Solving the subset sum problem by the quantum Ising model with variational quantum optimization based on conditional values at risk

Abstract

The subset sum problem is a combinatorial optimization problem, and its complexity belongs to the nondeterministic polynomial time complete (NP-Complete) class. This problem is widely used in encryption, planning or scheduling, and integer partitions. An accurate search algorithm with polynomial time complexity has not been found, which makes it challenging to be solved on classical computers. To effectively solve this problem, we translate it into the quantum Ising model and solve it with a variational quantum optimization method based on conditional values at risk. The proposed model needs only n qubits to encode 2ⁿ dimensional search space, which can effectively save the encoding quantum resources. The model inherits the advantages of variational quantum algorithms and can obtain good performance at shallow circuit depths while being robust to noise, and it is convenient to be deployed in the Noisy Intermediate Scale Quantum era. We investigate the effects of the scalability, the variational ansatz type, the variational depth, and noise on the model. Moreover, we also discuss the performance of the model under different conditional values at risk. Through computer simulation, the scale can reach more than nine qubits. By selecting the noise type, we construct simulators with different QVs and study the performance of the model with them. In addition, we deploy the model on a superconducting quantum computer of the Origin Quantum Technology Company and successfully solve the subset sum problem. This model provides a new perspective for solving the subset sum problem.