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**Contents:**

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 system, 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. The attraction of a neural network is that it can potentially approximate the wave function, or density matrix, with a lot less information.

The box is a parametrized function, and its parameters are optimized for a given task. Once the right guess is in hand, it can be used to calculate other physical properties, and with far fewer than 2 N parameters. In a paper, Carleo and Troyer showed that a neural network called a restricted Boltzmann machine [ 7 ] could—with a small tweak—be used to represent states of a closed system of many interacting spins [ 8 ]. First, they had to design a neural network function that represents a density matrix. Finally, they had to find a way to perform this minimization numerically.

Although they all worked independently, the four teams arrived at fairly similar solutions to these problems. These extra spins ensure that the output of the neural network has the mathematical properties of a density matrix. Yoshioka and Hamazaki, however, derived the density matrix without the ancillary spins and, in turn, relied on a mathematical trick when using the matrix to calculate physical quantities [ 2 ].

The groups also varied in how they defined the cost function. Two teams used an indirect approach, based on a so-called Hermitian operator, that has a guaranteed minimization procedure [ 2 , 5 ]. The two other teams opted for a more direct approach that requires more complex minimization methods [ 3 , 4 ]. Denis I.

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The wavelength is predicted to be nm. A useful constant in this computation is the Compton wavelength. Clearly the model has been pushed beyond range of quantitate validity, although the trend of increasing absorption band wavelength with increasing n is correctly predicted. Incidentally, a compound should be colored if its absorption includes any part of the visible range nm.

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