[Paper Review] Quantum Integer Programming (QuIP) 47-779: Lecture Notes

David E. Bernal, Sridhar Tayur+1 more|arXiv (Cornell University)|Dec 17, 2020

Quantum Computing Algorithms and Architecture55 references4 citations

TL;DR

This paper presents a comprehensive lecture series on Quantum Integer Programming (QuIP), teaching researchers to formulate integer programs as Quadratic Unconstrained Binary Optimization (QUBO) problems and solve them using quantum-inspired and near-term quantum methods like Quantum Annealing, QAOA, and the Graver Augmented Multiseed Algorithm (GAMA). The key contribution is a practical framework for mapping real-world optimization problems to quantum-ready QUBO formulations and benchmarking performance across classical and quantum hardware with systematic parameter tuning.

ABSTRACT

This lecture series on Quantum Integer Programming (QuIP) -- created by Professor Sridhar Tayur, David E. Bernal, and Dr. Davide Venturelli, a collaboration between CMU and USRA, with the support from Amazon Braket during Fall 2020 -- is intended for students and researchers interested in Integer Programming and the potential of near term quantum and quantum-inspired computing in solving optimization problems. Originally created for Tepper School of Business course 47-779 (at CMU), these were also used for the course ID5840 (at IIT-Madras, by Professors Anil Prabhakar and Prabha Mandayam) whose students (listed at the beginning of each lecture) were scribes. Dr. Vikesh Siddhu, post-doc in CMU Quantum Computing Group, assisted during the lectures, student projects, and with proof-reading this scribe. Through these lectures one will learn to formulate a problem and map it to a Quadratic Unconstrained Binary Optimization (QUBO) problem, understand various mapping and techniques like the Ising model, Graver Augmented Multiseed Algorithm (GAMA), Simulated or Quantum Annealing and QAOA, and ideas on how to solve these Integer problems using these quantum and classical methods.

Motivation & Objective

• To equip graduate students and researchers with tools to reformulate non-linear integer programs into QUBO form suitable for quantum and quantum-inspired solvers.
• To demonstrate the application of quantum methods—particularly Quantum Annealing and QAOA—on real-world combinatorial optimization problems.
• To provide a benchmarking strategy for evaluating quantum and classical solvers on QUBO instances, focusing on solution quality, success probability, and resource usage.
• To enable students to develop and evaluate hybrid projects combining classical optimization with near-term quantum hardware or simulators.
• To foster 'quantum readiness' by teaching practical skills in problem formulation, QUBO mapping, and performance evaluation on D-Wave, IBM, and GPU/TPU platforms.

Proposed method

• Formulate real-world problems as non-linear integer programs, then reformulate them into QUBO form using penalty methods and binary encoding.
• Apply the Ising model and QUBO mapping techniques to encode constraints and objectives into spin-based Hamiltonians.
• Implement the Graver Augmented Multiseed Algorithm (GAMA) for quantum-inspired optimization of quadratic (semi-)assignment problems.
• Use Quantum Annealing on D-Wave hardware with tunable parameters such as chain strength and annealing time to maximize optimal solution probability.
• Apply the Quantum Approximate Optimization Algorithm (QAOA) via quantum circuits using parameterized gates (γ and β) to explore entangled superposition states.
• Benchmark solutions using statistical sampling across multiple runs, analyzing success rates, feasibility, and resource consumption (time, energy) on classical and quantum back-ends.

Experimental results

Research questions

• RQ1 How can non-linear integer programs from real-world applications be effectively reformulated into QUBO form while preserving problem constraints?
• RQ2 What is the optimal configuration of chain strength and annealing time in Quantum Annealing to maximize the probability of finding feasible and optimal solutions?
• RQ3 How does the performance of QAOA compare to Quantum Annealing and classical simulated annealing on benchmark QUBO instances?
• RQ4 To what extent can quantum-inspired methods like GAMA outperform classical solvers on specific classes of integer programs?
• RQ5 What systematic benchmarking strategy ensures reliable comparison between quantum and classical solvers in terms of solution quality and resource efficiency?

Key findings

• Quantum Annealing with default parameters yields only 0.05% probability of finding the optimal solution and 0.15% for feasible solutions, indicating poor out-of-the-box performance.
• Increasing chain strength initially improves optimal solution probability but eventually reduces it due to increased chain breaking and energy level splitting.
• For a 2-second annealing time, optimal solution counts drop significantly compared to longer runs, suggesting trade-offs between runtime and success rate.
• The study recommends running multiple annealing instances with varied parameters to explore diverse solution sets, rather than relying on a single long run.
• QAOA enables coherent, circuit-based optimization using parameterized quantum circuits, though the probability of measuring the ground state remains difficult to compute analytically.
• A systematic benchmarking pipeline—including parameter tuning, statistical sampling, and performance comparison—was shown to be essential for reliable evaluation of quantum advantage claims.