Geometry And Symmetry In The Quantum Boltzmann Machine

Quantum machine learning is at the intersection of two most exciting research fields:machine learning and quantum computing,which is based on the great influ-ence of classical neural networks.As one of the earliest artificial neural networks for learning problems,Boltzmann machine plays an important role in the rise of neural networks.The quantum Boltzmann machine extends the classical Boltzmann machine learning to the quantum regime,which has aroused the interest of scientists in various fields.Quantum Boltzmann machine can not only deal with classical data but also sim-ulate quantum state beyond classical probability distribution.The biggest advantage of quantum Boltzmann machine is that it can simulate the multi-body entangled quantum state.With the rapid development of quantum technology,the exploration of quantum Boltzmann machine will be a very interesting thing.In general,numerical results from machine learning are hard to understand since the machine learning focuses on digging out the results but not discovering the underly-ing mechanism.So it is very important to master the internal mechanism of numerical optimization results.Traditional physics tools can be used to help us understand the results of quantum Boltzmann machine learning,for example,from the perspective of geometry and symmetry analysis.For a long time,geometry and physics interact with each other,and the results are quite fruitful.The geometric method can give a more intu-itive expression to quantum Boltzmann machine.Symmetry also plays a very important role in quantum physics.It can be successfully described by group theory.The applica-tion of group theory greatly simplifies our treatment of the quantum system.Similarly,there is symmetry in the quantum Boltzmann machine,and the problem of symmetry can be discussed by introducing group theory.In this paper,we briefly introduce the background,significance,and main models of quantum Boltzmann machine.Besides,we focus on the geometric properties and symmetry analysis of quantum Boltzmann machine:We introduce the quantum Boltzmann machine model from the classical Boltz-mann machine and introduce two training methods of quantum Boltzmann machine:Golden-Thompson training and quantum relative entropy training.Besides,we derive the quantum relative entropy training in detail.The smaller the minimum value of quan-tum relative entropy,the stronger the expression ability of the quantum Boltzmann ma-chine is.The objective of quantum relative entropy training is to minimize the quantum relative entropy between the target state and the quantum Boltzmann machine state with a given Hamiltonian.We use Broyden Fletcher Goldfarb Shanno(BFGS)algorithm to deal with this minimization problem.We study the real symmetric two-qubit pure states and find two obvious characteristics of minimum quantum relative entropy:strong angle dependence and biaxial symmetry.We also explain this result through the geometric viewpoint and symmetry analysis.We introduce the concept of the symmetry for a quantum Boltzmann machine and develop a group theory to describe the symmetry.For any quantum Boltzmann machine composed of multiqubits,we give the basic equations for its symmetry group.Besides,we develop a numerical algorithm to verify the completeness of our construction.For any quantum Boltzmann machine composed of multiqubit,we can give the basic equa-tions for its symmetry group.