Machine learning spatial geometry from entanglement features

Abstract

Motivated by the close relations of the renormalization group with both the holography duality and the deep learning, we propose that the holographic geometry can emerge from deep learning the entanglement feature of a quantum many-body state. We develop a concrete algorithm, call the entanglement feature learning (EFL), based on the random tensor network (RTN) model for the tensor network holography. We show that each RTN can be mapped to a Boltzmann machine, trained by the entanglement entropies over all subregions of a given quantum many-body state. The goal is to construct the optimal RTN that best reproduce the entanglement feature. The RTN geometry can then be interpreted as the emergent holographic geometry. We demonstrate the EFL algorithm on 1D free fermion system and observe the emergence of the hyperbolic geometry (AdS3 spatial geometry) as we tune the fermion system towards the gapless critical point (CFT2 point).

Entanglement area law for shallow and deep quantum neural network states

“…That is, the bulk geometry is given, and the neural network is tiled on the background bulk geometry, the structure of the neural network provides the required entanglement features on the boundary which is exactly the dual of the bulk geometry. Note that in reference [49], the inverse problem of the above problem is investigated, where they explored how to use given entanglement features of a state to determine the optimal holographic geometry. These topics will be left for our future studies.…”

“…If all the neurons of a neural network are local ε-connected with each other, we say the neural network is a local ε-connected neural network. Similar construction has been used in references [29,31,32,49] for exactly constructing neural network states of some physical systems. When the in a local ε-connected neural network, each neuron h (l) i only connects with K neurons both in (l − 1)th and (l + 1)th layers, we call it a K-local neural network.…”

“…, v n ) = j Φ j (v i : v i h j ). This kind of construction has be used in references [29,31,32,49] for investigating physical properties of complex systems.…”

“…The locality of the states is encoded in the connection pattern of the neural network, which agrees well with our intuition. This kind of construction will also be useful for understanding the entanglement-geometry correspondence [49], such as Ryu-Takayanagi formula in a discrete form [11,25,26].…”

“…Although many progress of RBM states has been made, the DBM states are less investigated [30]. There are several crucial reasons why we need deep neural network rather than shallow one: (i) the representational power of shallow network is limited, there exist states which can be efficiently represented by deep neural network while the shallow one can not represent [30]; (ii) aAny Boltzmann machine (BM) can be reduced into a DBM, this also makes some limits in usage of shallow BM (with just one hidden layer, viz, RBM) [30]; (iii) the hierarchy structure of deep neural is more suitable for encoding holography [49,55,56] and for procedure such as renormalization [57]. Now let us take a close look at the geometry of a DBM neural network.…”