Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2012.11382

Quantum Integer Programming (QuIP) 47-779: Lecture Notes

David E. Bernal, SRIDHAR R. TAYUR, Davide Venturelli

Published Dec 17, 2020• 3 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2012.11382

Related papers

Quantum annealing research at CMU: algorithms, hardware, applications

SRIDHAR R. TAYUR, Ananth Tenneti

In this mini-review, we introduce and summarize research from the Quantum Technologies Group (QTG) at Carnegie Mellon University related to computational experience with quantum annealing, performed in collaboration with several other institutions including IIT-Madras and NASA (QuAIL). We present a novel hybrid quantum-classical heuristic algorithm (GAMA, Graver Augmented Multi-seed Algorithm) for non-linear, integer optimization, and illustrate it on an application (in cancer genomics). We then present an algebraic geometry-based algorithm for embedding a problem onto a hardware that is not fully connected, along with a companion Integer Programming (IP) approach. Next, we discuss the performance of two photonic devices - the Temporal Multiplexed Ising Machine (TMIM) and the Spatial Photonic Ising Machine (SPIM) - on Max-Cut and Number Partitioning instances. We close with an outline of the current work.

Frontiers in Computer Science•Jan 5, 2024•3 citations

Quantum Optimization in Loc(Q)ation Science: QUBO Formulations, Benchmark Problems, and a Computational Study

Felix P. Broesamle, Stefan Nickel

Recent advances in quantum computing and the increasing availability of quantum hardware have substantially enhanced the practical relevance of quantum approaches to discrete optimization. Among these, the Quadratic Unconstrained Binary Optimization (QUBO) formulation provides a unifying modeling framework for a broad class of $\mathbf{NP}$-hard problems and is naturally suited to quantum computing and quantum-inspired algorithms. Location science, network design, and logistics represent core application domains of discrete optimization, combining high practical impact with substantial computational challenges. In this work, we develop QUBO formulations for several fundamental problems in these domains, including a nonlinear integer formulation of the Discrete Ordered Median Problem (DOMP). Beyond their modeling relevance, these QUBO formulations serve as representative benchmark problems for assessing quantum algorithms and quantum hardware. We further derive a tight bound for the penalty parameter ensuring equivalence between the QUBO formulation and its underlying integer program. Finally, we conduct a comprehensive computational study using QAOA, WS-QAOA, and classical heuristics for QUBO instances of the $p$-Median Problem and the Fixed-Charge Facility Location Problem (FCFLP), and introduce two effective warm-start strategies for WS-QAOA based on its linear programming relaxation.

quant-pharXiv:2602.10951•Feb 12, 2026•0 citations

Five Starter Problems: Solving Quadratic Unconstrained Binary Optimization Models on Quantum Computers

Arul Rhik Mazumder, SRIDHAR R. TAYUR

This tutorial offers a quick, hands-on introduction to solving Quadratic Unconstrained Binary Optimization (QUBO) models on currently available quantum computers and their simulators. We cover both IBM and D-Wave machines: IBM utilizes a gate-circuit architecture, and D-Wave is a quantum annealer. We provide examples of three canonical problems and two models from practical applications. The tutorial is structured to bridge the gap between theory and practice: we begin with an overview of QUBOs, explain their relevance and connection to quantum algorithms, introduce key quantum computing concepts, provide the foundations for two quantum heuristics, and provide detailed implementation guides. An associated GitHub repository provides the codes in five companion notebooks. In addition to reaching undergraduate and graduate students in computationally intensive disciplines, this article aims to reach working industry professionals seeking to explore the potential of near-term quantum applications. As our title indicates, this tutorial is intended to be a starting point in a journey towards solving more complex QUBOs on quantum computers.

arXiv:2401.08989•Oct 1, 2025•7 citations

Hybrid Quantum–Classical Framework for Fuzzy 0–1 Integer Linear Programming: α-Level Trade-off Analysis via Quadratic Unconstrained Binary Optimization Reformulation

Hamed Fazlollahtabar, Hadi Gholizadeh

This paper addressed the computational challenges of solving large-scale fuzzy 0-1 Integer Linear Programming (ILP) problems by proposing a Fuzzy-Hybrid Quantum-Classical Optimization Algorithm (F-HQCOA).The novel approach employed α-cut based transformations to convert fuzzy ILP models into a family of crisp Quadratic Unconstrained Binary Optimization (QUBO) problems, which were then solved using hybrid quantumclassical techniques.Unlike multi-objective formulations, the proposed framework performed a parametric analysis over α-levels to generate a spectrum of solutions that reflected trade-offs between objective optimality and feasibility satisfaction under uncertainty.The degree of satisfaction (µ) was adopted as an evaluation metric to assess solution quality rather than as an independent optimization objective.The method was evaluated on fuzzy Multi-Dimensional Knapsack and Set Covering problems.Results showed that the proposed approach could efficiently handle larger instances and provided a diverse set of solutions across α-levels.While quantum-assisted methods demonstrated reduced Time-to-Solution (TTS) in certain settings, comparisons were presented with careful consideration of differences in computational paradigms.The proposed framework offers a practical decision-support tool for optimization under uncertainty and establishes a foundation for future extensions toward true multi-objective fuzzy quantum optimization.

Journal of Industrial Intelligence•Sep 2, 2025•0 citations

Enhancing Quantum Algorithms for Quadratic Unconstrained Binary Optimization via Integer Programming

Friedrich Wagner, Jonas Nüßlein, Frauke Liers

To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing is realized, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem and for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of depth-1 QAOA, a parameterized quantum algorithm, by deriving a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.

arXiv:2302.05493•ACM Transactions on Quantum Computing•Jan 13, 2025•2 citations

Solving the resource constrained project scheduling problem with quantum annealing

Luis Fernando Pérez Armas, Stefan Creemers, Samuel Deleplanque

Quantum annealing emerges as a promising approach for tackling complex scheduling problems such as the resource-constrained project scheduling problem (RCPSP). This study represents the first application of quantum annealing to solve the RCPSP, analyzing 12 well-known mixed integer linear programming (MILP) formulations and converting the most qubit-efficient one into a quadratic unconstrained binary optimization (QUBO) model. We then solve this model using the D-wave advantage 6.3 quantum annealer, comparing its performance against classical computer solvers. Our results indicate significant potential, particularly for small to medium-sized instances. Further, we introduce time-to-target and Atos Q-score metrics to evaluate the effectiveness of quantum annealing and reverse quantum annealing. The paper also explores advanced quantum optimization techniques, such as customized anneal schedules, enhancing our understanding and application of quantum computing in operations research.

Scientific Reports•Jul 22, 2024•30 citations
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.
In this mini-review, we introduce and summarize research from the Quantum Technologies Group (QTG) at Carnegie Mellon University related to computational experience with quantum annealing, performed in collaboration with several other institutions including IIT-Madras and NASA (QuAIL). We present a novel hybrid quantum-classical heuristic algorithm (GAMA, Graver Augmented Multi-seed Algorithm) for non-linear, integer optimization, and illustrate it on an application (in cancer genomics). We then present an algebraic geometry-based algorithm for embedding a problem onto a hardware that is not fully connected, along with a companion Integer Programming (IP) approach. Next, we discuss the performance of two photonic devices - the Temporal Multiplexed Ising Machine (TMIM) and the Spatial Photonic Ising Machine (SPIM) - on Max-Cut and Number Partitioning instances. We close with an outline of the current work.
Recent advances in quantum computing and the increasing availability of quantum hardware have substantially enhanced the practical relevance of quantum approaches to discrete optimization. Among these, the Quadratic Unconstrained Binary Optimization (QUBO) formulation provides a unifying modeling framework for a broad class of NP\mathbf{NP}NP-hard problems and is naturally suited to quantum computing and quantum-inspired algorithms. Location science, network design, and logistics represent core application domains of discrete optimization, combining high practical impact with substantial computational challenges. In this work, we develop QUBO formulations for several fundamental problems in these domains, including a nonlinear integer formulation of the Discrete Ordered Median Problem (DOMP). Beyond their modeling relevance, these QUBO formulations serve as representative benchmark problems for assessing quantum algorithms and quantum hardware. We further derive a tight bound for the penalty parameter ensuring equivalence between the QUBO formulation and its underlying integer program. Finally, we conduct a comprehensive computational study using QAOA, WS-QAOA, and classical heuristics for QUBO instances of the ppp-Median Problem and the Fixed-Charge Facility Location Problem (FCFLP), and introduce two effective warm-start strategies for WS-QAOA based on its linear programming relaxation.
This tutorial offers a quick, hands-on introduction to solving Quadratic Unconstrained Binary Optimization (QUBO) models on currently available quantum computers and their simulators. We cover both IBM and D-Wave machines: IBM utilizes a gate-circuit architecture, and D-Wave is a quantum annealer. We provide examples of three canonical problems and two models from practical applications. The tutorial is structured to bridge the gap between theory and practice: we begin with an overview of QUBOs, explain their relevance and connection to quantum algorithms, introduce key quantum computing concepts, provide the foundations for two quantum heuristics, and provide detailed implementation guides. An associated GitHub repository provides the codes in five companion notebooks. In addition to reaching undergraduate and graduate students in computationally intensive disciplines, this article aims to reach working industry professionals seeking to explore the potential of near-term quantum applications. As our title indicates, this tutorial is intended to be a starting point in a journey towards solving more complex QUBOs on quantum computers.
This paper addressed the computational challenges of solving large-scale fuzzy 0-1 Integer Linear Programming (ILP) problems by proposing a Fuzzy-Hybrid Quantum-Classical Optimization Algorithm (F-HQCOA).The novel approach employed α-cut based transformations to convert fuzzy ILP models into a family of crisp Quadratic Unconstrained Binary Optimization (QUBO) problems, which were then solved using hybrid quantumclassical techniques.Unlike multi-objective formulations, the proposed framework performed a parametric analysis over α-levels to generate a spectrum of solutions that reflected trade-offs between objective optimality and feasibility satisfaction under uncertainty.The degree of satisfaction (µ) was adopted as an evaluation metric to assess solution quality rather than as an independent optimization objective.The method was evaluated on fuzzy Multi-Dimensional Knapsack and Set Covering problems.Results showed that the proposed approach could efficiently handle larger instances and provided a diverse set of solutions across α-levels.While quantum-assisted methods demonstrated reduced Time-to-Solution (TTS) in certain settings, comparisons were presented with careful consideration of differences in computational paradigms.The proposed framework offers a practical decision-support tool for optimization under uncertainty and establishes a foundation for future extensions toward true multi-objective fuzzy quantum optimization.
To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer programming. State-of-the-art integer programming algorithms can compute strong relaxation bounds even for hard instances, but may have to enumerate a large number of subproblems for determining an optimum solution. If the potential of quantum computing is realized, it can be expected that in particular finding high-quality solutions for hard problems can be done fast. Still, near-future quantum hardware considerably limits the size of treatable problems. In this work, we go one step into integrating the potentials of quantum and classical techniques for combinatorial optimization. We propose a hybrid heuristic for the weighted maximum-cut problem and for quadratic unconstrained binary optimization. The heuristic employs a linear programming relaxation, rendering it well-suited for integration into exact branch-and-cut algorithms. For large instances, we reduce the problem size according to a linear relaxation such that the reduced problem can be handled by quantum machines of limited size. Moreover, we improve the applicability of depth-1 QAOA, a parameterized quantum algorithm, by deriving a parameter estimate for arbitrary instances. We present numerous computational results from real quantum hardware.
Quantum annealing emerges as a promising approach for tackling complex scheduling problems such as the resource-constrained project scheduling problem (RCPSP). This study represents the first application of quantum annealing to solve the RCPSP, analyzing 12 well-known mixed integer linear programming (MILP) formulations and converting the most qubit-efficient one into a quadratic unconstrained binary optimization (QUBO) model. We then solve this model using the D-wave advantage 6.3 quantum annealer, comparing its performance against classical computer solvers. Our results indicate significant potential, particularly for small to medium-sized instances. Further, we introduce time-to-target and Atos Q-score metrics to evaluate the effectiveness of quantum annealing and reverse quantum annealing. The paper also explores advanced quantum optimization techniques, such as customized anneal schedules, enhancing our understanding and application of quantum computing in operations research.