A Scalable Distributed Quantum Optimization Framework via Factor Graph Paradigm

TL;DR

A structure-aware framework for distributed quantum optimization that resolves this complexity-resource trade-off and preserves Grover-like scaling, and extends the framework with a hierarchical divide-and-conquer strategy that scales to large-scale optimization problems.

Abstract

Distributed quantum computing (DQC) connects many small quantum processors into a single logical machine, offering a practical route to scalable quantum computation. However, most existing DQC paradigms are structure-agnostic. Circuit cutting proposed by Peng et al. in [Phys. Rev. Lett., Oct. 2020] reduces per-device qubits at the cost of exponential classical post-processing, while search-space partitioning proposed by Avron et al. in [Phys. Rev. A., Nov. 2021] distributes the workload but weakens Grover's ideal quadratic speedup. In this paper, we introduce a structure-aware framework for distributed quantum optimization that resolves this complexity-resource trade-off. We model the objective function as a factor graph and expose its sparse interaction structure. We cut the graph along its natural ``seams'', i.e., a separator of boundary variables, to obtain loosely coupled subproblems that fit on resource-constrained processors. We coordinate these subproblems with shared entanglement, so the network executes a single globally coherent search rather than independent local searches. We prove that this design preserves Grover-like scaling: for a search space of size $N$, our framework achieves $O(\sqrt{N})$ query complexity up to processors and separator dependent factors, while relaxing the qubit requirement of each processor. We extend the framework with a hierarchical divide-and-conquer strategy that scales to large-scale optimization problems and supports two operating modes: a fully coherent mode for fault-tolerant networks and a hybrid mode that inserts measurements to cap circuit depth on near-term devices. We validate the predicted query-entanglement trade-offs through simulations over diverse network topologies, and we show that structure-aware decomposition delivers a practical path to scalable distributed quantum optimization on quantum networks.

Figures (10)

• Figure 1: An illustration of an example quantum network, where $\mathrm{QPU}_{\mathrm{c}}$ is the coordinator processor and $\mathrm{QPU} _{1}, \mathrm{QPU} _{2}, \mathrm{QPU} _{3}$ are worker processors.
• Figure 2: Total number of leaf-oracle evaluations per execution of the distributed primitive versus the number of variables $| \mathcal{V} |$. The $y$-axis is logarithmic. The dashed curve is a log-linear fit (linear in $\log_{10}$).
• Figure 3: Total number of elementary EPR pairs consumed versus the coordinator-to-workers diameter (hops) under topology stretching. The $y$-axis is logarithmic.
• Figure 4: Total oracle-query count versus $| \mathcal{V} |$ (total number of variable nodes) on a two-level hierarchical benchmark, under four execution policies. The $y$-axis is logarithmic.
• Figure 6: The factor graph $\mathsf{G} _{1}$ for the example portfolio investment problem in Example \ref{['ex:local_portfolio_investment_problem_factor_graph']}, illustrating the relationships between asset decisions (circles) and financial objectives and constraints (squares).

Theorems & Definitions (41)

• Definition 1: Factor graph-based optimization problem
• Definition 2: Boundary-based split of a factor graph
• Definition 4: Amplitude amplification operator Brassard1997
• Theorem 5: Algorithm Performance Guarantee
• proof
• Proposition 6: Performance of Generic Quantum Maximization
• proof
• Definition 7: Local Oracles
• Lemma 8: Performance of $\mathbf{U}_{\mathrm{ini}}$ Construction
• proof