Hierarchical divide and conquer quantum approach to combinatorial optimization problems with tunable reduction

TL;DR

This work introduces a hierarchical divide-and-conquer strategy for quantum optimization that partitions a large problem into subgraphs, identifies a low-energy subspace in each, and then recombines these reduced representations. By using a tunable energy cut-off and binary encoding of local states, the method preserves the global optimum while drastically reducing qubit requirements, with iterative refinement possible. Numerical tests on sparse, weighted graphs show significant qubit reductions (up to roughly |V|/4 for |V|≈40) and near-perfect approximation ratios, especially when the retained energy range is modest (η≈0.5). The framework extends naturally to polynomial unconstrained binary optimization (PUBO) problems represented as hypergraphs, broadening its applicability to practical quantum optimization tasks.

Abstract

Combinatorial optimization is considered a promising class of problems in which quantum computers can show significant advantages. However, problems of practical relevance typically have more variables than current or foreseeable quantum computers have qubits. Here we introduce a divide and conquer approach that partitions the optimization problem into subgraphs that can be represented on smaller quantum processors. We then find all states of the subgraphs that can possibly be part of the solution to the entire problem by determining the cost or energy ranges in which the local subgraph energies of these states must be contained. This allows us to reduce the problem by only considering the subspace spanned by these states. We then recombine the system using a binary encoding for each subgraph with a local energy ordering. This process can be iterated until no further reduction is possible. We also find that the number of necessary qubits can be reduced further when only retaining states in a fraction of the relevant energy range at very little expense in terms of approximation ratio to the global ground state. In numerical simulations, we find that our approach allows us to solve combinatorial optimization problems on weighted random 3-regular graphs with $|\mathcal{V}|=40$ discrete variables on $\sim |\mathcal{V}| / 4$ qubits while retaining a possible approximation ratio of $\sim99.9\%$. We also observe an increasing reduction with larger system sizes.

Figures (11)

• Figure 1: Schematic representation of our approach. The combinatorial problem instance is represented as a graph. Using a community detection algorithm (\ref{['sec:clustering']}) this graph is divided into subgraphs, see step "Clustering" in the illustration. In a subroutine S (\ref{['fig:subroutine']}) the lowest-lying energy states within a predetermined energy range are determined and then represented by a new reduced System $\tilde{\mathscr{F}}$, see step "Reduction". If no further reduction is possible the whole system is solved using an optimizer $O_2$, otherwise it is again split into subgraphs and the process is iterated until no further reduction is possible.
• Figure 2: The local subroutine S of our algorithm. For a given Community $\mathop{\mathrm{\mathcal{C\mkern-3mu om}}}\nolimits_i$ we determine the cut-off range (Here for a quadratic Hamiltonian $\Delta_i=|J_{1_i,1_j}|+|J_{2_i,1_j}|+|J_{3_i,2_j}|$) and determine all low-lying states within ($\mathscr{F}_i$) using an optimizer $O_1$. Then we represent the local solutions as computational basis states using a local energy ordering with fewer qubits $\tilde{\mathscr{F}_i}$. If it is not the first iteration, a different optimizer $O_2$ might be used, see \ref{['subsec:iteration']}.
• Figure 3: The low lying set of eigenstates $\ket{\psi_i^\mu} \in \mathscr{F}_i$ of a community $\mathop{\mathrm{\mathcal{C\mkern-3mu om}}}\nolimits_i$ (left) is cast into the computational basis of a reduced number of qubits $\tilde{\mathscr{F}}_i$ (right) by using a binary encoding with a local energy ordering.
• Figure 4: Performance of the algorithm for $O_1=O_2=$F-VQE for a set of various weighted random graphs as indicated in the legend (see Sec. \ref{['sec:random_graphs']} for further explanations). Left column: qubit reduction $R$ as defined in Eq. \ref{['eq:qubit_red']}. Middle column: approximation ratio $\alpha$ as defined in Eq. \ref{['eq:approx_ratio']}. Right column: number of iterations in the algorithm $N_{\text{it}}$. Each row shows results for retaining different fractions $\eta$ of the energy ranges in the subsystems, where $\eta=0$ means that only the degenerate ground states are retained, where another optimization $O_2$ is still performed. Green lines correspond to graphs with underlying average degree 2, purple lines to graphs with degree 3 and orange lines to graphs with degree 4.
• Figure 5: Performance of the algorithm for $O_1=O_2=$brute force for a set of various random graphs as indicated in the legend (see Sec. \ref{['sec:random_graphs']} for further explanations). Left column: qubit reduction $R$ as defined in Eq. \ref{['eq:qubit_red']}. Middle column: approximation ratio $\alpha$ as defined in Eq. \ref{['eq:approx_ratio']}. Right column: number of iterations in the algorithm $N_{\text{it}}$. Each row shows results for retaining different fractions $\eta$ of the energy ranges in the subsystems, where $\eta=0$ means that only the degenerate ground states are retained, where another optimization $O_2$ is still performed. Green lines correspond to graphs with degree 2, purple lines to graphs with degree 3 and orange lines to graphs with degree 4 (The ground state energy was taken to be the energy found when $\eta=1$).

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• proof
• proof