Neural Networks Take on Open Quantum Systems

Abstract

Simulating a quantum system that exchanges energy with the outside world is notoriously hard, but the necessary computations might be easier with the help of neural networks. N eural networks are behind technologies that are revolutionizing our daily lives, such as face recognition, web searching, and medical diagnosis. These general problem solvers reach their solutions by being adapted or "trained" to capture correlations in real-world data. Having seen the success of neural networks, physicists are asking if the tools might also be useful in areas ranging from high-energy physics to quantum computing [1]. Four research groups now report on using neural network tools to tackle one of the most computationally challenging problems in condensed-matter physics-simulating the behavior of an open many-body quantum system [2-5]. This scenario describes a collection of particles-such as the qubits in a quantum computer-that both interact with each other and exchange energy with their environment. For certain open systems, the new work might allow accurate simulations to be performed with less computer power than existing methods. In an open quantum system, one typically wants to find the "steady states," which are states that do not evolve in time. A formal theory for determining such states already exists [6]. The computational difficulty arises when the system contains more than a few quantum particles. Consider a (closed) collection of N spins that can point either up or down. The amplitude of the wave function for a given spin configuration (say, up, down, up, up,.. .) is a complex number whose absolute value squared gives the probability of observing the configuration. To describe the entire spin sys-* Figure 1: Four teams have designed a neural network (right) that can find the stationary steady states for an ''open'' quantum system (left). Their approach is built on neural network models for closed systems, where the wave function was represented by a statistical distribution over ''visible spins'' connected to a number of ''hidden spins.'' To extend the idea to an open system, three of the teams [3-5] added in a third set of ''ancillary spins,'' which capture correlations between the system and environment. (APS/Alan Stonebraker) tem, a complex number has to be specified for each of the 2 N possible states. Simply storing this information for just 20 spins would take about 8 gigabytes of RAM, and the amount would double with each additional spin. Handling the same number of spins in an open system is even harder because the spins must be described by a "density matrix" ρ with 2 N × 2 N matrix elements. The attraction of a neural network is that it can potentially approximate the wave function, or density matrix, with a lot less information. A neural network is like a mathematical "box" that takes as its input a string of numbers (a vector or tensor) and outputs another string. The box is a parametrized function, and its parameters are optimized for a given task. For the specific case of simulating an N-body quantum system, the neural-network function serves as a "guess" for the wave function, and the states of the N objects serve as inputs. Researchers then optimize the function parameters by having the network "learn" from real or simulated measurement data or by minimizing a physical physics.aps.org