Computing the lowest eigenstate of tight-binding Hamiltonians using quantum walks

Finding or estimating the lowest eigenstate of quantum system Hamiltonians is an important problem for quantum computing, quantum physics, quantum chemistry, and material science. Several quantum computing approaches have been developed to address this problem. The most frequently used method is variational quantum eigensolver (VQE). Many quantum systems, and especially nanomaterials, are described using tight-binding Hamiltonians, but until now no quantum computation method has been developed to find the lowest eigenvalue of these specific, but very important, Hamiltonians. We address the problem of finding the lowest eigenstate of tight-binding Hamiltonians using quantum walks. Quantum walks is a universal model of quantum computation equivalent to the quantum gate model. Furthermore, quantum walks can be mapped to quantum circuits comprising qubits, quantum registers, and quantum gates and, consequently, executed on quantum computers. In our approach, probability distributions, derived from wave function probability amplitudes, enter our quantum algorithm as potential distributions in the space where the quantum walk evolves. Our results showed the quantum walker localization in the case of the lowest eigenvalue is distinctive and characteristic of this state. Our approach will be a valuable computation tool for studying quantum systems described by tight-binding Hamiltonians.