Let ρ(t) be the state of an evolving quantum system according to the dynamics https://www.w3.org/1998/Math/MathML"> i ρ ′ ( t ) = [ H 0 + V ( t ) , ρ ( t ) ] + θ ( ρ ( t ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353553/1603a02c-0268-4c9d-8b97-09135b05559e/content/math201_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
The current work addresses quantum machine learning in the context of Quantum Artificial Neural Networks such that the networks' processing is divided in two stages: the learning stage, where the network converges to a specific quantum circuit, and the backpropagation stage where the network effectively works as a self-programing quantum computing system that selects the quantum circuits to solve computing problems. The results are extended to general architectures including recurrent networks that interact with an environment, coupling with it in the neural links' activation order, and self-organizing in a dynamical regime that intermixes patterns of dynamical stochasticity and persistent quasiperiodic dynamics, making emerge a form of noise resilient dynamical record.
Let ρ, σ be two states and let spectral be https://www.w3.org/1998/Math/MathML"> ρ = ∑ a | e a > p ( a ) < e a | , σ = ∑ a | f a > q ( a ) < f a | https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353553/1603a02c-0268-4c9d-8b97-09135b05559e/content/math10_129_1_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> so that < e a | e b >= δ ( a, b ), < f a | f b >= δ ( a, b ). The relative entropy between these two states is https://www.w3.org/1998/Math/MathML"> D ( ρ | σ ) = T r ( ρ . ( log ( ρ ) − log ( σ ) ) ) = ∑ a p ( a ) log ( p ( a ) ) − ∑ a , b p ( a ) log ( q b ) | < e a | f b > | 2 = ∑ a , b ( p ( a ) | < e a | f b > | 2 ( log ( p ( a ) | < e a | f b > | 2 ) − log ( q b | < e a | f b > | 2 ) ) = ∑ a , b ( P ( a , b ) log ( P ( a , b ) ) − log ( Q ( a , b ) ) ) = D ( P | Q ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003353553/1603a02c-0268-4c9d-8b97-09135b05559e/content/math10_129_2_B.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
We study the non-Markovian dynamics of an open quantum system with machine learning. The observable physical quantities and their evolutions are generated by using the neural network. After the pre-training is completed, we fix the weights in the subsequent processes thus do not need the further gradient feedback. We find that the dynamical properties of physical quantities obtained by the dynamical learning are better than those obtained by the learning of Hamiltonian and time evolution operator. The dynamical learning can be applied to other quantum many-body systems, non-equilibrium statistics and random processes.
In the previous chapter, we did not examine the change of a state with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . It has been assumed that we have been examining a state at a given moment https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . In the general case, the state of a system changes with time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . For example, say the motion of an electron in a potential field, its wave function depends on time https://www.w3.org/1998/Math/MathML"> t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_4.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> : https://www.w3.org/1998/Math/MathML"> Ψ = Ψ q , t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_5.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> . Such a dependence has to be described by a differential equation in which we may have https://www.w3.org/1998/Math/MathML"> ∂ Ψ ∂ t , ∂ 2 Ψ ∂ t 2 , ⋯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math2_6.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> .