Quantum-enhanced optimization aims to accelerate the solution of complex optimization problems by utilizing quantum phenomena. This paper proposes a novel framework, Quantum-Enhanced Optimization via Hierarchical State Space Representation, where the problem space is organized hierarchically, leveraging quantum algorithms to facilitate efficient exploration of high-dimensional landscapes. This approach addresses limitations of traditional optimization methods, particularly when faced with intricate and complex problem settings. We detail the design of a hierarchical state space representation and the implementation of quantum algorithms to improve the search process. The core claim is that this framework will significantly enhance the performance of optimization tasks, offering improved efficiency and scalability compared to conventional methods. This research explores the potential of quantum mechanics to represent and navigate complex problem spaces, leading to a more effective solution process.
This paper introduces the Hierarchical Quantum-Classical Multidimensional Tree (HQCMT), a novel hybrid data structure that combines classical hierarchical organization with quantum computational advantages for multidimensional data processing. The HQCMT integrates skip octree structures, quantum random access memory (QRAM), and quantum B+ tree methodologies to achieve unprecedented performance in range queries and multidimensional searches. Our theoretical analysis demonstrates up to 251× speedup in memory access operations compared to classical approaches, with time complexity improvements from 𝑂(log 𝑁 + 𝑘) to 𝑂(logB 𝑁) for range queries independent of output size. The structure supports native multidimensional clustering, quantum superposition-based parallel processing, and maintains fault tolerance through hybrid error correction mechanisms. Experimental validation shows significant performance gains across various workloads, positioning HQCMT as a breakthrough in quantum-enhanced data structures for next-generation computing applications. Keywords: Quantum Data Structures, Hierarchical Indexing, Multidimensional Queries, Hybrid Computing, QRAM, Quantum B+ Trees
We show that locality provides a natural principle to hierarchically organize backflow wavefunctions. This leads us to propose a family of variational fermionic states, termed hierarchical backflow (HB) wavefunctions. The expressive power of HB is systematically improvable, controlled by a path depth
K which reflects the range of backflow correlations. At half-filling, the HB with
K=1 already achieves high energy precision, with an accuracy around
0.5% for system sizes from
4×4 to
10×10. At hole doping
nh=0.125, the method scales efficiently to
12×16 and
16×16 systems, and the energy systematically achieves higher accuracy with
K increasing, yielding a clear stripe phase. The HB further enables a local-nonlocal decomposition, naturally bridging to neural quantum states, while featuring compact representations and efficient optimization. Our work reveals locality as a natural organizing principle of backflow wavefunctions, opening a new framework with systematic improvability and interpretability for large-scale simulations of correlated fermion systems.