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A QUBO-QAOA Framework for Rhythmic Generation in Hindustani Music

Sonali Chawla, Urvashi Kumari, Dinesh Rai

Published Mar 3, 2026• 0 citations
Publisher

Abstract

Abstract This study presents a quantum-inspired optimization framework for generating rhythmic cycles (taals) in Hindustani classical music via Quadratic Unconstrained Binary Optimization (QUBO) formulations solved via the Quantum Approximate Optimization Algorithm (QAOA). By mapping tabla bols to binary decision variables, musical grammar – including rhythmic anchors like the Sam and Khali – is encoded as penalty terms with a cost Hamiltonian. Comparative benchmarking against classical autoregressive architectures, such as WaveNet and LSTM-RNN, reveals a significant efficiency gap. The proposed framework achieves a 97.8% constraint satisfaction rate using only 12 variational parameters, representing a reduction of 4-6 orders of magnitude in model complexity compared to deep learning baselines. Furthermore, the quantum approach facilitates zero-shot generation, deriving its power from rule-encoding rather than data-intensive training. Energy landscape analysis demonstrates that quantum tunnelling through the mixing operator allows the system to navigate sparse rhythmic manifolds more effectively than classical gradient-descent methods. These results establish a rigorous foundation for quantum-enhanced computational musicology, offering a scalable, interpretable, and parameter-efficient paradigm for the digital preservation and creative evolution of structured oral traditions.

DOI
10.21203/rs.3.rs-9003392/v1

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Abstract This study presents a quantum-inspired optimization framework for generating rhythmic cycles (taals) in Hindustani classical music via Quadratic Unconstrained Binary Optimization (QUBO) formulations solved via the Quantum Approximate Optimization Algorithm (QAOA). By mapping tabla bols to binary decision variables, musical grammar – including rhythmic anchors like the Sam and Khali – is encoded as penalty terms with a cost Hamiltonian. Comparative benchmarking against classical autoregressive architectures, such as WaveNet and LSTM-RNN, reveals a significant efficiency gap. The proposed framework achieves a 97.8% constraint satisfaction rate using only 12 variational parameters, representing a reduction of 4-6 orders of magnitude in model complexity compared to deep learning baselines. Furthermore, the quantum approach facilitates zero-shot generation, deriving its power from rule-encoding rather than data-intensive training. Energy landscape analysis demonstrates that quantum tunnelling through the mixing operator allows the system to navigate sparse rhythmic manifolds more effectively than classical gradient-descent methods. These results establish a rigorous foundation for quantum-enhanced computational musicology, offering a scalable, interpretable, and parameter-efficient paradigm for the digital preservation and creative evolution of structured oral traditions.
This study explores the integration of quantum computing techniques, specifically the Quantum Approximate Optimization Algorithm (QAOA), with classical portfolio optimization frameworks through the Quadratic Unconstrained Binary Optimization (QUBO) approach. Using monthly turnover data from the National Stock Exchange of India (NSE), this paper demonstrates the construction of a QUBO matrix by encoding expected returns and covariance-based risk profiles. Binary asset inclusion constraints are introduced into portfolios and then optimised using both brute force enumeration and quantum inspired solvers. The analysis reveals those combinations including Index Futures and Index Options exhibit superior risk-adjusted performance (Sharpe Ratio≈0.286\approx 0.286≈0.286). The hybrid approach suggests the promise of quantum inspired models as means to improve financial decision making under uncertainty. The results also give a robust template for scalable asset selection models that operate within the constraints of quantum computing infrastructure.
The advent of quantum computing can potentially revolutionize how complex problems are solved. This paper proposes a two-loop quantum-classical solution algorithm for generation scheduling by infusing quantum computing, machine learning, and distributed optimization. The aim is to facilitate employing noisy near-term quantum machines with a limited number of qubits to solve practical power system optimization problems such as generation scheduling. The outer loop is a 3-block quantum alternative direction method of multipliers (QADMM) algorithm that decomposes the generation scheduling problem into three subproblems, including one quadratically unconstrained binary optimization (QUBO) and two non-QUBOs. The inner loop is a trainable quantum approximate optimization algorithm (T-QAOA) for solving QUBO on a quantum computer. The proposed T-QAOA translates interactions of quantum-classical machines as sequential information and uses a recurrent neural network to estimate variational parameters of the quantum circuit with a proper sampling technique. T-QAOA determines the QUBO solution in a few quantum-learner iterations instead of hundreds of iterations needed for a quantum-classical solver. The outer 3-block ADMM coordinates QUBO and non-QUBO solutions to obtain the solution to the original problem. The conditions under which the proposed QADMM is guaranteed to converge are discussed. Two mathematical and three generation scheduling cases are studied. Analyses performed on quantum simulators and classical computers show the effectiveness of the proposed algorithm. The advantages of T-QAOA are discussed and numerically compared with QAOA which uses a stochastic gradient descent-based optimizer.
The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational quantum algorithm introduced to tackle classically intractable combinatorial optimization problems. This tutorial offers a comprehensive, first-principles introduction to QAOA and its properties, focusing on its application to Quadratic and Polynomial Unconstrained Binary Optimization (QUBO and PUBO) problems. The tutorial begins by outlining variational quantum circuits and QUBO problems, focusing on their key properties and the encoding of problem constraints through quadratic penalty terms. Next, it explores the QAOA in detail, covering its Hamiltonian formulation, gate decomposition, and example applications, along with their implementation and performance results. This is followed by an analysis of the algorithm's energy landscape, where proofs are provided for its symmetry and periodicity, and where a resulting parameter space reduction is proposed. Finally, the tutorial extends these concepts to PUBO problems by generalizing the results to higher-order Hamiltonians and discussing the associated symmetries and circuit construction.
Combinatorial optimization problems are typically formulated using Quadratic Unconstrained Binary Optimization (QUBO), where constraints are enforced through penalty terms that introduce auxiliary variables and rapidly increase Hamiltonian complexity, limiting scalability on near term quantum devices. In this work, we systematically study Quadratic Unconstrained D-ary Optimization (QUDO) as an alternative formulation in which decision variables are encoded directly in higher dimensional Hilbert spaces. We demonstrate that QUDO naturally captures structural constraints across a range of problem classes, including the Traveling Salesman Problem, two variants of the Vehicle Routing Problem, graph coloring, job scheduling, and Max-K-Cut, without the need for extensive penalty constructions. Using a qudit-level implementation of the Quantum Approximate Optimization Algorithm (qudit QAOA), we benchmark these formulations against their binary QUBO counterparts and exact classical solutions. Our study show consistently improved approximation ratios and substantially reduced computational overhead at comparable circuit depths, highlighting QUDO as a scalable and expressive representation for quantum combinatorial optimization.
The advent of quantum computing can potentially revolutionize how complex problems are solved. This paper proposes a two-loop quantum-classical solution algorithm for generation scheduling by infusing quantum computing, machine learning, and distributed optimization. The aim is to facilitate employing noisy near-term quantum machines with a limited number of qubits to solve practical power system optimization problems such as generation scheduling. The outer loop is a 3-block quantum alternative direction method of multipliers (QADMM) algorithm that decomposes the generation scheduling problem into three subproblems, including one quadratically unconstrained binary optimization (QUBO) and two non-QUBOs. The inner loop is a trainable quantum approximate optimization algorithm (T-QAOA) for solving QUBO on a quantum computer. The proposed T-QAOA translates interactions of quantum classical machines as sequential information and uses a recurrent neural network to estimate variational parameters of the quantum circuit with a proper sampling technique. T-QAOA determines the QUBO solution in a few quantum-learner iterations instead of hundreds of iterations needed for a quantum-classical solver. The outer 3-block ADMM coordinates QUBO and non-QUBO solutions to obtain the solution to the original problem. The conditions under which the proposed QADMM is guaranteed to converge are discussed. Two mathematical and three generation scheduling cases are studied. Analyses performed on quantum simulators and classical computers show the effectiveness of the proposed algorithm. The advantages of T-QAOA are discussed and numerically compared with QAOA which uses a stochastic gradient descent-based optimizer.
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.