Lagrangian Duality in Quantum Optimization: Overcoming QUBO Limitations for Constrained Problems

TL;DR

This paper introduces a quantum optimization method leveraging Lagrangian duality to efficiently solve constrained problems, outperforming traditional QUBO-based approaches in circuit depth and resource usage.

Contribution

The authors develop a Lagrangian duality-based quantum approach that overcomes QUBO limitations, achieving quadratic circuit depth improvement and constraint independence.

Findings

Quadratic reduction in circuit depth compared to QUBO methods

Resource-efficient quantum circuits for constrained problems

Successful benchmarking on the binary knapsack problem

Abstract

We propose an approach to solving constrained combinatorial optimization problems based on embedding the concept of Lagrangian duality into the framework of adiabatic quantum computation. Within the setting of circuit-model fault-tolerant quantum computation, we demonstrate that this approach achieves a quadratic improvement in circuit depth and maintains a constraint-independent circuit width in contrast to the prevalent approach of solving constrained problems via reformulations based on the quadratic unconstrained binary optimization (QUBO) framework. Our study includes a detailed review of the limitations encountered when using QUBO for constrained optimization. We show that the proposed method overcomes these limitations by encoding the optimal solution at an energetically elevated level of a simpler problem Hamiltonian, which results in substantially more resource-efficient quantum circuits. We consolidate our strategy with a detailed analysis on how the concepts of Lagrangian duality such as duality gap and complementary slackness relate to the success probability of sampling the optimal solution. Our findings are illustrated by benchmarking the Lagrangian dual approach against the QUBO approach using the NP-complete binary knapsack problem.