#### Short Background Quantum computing is often heralded as the future of computational power, but the reality is that practical applications remain elusive. Classical computing methods, particularly tensor network algorithms, can efficiently simulate many quantum systems, making it unclear where quantum advantage truly lies. This paper introduces _Tensor Quantum Programming_ (TQP), a novel hybrid approach that integrates tensor networks with quantum computation to overcome existing computational barriers. #### Key Insights 1. **Bridging Classical and Quantum Methods** The primary limitation of tensor network algorithms lies in their high _bond dimensions_, which restrict scalability. Quantum computers can represent tensors with arbitrarily high ranks, allowing for computational advantages in problems with complex correlations, such as optimization and quantum chemistry. 2. **Efficient Matrix Encoding for Quantum Computation** The authors propose a method for encoding _Matrix Product Operators_ (MPOs) into quantum circuits with a linear-depth scaling relative to the number of qubits. This makes quantum implementations more practical by reducing the computational cost of matrix-vector multiplications—a fundamental operation in quantum algorithms. 3. **Hybrid Tensor Quantum Programming Workflow** The proposed TQP framework dynamically switches between classical and quantum computing, depending on the complexity of tensor network ranks. Low-rank computations are handled classically, while high-rank calculations leverage quantum computing power. This selective use of quantum resources ensures efficiency and minimizes computational overhead. #### So What? TQP offers a concrete pathway for harnessing quantum computing in real-world applications, particularly in domains such as differential equations, optimization, machine learning, and quantum chemistry. By integrating classical tensor methods with quantum processing, this approach circumvents the bottlenecks that have limited quantum computing’s practical impact. If quantum advantage is to be realized in the near term, hybrid approaches like TQP could be the key to unlocking scalable quantum solutions. ### Dive deeper - [[Tensor Networks 101]] - [[Quantum Algorithms x Tensor Networks]]