Researches On Theories And Applications Of Variational Quantum Algorithms

Quantum computing is a computing method that utilizes the basic principles of quantum mechanics.Quantum algorithms can achieve a speedup relative to classical computing when solving cerrtain specific problems,which is "quantum advantage".In the past decades,the development of quantum computing has attracted much attention.Researchers have proposed quantum algorithms such as Shor’s algorithm and Grover’s algorithm that can demonstrate quantum advantages in terms of computational complexity.However,due to problems such as decoherence noise in quantum computers,applying these quantum algorithms to solve practical problems requiring the support of fault-tolerant quantum computing technology.The quantum hardware resources needed,such as the number of qubits and the depth of quantum circuits,are far from being achieved for today’s noisy intermediate-scale quantum(NISQ)devices.Therefore,how to use NISQ devices to solve practical problems and demonstrate quantum advantages has become one of the key issues in the current stage of quantum computing researchIn 2014,Alberto Peruzzo et al.proposed the variational quantum eigensolver algorithm for solving the ground state of molecular systems and demonstrated it on an optical quantum computer,opening the era of variational quantum algorithms.The algorithm mainly solves optimization problems.Based on the high expressibility of quantum states and the parallelism of quantum operations in quantum computing,the variational quantum algorithm performs ansatz preparations and loss function acquisitions on the quantum computer,while the parameter optimization is realized by the classical computer.As one of the most important algorithms in the NISQ era,the variational quantum algorithm has developed rapidly after its introduction.Its potential application scenarios have also expanded from solving the ground state to numerical computing,machine learning,biochemical and other fields.As a primary goal of studying quantum algorithms,researchers hope to demonstrate quantum advantage using variational quantum algorithms.However,with the deepening of the research on the variational quantum algorithm,some limitations of the algorithm have gradually been paid attention to.For example,the training process in the variational quantum algorithm is proved to be an NP-hard problem,and one of the key challenging factors is the barren plateau phenomenon,that is,the gradient of the loss function decays exponentially with the number of qubits,resulting in the number of samples required to obtain the loss function increasing exponentially with the problem size;the reachability of the variational quantum algorithm also determines whether the variational quantum algorithm can successfully solve the problem.At present,there is no clear conclusion on this question in general.The main content of this paper is the research on the application extension and algorithm theory of variational quantum algorithm.The main research results and innovations introduced in this paper are:1.Based on purifications of density matrices,the application scenario of variational quantum algorithms is extended to solve the steady state of open quantum systems.When dealing with an open quantum system,it is a conventional and reasonable way to regard the whole of the system and the environment as a closed system,and the system itself as the reduced density matrix of the whole system.Since the time evolution of an open quantum system is a non-unitary transformation process,previous work related to using variational quantum algorithms to solve steady states failed to use quantum operations to obtain relevant information,and was forced to choose the form of vectorized matrices for model building,introducing some unnecessary and tricky questions.We successfully realized solving the steady state of the open quantum system using the variational quantum algorithm by constructing the purification of a density matrix.2.Use transfer learning to alleviate the barren plateau problem in the variational quantum algorithm,and improve the trainability and scalability of the variational quantum algorithm.Transfer learning is a widely used method in machine learning to save training resource consumption.The barren plateaus problem is a significant obstacle to the trainability of variational quantum algorithms.We innovatively established the connection between transfer learning and the barren plateaus problem in the variational quantum algorithm,and used transfer learning to set initialization parameters for the variational quantum algorithm to mitigate the barren plateaus problem.3.With translation invariant systems,the feasibility of using transfer learning to alleviate the barren plateau problem in the variational quantum algorithm is given.When using the variational quantum algorithm to solve the ground state of the translation invariant system,we give a feasibility analysis of using transfer learning to alleviate the barren plateau problem,and explain the highly open issue of the interpretability of transfer learning.4.Point out that the amount of parameters is an important factor limiting the scalability of variational quantum algorithms,which in turn makes it difficult for variational quantum algorithms to demonstrate quantum advantages.We prove a theorem stating that in variational quantum algorithms,the gradient acquisition cost when training parameters is highly dependent on the number of parameters.Compared with the characteristic that the cost of solving the gradient is basically independent of the number of parameters in the backpropagation algorithm for calculating the gradient of the classical neural network,the number of parameters in the variational quantum algorithm severely limits its scalability.On this basis,we analyze the time consumption of executing variational quantum algorithms,and point out that too many parameters and too many samples make it difficult for the current structure to show quantum advantages.Overall,we have conducted a relatively comprehensive study of variational quantum algorithms and made contributions in some areas.These studies have a positive effect on the improvement and development of variational quantum algorithms and quantum computing in the NISQ era.