Differentiable programming of isometric tensor networks

Abstract

Differentiable programming is a new programming paradigm which enables large scale optimization through automatic calculation of gradients also known as auto-differentiation. This concept emerges from deep learning, and has also been generalized to tensor network optimizations. Here, we extend the differentiable programming to tensor networks with isometric constraints with applications to multiscale entanglement renormalization ansatz (MERA) and tensor network renormalization (TNR). By introducing several gradient-based optimization methods for the isometric tensor network and comparing with Evenbly-Vidal method, we show that auto-differentiation has a better performance for both stability and accuracy. We numerically tested our methods on 1D critical quantum Ising spin chain and 2D classical Ising model. We calculate the ground state energy for the 1D quantum model and internal energy for the classical model, and scaling dimensions of scaling operators and find they all agree with the theory well.

The ITensor Software Library for Tensor Network Calculations

“…For example, AD can be used for state-of-the-art infinite PEPS calculations and for calculating critical properties of classical systems [74]. In addition, it has proven useful for optimizing tensor networks with unitary or isometric constraints like quantum circuits, MERA, and gauged MPS [75][76][77][78] as well as for computing excitations and structure factors of MPS and PEPS [79,80]. The unique index system and generic high level interface makes ITensor ideal for defining differentiation through a variety of ITensor operations.…”

Tensor networks for quantum machine learning

“…First, DMRG is hard to implement in standard ML frameworks, especially when combining TNs and neuronal layers [49]. The algorithm has to be handcrafted for each problem [23]. Second, generalization to higher dimensions is possible [22,50] but not as efficient as for MPS due to entropy scaling [9].…”

“…Tools from differential geometry can be used for analysing the TN on the space of entanglement patterns [55] and optimizing on loss manifolds [56]. This kind of optimization performs well on high dimensional parameter spaces, especially in combination with stochastic gradient descent [57] and auto-differentiation on individual nodes [58,59] or whole layers [23].…”

“…First of all, isometric tensors automatically fulfil a normalization condition AaμAaμfalse~†=δμfalse~μ which enables the application of optimization schemes (see §2c). Second, it is mandatory for techniques that require directionality [22] or make use of the properties of isometries [23]. In particular, mapping a TN to a quantum circuit requires the tensors to be at least isometric (see §3).…”

Scalable and Flexible Classical Shadow Tomography with Tensor Networks

“…[26] was nine qubits. Facing this challenge, in this work, we further develop efficient numerical methods based on matrix product state (MPS) [33][34][35][36] for calculating entanglement features and solving reconstruction coefficients. The key idea is to pack the entanglement feature (purities in different entanglement regions) of the classical snapshot state U † |b into a fictitious quantum many-body state and represent this entanglement feature state by an MPS.…”