Combinatorial optimization problems are foundational in computational science, yet their NP-complete nature renders them classically intractable as problem scales increase. Many such problems can be successfully mapped onto the ground-state search of an Ising Hamiltonian. In this work, we propose a full-quantum iterative algorithm based on the quantum gradient descent (QGD) framework, implemented via the linear combination of unitaries, to solve the standard exact cover problem. By reformulating discrete logical constraints into a quadratic unconstrained binary optimization model, the algorithm evolves an initial uniform superposition state directly within the Hilbert space toward the ground-state subspace. Numerical simulations on instances ranging from different qubits demonstrate that the algorithm achieves high-fidelity convergence with a logarithmic scaling of iteration depth relative to solution precision. Notably, our results reveal that the QGD framework effectively preserves the symmetry of the solution space, allowing for the simultaneous identification of the entire set of optimal results in the presence of ground-state degeneracy. By bypassing complex classical-quantum optimization loops and the associated barren plateau problem, this work provides a robust and scalable full-quantum pathway for resolving high-dimensional logic constraints on future universal quantum computing platforms.
Combinatorial optimization problems are foundational in computational science, yet their NP-complete nature renders them classically intractable as problem scales increase. Many such problems can be successfully mapped onto the ground-state search of an Ising Hamiltonian. In this work, we propose a full-quantum iterative algorithm based on the quantum gradient descent (QGD) framework, implemented via the linear combination of unitaries, to solve the standard exact cover problem. By reformulating discrete logical constraints into a quadratic unconstrained binary optimization model, the algorithm evolves an initial uniform superposition state directly within the Hilbert space toward the ground-state subspace. Numerical simulations on instances ranging from different qubits demonstrate that the algorithm achieves high-fidelity convergence with a logarithmic scaling of iteration depth relative to solution precision. Notably, our results reveal that the QGD framework effectively preserves the symmetry of the solution space, allowing for the simultaneous identification of the entire set of optimal results in the presence of ground-state degeneracy. By bypassing complex classical-quantum optimization loops and the associated barren plateau problem, this work provides a robust and scalable full-quantum pathway for resolving high-dimensional logic constraints on future universal quantum computing platforms.
Abstract Combinatorial optimization problems are pervasive across various real-world industries, yet solving them remains classically intractable due to their nondeterministic polynomial (NP)-hard nature. This paper presents an iterative quantum algorithm based on full quantum gradient descent (QGD) to solve the exact cover problem, a fundamental NP-complete problem serving as a representative scenario of airline tail assignment. We formulate the practical optimization task into a set partitioning framework and subsequently map it onto the ground-state search of an Ising Hamiltonian. By employing a QGD-driven iterative mechanism, the proposed algorithm can evolve the initial state in a quantum register towards a high-fidelity approximation of the Ising ground state within only a few iterations. Numerical simulations validate the algorithm’s effectiveness across various problem scales, encompassing both sparsely and nearly fully connected graph configurations. Results demonstrate that the algorithm achieves high precision with a relatively few number of iterative steps. Compared to existing approaches, this work bypasses complex classical optimization loops and requires significantly less pre-computation overhead, thereby offering a promising and scalable pathway for solving combinatorial optimization problems on future universal quantum computing platforms.
Combinatorial optimization has wide and high-value applications in many fields of science and industry, but solving general combinatorial optimization problems is non-deterministic polynomial time (NP) hard. Many such problems can be mapped onto the ground-state-search problems of the Ising model. Here, an iterative quantum algorithm based on quantum gradient descent to solve combinatorial optimization problems is introduced, where the initial state of a quantum register evolves over several iterations to a good approximation of the Ising-Hamiltonian ground state. We verified the effectiveness of the proposed algorithm in solving the MaxCut problem for different types of undirected graphs by numerical simulations, and analyzed the robustness of the algorithm to errors by simulating random error and Gaussian error. We compared the performance of the algorithm with the quantum approximate optimization algorithm, and the results indicate that the proposed algorithm has comparable convergence performance. We also verified the feasibility of the algorithm by conducting experiments on a real quantum computer through the quantum cloud platform. Our work provides a potential method for solving combinatorial optimization problems on future quantum devices without the use of complex classical optimization loops.
Quantum-Enhanced Gradient Descent (QGD) presents a novel algorithmic approach to optimization, leveraging quantum computing to accelerate the gradient descent process. This paper explores the potential of quantum algorithms to significantly expedite convergence in machine learning and other optimization problems. We detail the core mechanism of QGD, outlining its integration of quantum parallelism and its impact on the traditional iterative nature of gradient descent. We provide a comprehensive discussion of the theoretical foundations, potential implementation challenges, and preliminary experimental results demonstrating the efficacy of QGD. The core claim is that QGD can provide a substantial speedup compared to classical gradient descent, particularly in scenarios involving complex landscapes and large datasets.
Quantum Gradient Descent (QGD) presents a novel approach to optimization, leveraging the inherent quantum properties of systems to achieve solutions beyond the limitations of classical methods. This paper explores the potential of QGD, focusing on the implementation of a "Quantum State Collapse" mechanism to dynamically shift and explore multiple solution spaces simultaneously. We detail the theoretical framework underpinning this approach, outlining the challenges and opportunities presented by this paradigm shift. The core claim is that QGD offers a fundamentally different path toward finding optimal solutions, potentially unlocking new capabilities in complex optimization problems across diverse fields.
Riemannian optimization provides a natural geometric framework for designing quantum algorithms for ground-state preparation, most notably through quantum imaginary-time evolution. However, existing geometry-aware approaches primarily employ steepest-descent updates, leaving more advanced methods from classical Riemannian optimization largely unexplored. In this work, we introduce quantum conjugate gradient descent (QCGD), a quantum-compatible implementation of Riemannian conjugate gradient that combines the steepest descent direction with information retained from previous search directions. In our numerical experiments, the proposed approach achieves faster convergence per iteration than the baseline double-bracket quantum imaginary-time evolution (DB-QITE) algorithm, and we quantify the associated circuit cost with gate counts produced by Qrisp. This improvement, however, comes at the cost of increased circuit depth for each update step, revealing a trade-off between convergence rate and implementation complexity. Nevertheless, since the circuit depth for those algorithms grows exponentially with the number of iterations, the proposed method may offer practical advantages, particularly in the early fault-tolerant regime.
Quadratic Unconstrained Binary Optimization (QUBO) provides a versatile framework for representing NP-hard combinatorial problems, yet existing solvers often face trade-offs among speed, accuracy, and scalability. In this work, we introduce a quantum-inspired solver (QIS) that unites branch-and-bound pruning, continuous gradient-descent refinement, and quantum-inspired heuristics within a fully adaptive control architecture. We benchmark QIS3 against eight state-of-the-art solvers, including genetic algorithms, coherent Ising machines, simulated bifurcation, parallel tempering, simulated annealing, our prior QIS2 version, D-Wave's simulated-annealing (Neal), and Gurobi on three canonical QUBO problem classes: Max-Cut, NAE-3SAT, and Sherrington-Kirkpatrick spin glass problems. Under a uniform runtime budget, QIS3 attains the best solution on nearly all instances, achieving optimality in 94% of max-cut instances. These results establish QIS3 as a robust, high-performance solver that bridges classical exact strategies and quantum-inspired heuristics for scalable QUBO optimization.