Neural Quantum States

How neural networks can solve highly complex problems in quantum mechanics

... Restricted Boltzmann Machines (RBMs), a simple type of artificial neural network, can be used to compute with extremely high accuracy the ground-state energy of quantum systems of many particles.


Some trajectories of a harmonic oscillatoraccording to Newton's laws of classical mechanics(A–B), and according to the Schrödinger equation of quantum mechanics (C–H). In A–B, the particle (represented as a ball attached to a spring) oscillates back and forth. In C–H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. C, D, E, F, but not G, H, are energy eigenstates. H is a coherent state—a quantum state that approximates the classical trajectory.

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What does Schrödinger equation say?
Hummm… That’s a tough one. I have a lot of trouble making sense out of it. Here’s the best explanation I’ve come up with. At any position of space, a wave function has a particular energy which is deduced from its global structure. This local energy, also known as the Hamiltonian, is a composition of some potential energy which is due to external forces like electromagnetism, and some kinetic energy which depends on the superposition of momenta which makes up the wave function. What’s particularly weird about this kinetic energy is that it can have a complex value. Now, what Schrödinger equation says is that, at every position, the wave function rotates around the origin of the complex plane at a speed proportional to the Hamiltonian. This corresponds to the following formula:

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