Unit commitment problem (UC) is a nonlinear problem of mixed integer programming used to minimize the operating costs of generating units in the power system. It can be described as a procedure to turn the generator on and off to increase output power and meet market demand. When deciding which of the generators will be included and which will not, a number of limitations of the power system must be met. At the beginning of the paper, we get acquainted with the work and basic concepts of quantum computing such as quantum bit or qubit, and superposition. We are also introduced to the definition of the Unit Commitment Problem and the most well-known algorithm for its solution - the particle swarm optimization (PSO) algorithm. The PSO algorithm is used to find the maximum or minimum of a function defined in multidimensional vector space. It consists of a swarm of individuals (particles) moving in the search space to find a solution to a given problem. The movement of each particle according to the velocity and past experience of the particle is taken into account, as well as the experience of neighboring particles. Furthermore, a binary particle swarm optimization algorithm or BPSO is proposed. The difference between the PSO and BPSO approaches is in defining the particle position vector, which is binary in BPSO. In order to successfully describe the work of the LAQPSO algorithm, we introduce the concept of a quantum-inspired evolutionary algorithm - QEA. It is based on the concepts and principles of quantum computing such as quantum bit and state superposition. It is characterized by the selection (representation) of the individual, the evaluation function and the dynamics of the population. A new, probabilistic notation for representation is introduced, which is based on the concept of qubits and the notion of the qubit of an individual as a series of qubits. Although BPSO-based approaches have been successfully applied to the combinatorics of optimization problems in various fields, the BPSO algorithm has several shortcomings, one of which is premature convergence in solving more difficult problems. In order to overcome the mentioned shortcomings, a new binary PSO algorithm inspired by quantum computing - QBPSO algorithm - was introduced. The proposed QBPSO combines conventional BPSO with concepts of quantum computing. QBPSO uses the term qubit individual for probabilistic representation, which replaces the speed update process in particle swarm optimization. A new rotary gate is also proposed, ie a gate for the rotation of coordinates to update the qubit of individuals in combination with a dynamic rotation angle to determine the size of the angle. In order to improve performance and reduce the overall minimum operating cost, the local attraction quantum particle swarm algorithm - LAQPSO for UC problem solving is based on the quantum evolution algorithm (QEA) and the PSO algorithm. A new quantum bit expression mechanism called quantum angle is used to encode the solution to the particle, and the notion of a local attractor is introduced which is in charge of automatically determining the rotational angle of a quantum rotating gate. During the process of finding a global solution, the magnitude of the rotation angle is adjusted to an important parameter called the contraction coefficient, which can quantify the trade-off between exploration capability and exploitation capability. The proposed algorithm is applied to solve UC problems for a 26-unit (generator) power system. It is compared with binary PSO (BPSO), improved quantum BPSO (IQBPSO) and other techniques to demonstrate the efficiency and accuracy of the proposed algorithm. The results show the superior performance of the LAQPSO algorithm to minimize total costs compared to the other listed algorithms.