A Rigorous Quantum Framework for Inequality-Constrained and Multi-Objective Binary Optimization: Quadratic Cost Functions and Empirical Evaluations

TL;DR

This work introduces MOQA, a rigorous framework to solve inequality-constrained and multi-objective binary optimization by mapping to Ising-type Hamiltonians. It extends QUBO formulations by approximating the max of multiple objectives with a polynomial $h_{(p)}(\mathbf{b})=\sum_{m=1}^M h_m^p(\mathbf{b})$, and provides a method to shift and bound the Hamiltonians so that $h_{(p)}$ preserves the optimization landscape with a relative error governed by $M^{-1/p}$. The authors supply a concrete construction algorithm that computes coefficients $C_{(p)}(\mathbf{x})$ for $\hat{H}_{(p)}$ from $p$-th powers of constituent QUBO Hamiltonians, achieving a sparse representation with at most $\mathcal{O}(n^{2p})$ nonzero terms, enabling polynomial preprocessing and subexponential quantum resource growth for moderate $p$ and $M$. Numerical experiments across generic objectives, partitioning, and inequality constraints show MOQA yields near-optimal solutions with errors diminishing as $p$ increases, while exploiting sparsity to keep the quantum-cost manageable. Overall, MOQA broadens the applicability of quantum solvers (AQC, QA, QAOA) to a wider class of combinatorial problems, providing a scalable pathway toward practical quantum- and quantum-inspired optimization with inequality constraints and multiple objectives.

Abstract

The prospect of quantum solutions for complicated optimization problems is contingent on mapping the original problem onto a tractable quantum energy landscape, e.g. an Ising-type Hamiltonian. Subsequently, techniques like adiabatic optimization, quantum annealing, and the Quantum Approximate Optimization Algorithm (QAOA) can be used to find the ground state of this Hamiltonian. Quadratic Unconstrained Binary Optimization (QUBO) is one prominent problem class for which this entire pipeline is well understood and has received considerable attention over the past years. In this work, we provide novel, tractable mappings for the maxima of multiple QUBO problems. Termed Multi-Objective Quantum Approximations, or MOQA for short, our framework allows us to recast new types of classical binary optimization problems as ground state problems of a tractable Ising-type Hamiltonian. This, in turn, opens the possibility of new quantum- and quantum-inspired solutions to a variety of problems that frequently occur in practical applications. In particular, MOQA can handle various types of routing and partitioning problems, as well as inequality-constrained binary optimization problems.

Figures (8)

• Figure 1: Random undirected graph. Example graph for a partitioning task, with randomly distributed and valued vertices $\bm{v}$ whose distance corresponds to the adjacency matrix $\bm{W}$. For example, this graph could represent a protein network, where edges reflect protein–protein interactions and vertex values indicate if a protein should be contained or avoided Sun_2017.
• Figure 2: Bound on relative errors. The relative difference between the upper and lower bounds of the sandwich inequality (Eq. \ref{['eq: eigenvalue-sandwich']}) is given by $M^{-1/p}$. This term is visualized as a function of the approximation level $p$ for various numbers of objectives $M$. To bound up to ${M = 105}$ objectives in an interval of $\pm 5\%$ around the true multi-objective, $p\leq 100$ is required.
• Figure 3: Comparison of runtimes. The $Mn^22^n$ steps of a classical brute force algorithm are compared to the requirements of MOQA. Those encompass $(M+3)n^2\binom{p + (n^2+n)/2}{p}$ classical steps and $\sum_{k=1}^{2p}\binom{n}{k}$ quantum gates. The comparison is done for $M=10$ QUBOs and an approximation-level of $p=4$ as a function of the problem size $n$.
• Figure 4: Visualization of the MOQA framework for 4 random QUBOs in $n=6$ variables. We present the diagonal matrix of Ising Hamiltonians as functions $h(\bm{b})$ of their indices. We sample 4 generic QUBOs shown as dotted green lines. In the multi-objective task, we take the maximum value per $\bm{b}$ across those 4 Hamiltonians, yielding the desired $h_{\max}$, shown as the solid red line. $h_{\max}$ is consequently approximated via $h_{(p)}$ whose $p$-th root displayed as blue dashed lines. They become more tightly bound to $h_{\max}$ as $p$ increases, leading to alignment of their global minima, as shown by the dots.
• Figure 5: Average errors across multiple random tasks. The approximation error of $h_{(p)}$ to $h_{\max}$ as a function of $p$ is analyzed over $N_S = 10000$ random tasks per problem size $n\in\{4,8,12,16,20\}$ and either $M=2$ or $M=8$ random quadratic objectives. The absolute error $\epsilon$ shown in blue is the proportion of problems where the minima of $h_{\max}$ and $h_{(p)}$ align. The orange lines represent the relative error between these minima.

Theorems & Definitions (5)

• Proposition 1: runtime demands of Algorithm \ref{['alg:compute_cp']}
• proof : Proof of Proposition \ref{['prop: runtime']}
• Lemma 1: maximum number of non-zero terms in a MOQA Hamiltonian
• Proposition 2
• proof : Proof of Proposition \ref{['prop:ineq']}