A Rigorous Quantum Framework for Inequality-Constrained and Multi-Objective Binary Optimization

TL;DR

It is shown that including inequality constraints is equivalent to solving a multi-objective optimization and motivates the Multi-Objective Quantum Approximation (MOQA) framework, which approximates the maximum via smaller $p$-norms and comes with rigorous performance guarantees.

Abstract

Encoding combinatorial optimization problems into physically meaningful Hamiltonians with tractable energy landscapes forms the foundation of quantum optimization. Numerous works have studied such efficient encodings for the class of Quadratic Unconstrained Binary Optimization (QUBO) problems. However, many real-world tasks are constrained, and handling equality and, in particular, inequality constraints on quantum computers remains a major challenge. In this letter, we show that including inequality constraints is equivalent to solving a multi-objective optimization. This insight motivates the Multi-Objective Quantum Approximation (MOQA) framework, which approximates the maximum via smaller $p$-norms and comes with rigorous performance guarantees. MOQA operates directly at the Hamiltonian level and is compatible with, but not restricted to, ground-state solvers such as quantum adiabatic annealing, the Quantum Approximate Optimization Algorithm (QAOA), or imaginary-time evolution. Moreover, it is not limited to quadratic functions.

Figures (3)

• Figure 1: Concept and Performance of the MOQA framework. Top: Summary of the framework. We start with a problem described by multiple individual objectives. Evaluating their maximum can be done either exactly using exponential resources or approximately with only a polynomial overhead using MOQA. Theorem \ref{['thm: theorem']} relates these two approaches. Bottom left: Visualization of a QUBO in $n=6$ variables with a single linear inequality constraint ($\gamma=120$). The resulting $h_{\max}(\bm{b})$ is shown as the solid red line. This function is consequently approximated via $h_{(p)}(\bm{b})$ displayed by their $p$-th root as blue dashed lines. They become more tightly bound to $h_{\max}(\bm{b})$ as $p$ increases, leading to alignment of their global minima, as shown by the dots. Bottom right: Sampling 10000 random instances of such problems allows for an analysis of the two error statistics $\delta$ and $\epsilon$ as a function of $p$.
• Figure 2: Approximation error over spectral gap ratio. A total of $N_s=10000$ random QUBOs in $n=20$ variables combined with a linear inequality constraint ($\gamma=6$) are analyzed. They are collected into bins based on the resulting spectral gap ratio $r (\hat{H}_{\max})$. For the three approximation levels $p\in\{5,10,20\}$, the absolute difference $\epsilon$ is shown per bin. In addition, the corresponding vertical lines indicate the threshold, after which Theorem \ref{['thm: theorem']} guarantees that this error is 0.
• Figure 3: (Non)degenerate ground space. On the left, we show a schematic of the ground space of some $\hat{H}_{\max}$ with a degenerate minimum. If we think of the energy ($y$-axis) in units of $\lambda_1$, we can show the spectral gap ratio $r_{\max}$ as a shift between the degenerate minima and the first excited state. On the right, we show the schematic ground space of $\hat{H}_{(p)}$. The approximation will likely break that symmetry, leaving only one global minimum. Thus, we distinguish between the gap ratio $r_{(p)}$ between the levels that are ideal solutions to the original problem and the gap ratio $\tilde{r}_{(p)}$ that resembles the difference to the first non-ideal state.

Theorems & Definitions (4)

• Proposition 1
• proof
• Theorem 1
• proof : Proof of Theorem \ref{['thm: theorem']}