A Linear Algebraic Framework for Dynamic Scheduling Over Memory-Equipped Quantum Networks

TL;DR

The paper addresses dynamic scheduling in memory-enabled quantum networks supporting entanglement swapping across arbitrary topologies and multiple user demands. It develops a linear algebraic framework that couples ebit queues and demand queues, and applies Lyapunov Drift Minimization with a stability-focused objective to derive quadratic scheduling policies, alongside Max-Weight–inspired linear variants. It analyzes information availability levels (fully, partially, and node-local) and validates policies via an ad-hoc Python simulator over various topologies, showing that linear Max-Weight variants closely approach throughput-optimal performance while offering reduced computation. The work highlights memory-assisted intermediate links as a core mechanism and provides practical design insights for scalable quantum networks, including policy-selection trade-offs and routing considerations for a potential Quantum Internet.

Abstract

Quantum Internetworking is a recent field that promises numerous interesting applications, many of which require the distribution of entanglement between arbitrary pairs of users. This work deals with the problem of scheduling in an arbitrary entanglement swapping quantum network - often called first generation quantum network - in its general topology, multicommodity, loss-aware formulation. We introduce a linear algebraic framework that exploits quantum memory through the creation of intermediate entangled links. The framework is then employed to apply Lyapunov Drift Minimization (a standard technique in classical network science) to mathematically derive a natural class of scheduling policies for quantum networks minimizing the square norm of the user demand backlog. Moreover, an additional class of Max-Weight inspired policies is proposed and benchmarked, reducing significantly the computation cost at the price of a slight performance degradation. The policies are compared in terms of information availability, localization and overall network performance through an ad-hoc simulator that admits user-provided network topologies and scheduling policies in order to showcase the potential application of the provided tools to quantum network design.

Figures (5)

• Figure 1: The scheme of our rank system for an $ABCDE$ chain topology. Every square with only two letters inside (e.g. $DE$) represents a consumption operation along a given link, while three-letter squares (e.g. $C[D]E$) represent swapping transitions. A set of squares at the same height are grouped in one rank, starting from zero at the top (direct consumption from physical queues) and increasing going down. Arrows represent the "paths" to follow to obtain one of the final, user-requested pairs. Focusing on the bright red squares in this scheme, which all involve node $C$ in some way, we can provide an example of how the conflict-management system works. Whenever it needs to apply a scheduling order, node $C$ will sequentially: Perform transition $B[C]D$ (rank $1$) as many times as requested;Satisfy consumption orders along $CE$ and $AC$ in a random order, since they are competing rank $2$ operations;Perform transitions $B[C]E$ and $A[C]D$ in a random order, since they are competing rank $3$ operations.As discussed in the main text, $CE$ and $AC$'s consumption orders are satisfied in a random order, but always after the upstream $B[C]D$ transition and always before the downstream $B[C]E$ and $A[C]D$ transistions.
• Figure 2: Comparison of the performance of the linear scheduling policies we presented in the main text and their quadratic counterparts. For brevity, we only report results for the Grid topology shown in fig. \ref{['fig:gridtopo']}, while stating that the same phenomenon is observed for all topologies: the margin of performance between Max-Weight and Quadratic schedulers is almost imperceptible in our tests. This figure was calculated with an additional set of eight random parasitic pairs, whose average load was fixed at $100$ kHz. Analogously to the main results in fig. \ref{['fig:stabgrids']}, simulations were run for $1000$ time steps of $1 \mu s$, discarding the first $100$ observations to reduce the impact of transients. The white points were skipped by the simulator and directly deemed unstable, since one or more strictly lower-load points were found to be unstable. More details on this computational economy technique can be found in the main text.
• Figure 3: The four topologies analyzed in this work. The main service pairs and the routes connecting them have been highlighted in red and blue, with purple representing shared edges, i.e. edges that appear in both pairs' service routes. In green, we provide a visual example of the random parasitic pairs: every green node is paired with another colored node and requests entanglement with a fixed rate. At every run of the simulator, we redraw the green pairs to study the effect of traffic without bias towards a specific configuration.
• Figure 4: Global summary of all the network performance metrics that can be analyzed through our simulator, when running the full information Max Weight scheduler over the grid topology from fig. \ref{['fig:gridtopo']}. Inside each cell: The plot shows the temporal evolution of total demand from start to finish; it allows to easily distinguish stable regime (with finite excursion) from the unstable one (with a linear trend);The background color represents the average demand backlog throughout a simulation run;The top-left number is the maximum excursion of the total demand in the network; in the stable regime, it can be seen as a rough upper bound on the amount of quantum memory required at each node to achieve this performance level.
• Figure 5: For each of the four topologies, we provide a grid of plots obtained by simulating different operating points. As mentioned in the main text, there are ten pairs of users, of which two are fixed and eight randomized. Each cell of the grids is a plot of the average demand backlog vs. the load across the two main pairs (reported in $kHz$ on the small axes of the individual cells) under certain operating conditions. The conditions in which every plot was calculated are fixed by the Information Availability and Parasitic Load meta-axes, the former indicating which scheduler was employed to control the network (Greedy to Full Information, in increasing order of available information), the latter the load placed upon the randomized parasitic pairs in $kHz$. As discussed in the main text, a dark blue point is deemed stable and a yellow one unstable, while the middle grounds are somewhat ambiguous due to the finite-time nature of the simulation. The white points have not been calculated by the simulator to save time, since a point at a lower load was found to be unequivocally unstable and the stability region is expected to be a Pareto bound. We recall that every cell in the grids comes from averaging ten different traffic configurations, where a configuration consists of the same two main pairs and a fresh set of eight parasitic ones. The shape of each stability region may be seen as a measure of competition between user pairs: the more diagonal the boundary of the dark blue region, the higher the direct competition between the main pairs. The difference in area of regions along one given column is a direct measurement of how the main and parasitic pairs compete (and therefore how the network serves requests under increasing stress), while the differences along one row show how well the scheduler leverages additional information.