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.

NOVA The Fabric of The Cosmos: Quantum Leap

Dynamics of the Wave Function: Heisenberg, Schrödinger, Collapse

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:

The Mathematics of Quantum Computers

Heisenberg's Uncertainty Principle Explained

Quantum computing explained with a deck of cards

Simulations of quantum transport: Universal spreading laws confirmed

The work deals with two of the most fundamental phenomena of condensed matter: interaction and disorder. Think about ultra-cold atomic gases. One atom from the gas is a quantum particle, and thus a quantum wave as well, which has both amplitude and phase. When such quantum particles, i.e. waves fail to propagate in a disordered medium, they get trapped and come to a complete halt. This destructive interference of propagating waves is Anderson localization.
Microscopic particles, described by quantum mechanics, interact when approaching each other. The presence of interaction, at least initially, destroys localization in a cloud of quantum particles, and allows the cloud to escape and smear out, though very slowly and subdiffusively. When atoms interact (collide) they exchange not only energy and momentum, but change their phases as well. The interaction destroys regular wave patterns, leading to the loss of the phase information. As time goes on the cloud spreads and thins out.
Hot debates over the past decade were devoted to the question whether the process will stop because the effective strength of interaction becomes too low, or not.

... quantum particles continue to spread even when particle to particle interactions originally deemed to be the activator of the spreading, exert almost no strength.