Sparse QUBO Formulation for Efficient Embedding via Network-Based Decomposition of Equality and Inequality Constraints

TL;DR

The paper tackles the embedding bottleneck in quantum annealing by reformulating equality and inequality constraints into sparse QUBOs via network-based constraint decomposition. By introducing auxiliary variables and a divide-and-conquer network, it reduces dense $O(N^2)$ constraint interactions to $O(N)$ (one-hot) or $O(N\log N)$ (general equality), improving embedding onto Pegasus and Zephyr graphs. Empirical results on D-Wave hardware show substantial reductions in required qubits, shorter chain lengths, lower chain-break rates, and higher feasible solution rates compared to conventional formulations. The approach offers a practical pathway to solve constrained optimization problems on current and future quantum annealers, with potential extensions to broader constraint classes and network designs.

Abstract

Quantum annealing is a promising approach for solving combinatorial optimization problems. However, its performance is often limited by the overhead of additional qubits required for embedding logical QUBO models onto quantum annealers. This overhead becomes severe when logical QUBO models have dense connectivity. Such dense structures frequently arise when formulating equality and inequality constraints. To address this issue, we propose a method to construct a significantly sparser logical QUBO model for these constraints. By adding auxiliary variables based on specific network structures, our approach decomposes the original constraint into smaller, more manageable constraints. We demonstrate that this method reduces the number of edges (quadratic terms) from $O(N^2)$ to $O(N)$ for the one-hot constraint and to $O(N\log N)$ in the worst case for general equality constraints, where $N$ is the number of variables. Experimental results on D-Wave's hardware show that our formulation leads to substantial reductions in the number of qubits required for embedding, shorter average chain lengths, lower chain break rates, and higher feasible solution rates compared to conventional methods. This work provides a practical tool for efficiently solving constrained optimization problems on current and future quantum annealers.

Figures (10)

• Figure 1: Comparison of the conventional and proposed formulations for a constraint $\sum x_i = 1$. The conventional method (top) generates a dense QUBO model (clique), resulting in a complex embedding that requires many physical qubits and is prone to chain breaks. In contrast, the proposed method (bottom) introduces auxiliary variables to construct a sparse logical QUBO model. This sparsity significantly reduces the number of physical qubits required for embedding and suppresses chain break.
• Figure 2: Examples of network structures for $N=4$ variables. (a) The network representation of the conventional method (clique). In this framework, the conventional method can be interpreted as a special case where no auxiliary variables are introduced (i.e., $\boldsymbol{Y}=\emptyset$) and the sub-constraint consists of a single constraint $\boldsymbol{S}=[\sum x_i = \sum c_i]$. A single switch connects all wires simultaneously, directly enforcing $\sum x_i = \sum c_i$. (b) The proposed sparse network structure for a one-hot constraint ($\sum x_i = 1$). Binary auxiliary variables $\boldsymbol{Y}$ are assigned to the intermediate wire segments. The original constraint is decomposed into sub-constraints $\boldsymbol{S}$ at each switch. For example, the top-left sub-constraint $S_1$ enforces $x_1+x_2=c_1+y_1$, where $[x_1, x_2]$ are the inputs and $[c_1, y_1]$ are the outputs to the switch.
• Figure 3: An illustrative example of the network operation for a specific input configuration ($x_1=0, x_2=1, x_3=0, x_4=0$) that satisfies the constraint $x_1+x_2+x_3+x_4=1$. The red labels indicate the state of each switch: "straight" means the values pass through, and "swap" means the two values are exchanged. The resulting values of the auxiliary variables ($y_1=1, y_2=1$) are also shown. Through this sequence of local operations, the network correctly permutes the set of input values to match the set of output constants at the right side.
• Figure 4: Stepwise visualization of the recursive decomposition process for an equality constraint on 8 variables using the divide-and-conquer algorithm. As the algorithm proceeds, the constraints are recursively divided into smaller sub-constraints by introducing auxiliary variables, forming a hierarchical tree structure of switches.
• Figure 5: Network structures for equality constraints on 10 variables with varying target values $K$. The network complexity depends on $K$, reaching a maximum at $K=N/2$. Consequently, the number of auxiliary variables and edges in the QUBO model also increases as $K$ approaches $N/2$.