Towards Quantum Algorithms for the Optimization of Spanning Trees: The Power Distribution Grids Use Case

TL;DR

This work tackles reconfiguration of radial distribution grids to minimize Ohmic losses by mapping the problem to MDST, a spanning-tree optimization that is NP-hard and non-locally dependent on topology. It proposes two quantum primitives based on the Quantum Alternating Operator Ansatz (QAOA): a penalty-based sampling method and an invariant feasible-subspace approach that uses partial edge-rotation mixers to navigate only between feasible spanning trees. The authors provide a rigorous quantum encoding using binary variables $y_{e,n}$, develop complete edge-rotation moves, derive resource estimates, and perform QAOA simulations showing the invariant-subspace method can outperform penalty-based sampling on small instances, albeit with deeper circuits and sensitivity to hyperparameters. By mapping network reconfiguration to MDST+ on reduced graphs, the paper links a hard classical problem to quantum optimization strategies, offering pathways for future hardware experiments and potential benefits for energy-grid planning and resilience.

Abstract

Optimizing the topology of networks is an important challenge across engineering disciplines. In energy systems, network reconfiguration can substantially reduce losses and costs and thus support the energy transition. Unfortunately, many related optimization problems are NP hard, restricting practical applications. In this article, we address the problem of minimizing losses in radial networks, a problem that routinely arises in distribution grid operation. We show that even the computation of approximate solutions is computationally hard and propose quantum optimization as a promising alternative. We derive two quantum algorithmic primitives based on the Quantum Alternating Operator Ansatz (QAOA) that differ in the sampling of network topologies: a tailored sampling of radial topologies and simple sampling with penalty terms to suppress non-radial topologies. We show how to apply these algorithmic primitives to distribution grid reconfiguration and quantify the necessary quantum resources.

Figures (19)

• Figure 1: One-to-one correspondence between a feasible configuration of switches in an electrical distribution grid $\mathcal{G}_{\mathrm{grid}}$ (left) and a spanning tree $\mathcal{T}$ with root $v_0$ in the reduced graph $\mathcal{G}_{\mathrm{red}}$ (right), whose edges represent the switches in the distribution grid. Note that buses $7$ and $8$ in $\mathcal{G}_{\mathrm{grid}}$ can be reduced to bus $6$, since the currents $i_e$ on $e=(6,7)$ and $e=(7,8)$ are nor affected by any reconfiguration.
• Figure 2: Comparison of the two quantum algorithms for sampling spanning trees for a simple graph with three nodes and three edges. The root is set as $r=0$. a: For the transverse field mixer $U_{\mathrm{TF}}(\beta)$, all 64 configurations can be reached; however, only three of them are feasible (highlighted in light blue). The graph corresponds to the Hamiltonian $H_{\mathrm{TF}} = \sum_j X_j$, the edges correspond to the possible transitions according to the Hamiltonian $H_{\mathrm{TF}}$. Blue edges show (potential) shortest paths between the feasible configurations, b: Partial mixers $U^{\mathrm{PM}}_{r: e \leftrightarrow e^\prime}(\beta)$ implement transition only between two feasible configurations, that is, spanning trees $\mathcal{T}$.
• Figure 3: Comparison of the performance of LR-QAOA using the penalty method and RevLR-QAOA employing the invariant feasible subspace approach for a simple MDST instance based on the graph topology shown in Fig. \ref{['paper:fig:comparison_ctqw']}. The problem instance is $\alpha_0 = \alpha_1 = 1$ and $\alpha_2=10$; $\mathfrak{f}_0=-3$, $\mathfrak{f}_1=1$ and $\mathfrak{f}_2=2$. The optimal bit string solution is given by $110100$. a: QAOA schedules $(\gamma_k, \beta_k)$ for LR-QAOA and RevLR-QAOA. The annealing time $T_{\mathrm{A}}$ and the number of layers $K$ are (hyper)-parameters, that define the values of the angles $\beta_k$ and $\gamma_k$ (cf. Methods). b: Final measurement statistics for the best found parameter configurations in a grid search. For LR-QAOA we have $K=200$, $T_{\mathrm{A}}=1$, for RevLR-QAOA $K=200$, $T_{\mathrm{A}} = 0.54$. c: Performance measured by the approximation ratio as a function of the annealing time $T_A$ for fixed $K$. The approximation ratio is defined as the ratio between the energy expectation value of the final state and the ground state (optimal) energy. Lower values of the approximation ratio indicate better performance, with the theoretical lower bound (best achievable value) being $1$. As a benchmark, we compare the performance to picking any feasible state completely at random (vertical black line) and any spin configuration at random (vertical dashed black line).
• Figure 4: Two spanning trees and fundamental cycle basis for an elementary six node graph $\mathcal{G}$ (light blue). A spanning tree $\mathcal{T}$ is a subgraph of $\mathcal{G}$ such that all nodes are connected and there are no cycles. The edges $e, e^\prime$ and $e^{\prime\prime}$ not in $\mathcal{T}$ each define a fundamental cycle, and all fundamental cycles together define a cycle basis of $\mathcal{G}$. The two drawn spanning trees $\mathcal{T}$ (red) and $\mathcal{T}^\prime$ (yellow) define the same fundamental cycle basis $\{\mathcal{C}_e, \mathcal{C}_{e^\prime}, \mathcal{C}_{e^{\prime \prime}} \}$.
• Figure 5: The flows on an edge $e \in \mathcal{T}$ can be readily computed by summing over all downward demands/supplies, respecting the signs. For illustration, we make the signs that indicate the direction of the flow on an edge explicit.

Theorems & Definitions (23)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Corollary 1
• proof