to ensure photons arrive simultaneously, and a manual polarization controller (PC) to maintain photon polarizations in fiber. Finally, the entangled photons propagate to the receiver. The analyzers set polarization bases {𝜑𝐴 , 𝜑 𝐵 }, and detection events at each SNSPD are timestamped with a time tagger (not shown) to count coincident detections. Through multiple measurements in different bases [2], the receiver fully reconstructs the state of the polarization-entangled photons. B

DETAILS OF ENTANGLEMENT SOURCE MODELING

This Appendix includes more complete data used for SPDC and four-wave mixing source characterizations and development of the corresponding source models. B.1

SPDC Source

Extending the main body plots from Figure 4a, additional information from the same data sets is shown in Figure 11a. This includes measurements at lower powers 5 mW and 10 mW, and raw count rates are shown without normalization. The four shown data sets correspond to measurement sequences taken on different days with slightly different conditions: green squares and orange circles on one day; purple diamonds and red triangles at a later date. Variations between trials are related to unstabilized polarization drift in the source-detector setup shown in Figure 10, which is elaborated upon below. Entanglement generation rates are clearly consistent between data sets taken subsequent trials on the same day, but not on different days. This indicates different overall loss between source

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1.000 0.995

Fidelity

Fidelity

0.95 0.90 0.85

0.990 0.985 0.980

0.80

Entanglement Generation Rate (pairs/s)

Entanglement Generation Rate (pairs/s)

×106 6000 4000 2000

15

10

5

0 0

50

100

150 200 Pump Power (mW)

250

(a) Full SPDC source characterization.

300

0

1

2 3 Pump Power (mW)

4

5

(b) Full commercial four-wave mixing source characterization.

Fig. 11. Extension of source characterizations in Figure 4 including all collected data.

and receiver, largely attributed to the polarization-dependent SNSPD detection efficiency. The fibers between the analyzers and detectors are relatively stable over the course of one day, but not over longer time scales. The same is true (to a lesser extent) of the coupling efficiency from free space into fiber, which may have also reduced the overall efficiency in the later data sets. This combined effect simulates different link losses between source and receiver, so normalization is used to represent the underlying source dependence on pump power in Figure 4a. State fidelity is also largely consistent between data sets, but with some increased variability between measurements taken on the same day. This is because entanglement is generated from quantum interference at a 50:50 beamsplitter, which is highly sensitive to the polarization of the incoming photons — more sensitive than the SNSPD efficiency dependence. We characterize the quality of interference with the Hong-Ou-Mandel (HOM) effect [20], observing about 96% and 93% visibility for the two different days. This value is limited by the ability to manually match polarization states in fiber via polarization controller (PC). This in turn limits the state fidelity at low powers, as seen at the top of Figure 10, and is also affected by drift, causing higher variability between trials. Altogether, rates exhibit a clear linear trend over all measured pump powers, while fidelity shows no significant change until around 25 mW (for a given data set), above which multipair effects become significant and degrade fidelity. This defines an operable regime (25 mW to 300 mW) over which decreasing rate returns a consistent increase in state fidelity, offering a useful tradeoff that can be leveraged in our control protocol. B.2

Four-Wave Mixing Source

In Figure 4b we presented source characterization data for a Qunnect Qu-SRC [11] which generates pairs of polarization-entangled photons via four-wave mixing (FWM) in warm atomic vapor. We received this characterization data directly from the device manufacturer. The full (non-normalized) data is presented in Figure 11b.

Rate-Fidelity Control for Wide-Area Quantum Links Experimental Modeled

0.006 0.005 Density

25

0.004 0.003 0.002 0.001 0.000

0

500 1000 1500 Compensation Time (ms)

2000

Fig. 12. Histograms of APC compensation times generated by Algorithm 1 and experiment.

Note the given SPDC rates include all optical losses incurred by the apparatus in Fig. 10, while this commercial FWM source is highly optimized. As a result, the commercial source emits photonic Bell states at a rate several orders of magnitude higher and an error rate one order of magnitude lower than our in-house SPDC-based source. The output photons also have much different wavelength characteristics: the SPDC source emits degenerate photons with the same center wavelength (1550 nm); the FWM source emits non-degenerate photon pairs with wavelengths 795 nm and 1324 nm, each with a much narrower spectral band. These spectral properties will result in completely different polarization drift due to the dispersive properties of optical fiber. To translate our model to a new source, drift characterization similar to Fig. 7 would be required for the relevant wavelengths, although we can expect similar dependence of drift rate on environmental conditions. C

FIT OF APC MODEL TO EXPERIMENTAL DATA

Our empirically-derived model of polarization compensation in the Qunnect APC, presented in Section 3.3, assumes Riemannian gradient descent and a noise term that counteracts the improvements gained in each gradient descent step. We fit our model to 70 hours of experimental data over a similar link and determined a gradient descent step size of 𝜂 = 0.062 and a step duration of Δ = 27.7ms. Figure 12 compares the histogram of compensation durations from the experimental APC dataset with a histogram derived from our simulations, which use our model on the data shown in Figure 5. Visual comparison demonstrates reasonable empirical fit. D

COMPUTING THE FIDELITY CHECK CONDITION

Here we discuss the computation of the quantity Λ1 −Λ2 introduced in Section 5.3. Λ1 −Λ2 quantifies the throughput gain of performing a fidelity check: Λ1 is the expected throughput if we perform the check, and Λ2 is the throughput if we don’t perform the check. Let 𝑒 denote the time elapsed since the last fidelity check and 𝑇eff the expected time until the next fidelity check. Let 𝑟 (𝐹 pol ) denote the rate achieved when the polarization fidelity is predicted to be 𝐹 pol . Let 𝑡 check denote the duration of a fidelity check. To ease the notation in this discussion, we define 𝑟¤ (𝑡) := 𝑟 (𝑡 0 + 𝑡) where the most recent fidelity check occurred at time 𝑡 0 — i.e., 𝑟¤ (𝑡) is the rate 𝑡 seconds after the last check.

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The first important realization is that we can always compute the rate 𝑟¤ (𝑡) for 𝑡 > 0 up until the next fidelity check. Intuitively this must be true: we learn no information about the system between fidelity checks, so we should be able to predict all protocol behavior until the next check. Indeed, 𝑟¤ (𝑡) is determined by the pump power, which in turn is determined by our prediction of 𝐹 pol at time 𝑡 (see Section 5.2), so we can project 𝑟¤ (𝑡) by querying our polarization drift prediction model. We employ this rate projection to estimate 𝑇eff by finding the time at which the condition in Equation 7 is met and taking the minimum of this quantity and the time since the last check. Given this, Λ2 is easy to compute by noticing that ∫ 𝑒+𝑇eff Λ2 = 𝑟¤ (𝑡)𝑑𝑡 (8) 𝑒

Next, we estimate Λ1 . We assume that the rate curve 𝑟¤after-check (𝑡) after a hypothetical fidelity check would follow the same trend as the current 𝑟¤ (𝑡) with an offset depending on the expected true polarization fidelity E[𝐹 pol ]. We compute E[𝐹 pol ] by querying our drift prediction model with 𝛿 = 0.5 (50th percentile drift). We can now estimate Λ1 : ∫ 𝑇eff −𝑡check Λ1 = 𝑟¤after-check (𝑡)𝑑𝑡 (9) 0 ∫ 𝑇eff −𝑡check   ≈ 𝑟¤ (𝑡) − 𝑟¤ (0) − 𝑟¤ (E[𝐹 pol ]) 𝑑𝑡 (10) 0

This uses the same projection of 𝑟¤ (𝑡) that we used to compute Λ2 . Note that we only need to perform the calculations here once per fidelity check. E

DERIVATION OF OPTIMAL COMPENSATION TIMING

Here we derive Equation 7, which states the condition under which a protocol should initiate a polarization compensation routine. First, we approximate the objective of maximizing end-to-end rate by a greedy strategy of maximizing the rate of each period individually (from the end of one calibration routine to the end of the next). Suppose each compensation routine takes time 𝑐, the previous routine ended at time 𝑡 0 , 𝑟 (𝑡) is the entanglement distribution rate at time 𝑡, and 𝑟¯(𝑡) is the average rate of the period if we were to perform a compensation routine at time 𝑡. Then the objective is to maximize ∫ 𝑡 1 𝑟¯(𝑡) = 𝑟 (𝑡)𝑑𝑡 (11) (𝑡 − 𝑡 0 ) + 𝑐 𝑡0 A maximum must satisfy 0= =

𝑑 𝑟¯(𝑡) 𝑑𝑡 ∫𝑡 (𝑐 + 𝑡 − 𝑡 0 )𝑟 (𝑡) − 𝑡 𝑟 (𝑡)𝑑𝑡 0

(𝑐 + 𝑡 − 𝑡 0 ) 2

which has a unique solution at 𝑟 (𝑡) =

1 𝑐 + 𝑡 − 𝑡0

∫ 𝑡 𝑟 (𝑡)𝑑𝑡 = 𝑟¯(𝑡)

(12)

𝑡0

𝑟¯(𝑡) is increasing at 𝑡 = 0, so assuming decreasing 𝑟 (𝑡), which is generally true as we generally lower pump power and therefore 𝑟 (𝑡) throughout a period, 𝑟¯(𝑡) is maximized by 𝑡 satisfying Equation 12 maximizes.

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