Harnessing Deep Neural Networks to Solve Inverse Problems in Quantum Dynamics: Machine-Learned Predictions of Time-Dependent Optimal Control Fields

Inverse problems continue to garner immense interest in the physical sciences, particularly in the context of controlling desired phenomena in non-equilibrium systems. In this work, we utilize a series of deep neural networks for predicting time-dependent optimal control fields, <i>E(t)</i>, that enable desired electronic transitions in reduced-dimensional quantum dynamical systems. To solve this inverse problem, we investigated two independent machine learning approaches: (1) a feedforward neural network for predicting the frequency and amplitude content of the power spectrum in the frequency domain (i.e., the Fourier transform of <i>E(t)</i>), and (2) a cross-correlation neural network approach for directly predicting <i>E(t)</i> in the time domain. Both of these machine learning methods give complementary approaches for probing the underlying quantum dynamics and also exhibit impressive performance in accurately predicting both the frequency and strength of the optimal control field. We provide detailed architectures and hyperparameters for these deep neural networks as well as performance metrics for each of our machine-learned models. From these results, we show that machine learning approaches, particularly deep neural networks, can be employed as a cost-effective statistical approach for designing electromagnetic fields to enable desired transitions in these quantum dynamical systems.

Harnessing Deep Neural Networks to Solve Inverse Problems in Quantum Dynamics: Machine-Learned Predictions of Time-Dependent Optimal Control Fields

Inverse problems continue to garner immense interest in the physical sciences, particularly in the context of controlling desired phenomena in non-equilibrium systems. In this work, we utilize a series of deep neural networks for predicting time-dependent optimal control fields, <i>E(t)</i>, that enable desired electronic transitions in reduced-dimensional quantum dynamical systems. To solve this inverse problem, we investigated two independent machine learning approaches: (1) a feedforward neural network for predicting the frequency and amplitude content of the power spectrum in the frequency domain (i.e., the Fourier transform of <i>E(t)</i>), and (2) a cross-correlation neural network approach for directly predicting <i>E(t)</i> in the time domain. Both of these machine learning methods give complementary approaches for probing the underlying quantum dynamics and also exhibit impressive performance in accurately predicting both the frequency and strength of the optimal control field. We provide detailed architectures and hyperparameters for these deep neural networks as well as performance metrics for each of our machine-learned models. From these results, we show that machine learning approaches, particularly deep neural networks, can be employed as a cost-effective statistical approach for designing electromagnetic fields to enable desired transitions in these quantum dynamical systems.

Ergodic properties of the Anzai skew-product for the non-commutative torus

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product $\unicode[STIX]{x1D6F7}$ on the non-commutative $2$ -torus $\mathbb{A}_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ , we investigate the pointwise limit, $\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{x1D6F7}^{k}(x)$ , for $x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D706}$ a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.

Reachability in Infinite-Dimensional Unital Open Quantum Systems with Switchable GKS–Lindblad Generators

In quantum systems theory one of the fundamental problems boils down to: given an initial state, which final states can be reached by the dynamic system in question. Here we consider infinite-dimensional open quantum dynamical systems following a unital Kossakowski–Lindblad master equation extended by controls. More precisely, their time evolution shall be governed by an inevitable potentially unbounded Hamiltonian drift term H0, finitely many bounded control Hamiltonians Hj allowing for (at least) piecewise constant control amplitudes [Formula: see text] plus a bang-bang (i.e., on-off) switchable noise term ГV in Kossakowski–Lindblad form. Generalizing standard majorization results from finite to infinite dimensions, we show that such bilinear quantum control systems allow to approximately reach any target state majorized by the initial one as up to now it only has been known in finite dimensional analogues. The proof of the result is currently limited to the bounded control Hamiltonians Hj and for noise terms ГV with compact normal V.

Note on transmitted complexity for quantum dynamical systems

Transmitted complexity (mutual entropy) is one of the important measures for quantum information theory developed recently in several ways. We will review the fundamental concepts of the Kossakowski, Ohya and Watanabe entropy and define a transmitted complexity for quantum dynamical systems. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.

Limit Theorems

In this chapter we introduce selected limit theorems on fuzzy quantum space, namely Egorov’s theorem, Central limit theorem, Weak and strong law of large numbers, and extreme value theorems for fuzzy quantum space. We also study here the Ergodic theory for fuzzy quantum space and Ergodic theorems and Poincaré recurrence theorems for fuzzy quantum dynamical systems, the Hahn-Jordan decomposition and Lebesgue decomposition for fuzzy quantum space.

Note on entropies for quantum dynamical systems

Quantum entropy and channel are fundamental concepts for quantum information theory progressed recently in various directions. We will review the fundamental aspects of mean entropy and mean mutual entropy and calculate them for open system dynamics.