Evaluation of the spectrum of a quantum system using machine learning based on incomplete information about the wavefunctions

Abstract

We propose an effective approach to rapid estimation of the energy spectrum of quantum systems with the use of machine learning (ML) algorithm. In the ML approach (back propagation), the wavefunction data known from experiments is interpreted as the attributes class (input data), while the spectrum of quantum numbers establishes the label class (output data). To evaluate this approach, we employ two exactly solvable models with the random modulated wavefunction amplitude. The random factor allows modeling the incompleteness of information about the state of quantum system. The trial wave functions fed into the neural network, with the goal of making prediction about the spectrum of quantum numbers. We found that in such configuration, the training process occurs with rapid convergence if the number of analyzed quantum states is not too large. The two qubits entanglement is studied as well. The accuracy of the test prediction (after training) reached 98 percent. Considered ML approach opens up important perspectives to plane the quantum measurements and optimal monitoring of complex quantum objects.Introduction. The applications of intelligent machines in various context of scientific research recently become an area of active investigations 1-14 . One of the important perspective directions of quantum physics is the measurement of wave functions and the energy spectrum of quantum objects. At present, the wavefunction is determined using the tomographic methods 9 -13 , which evaluate the wavefunction that is most compatible with a diverse set of measurements. The indirectness of these methods compounds the problem of direct determining the wavefunction. To overcome this problem, it was shown 1 that the photon wavefunction can be measured directly by sequentially measuring two additional system variables. As result, the components of the wavefunction appear directly on the measuring apparatus. The alternative approach can be used to determine the polarization quantum state 7 . In contrast to various works motivated by physically oriented approaches to artificial intelligence, there are much fewer practical studies estimating the energy spectrum (spectral numbers) for a quantum system based on incomplete information about the wavefunction with the use of the machine learning (ML) approaches. The information on the quantum object structure normally is obtained as the result of measurements of the wavefunction in series of experiments. By the measurements we mean the definition values of wavefunction at a number of spatial points x j , see Ref. 1 Fig.2(a). According to the Heisenberg uncertainty principle, in quantum theory, an exact measurement of position X violates the wavefunction of the particle and forces the subsequent measurement of momentum P to become random. To include such a factor into consideration in the machine learning (ML), we model uncertainty as a random modulation of the measured amplitude of the wavefunction. Thus an incomplete wavefunction can be written as Ψ k (x i |n j ) = γΨ(x i |n j ), whe...

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