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Learning Hidden Structures in Open Quantum Dynamics

Alexander Teretenkov, Sergey Kuznetsov, Alexander Pechen

Abstract

We introduce a machine-learning approach for identifying hidden structural features of open quantum dynamics under restricted experimental access. Unlike most existing data-driven methods which focus on detection or prediction of dynamical behavior, our framework targets the inference of invariant algebraic structures underlying the effective Markovian evolution. Measurement limitations, symmetries, and superselection rules are incorporated through a $*$-algebraic description of accessible observables. The learning problem is formulated as maximum-likelihood estimation from multi-time measurement sequences, where the algebraic type of an invariant subalgebra - articularly a decoherence-free subalgebra - is treated as a discrete structural hypothesis. The feasibility of the approach is illustrated on multiple synthetic models and a waveguide quantum electrodynamics system, where nontrivial intermediate algebraic structures are identified directly from measurement data.

Learning Hidden Structures in Open Quantum Dynamics

Abstract

We introduce a machine-learning approach for identifying hidden structural features of open quantum dynamics under restricted experimental access. Unlike most existing data-driven methods which focus on detection or prediction of dynamical behavior, our framework targets the inference of invariant algebraic structures underlying the effective Markovian evolution. Measurement limitations, symmetries, and superselection rules are incorporated through a -algebraic description of accessible observables. The learning problem is formulated as maximum-likelihood estimation from multi-time measurement sequences, where the algebraic type of an invariant subalgebra - articularly a decoherence-free subalgebra - is treated as a discrete structural hypothesis. The feasibility of the approach is illustrated on multiple synthetic models and a waveguide quantum electrodynamics system, where nontrivial intermediate algebraic structures are identified directly from measurement data.

Paper Structure

This paper contains 11 sections, 2 theorems, 43 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Algebras of observables and their parameterization
  3. Markovian quantum dynamics and the probability of multi-time measurement outcomes
  4. Decoherence-free subalgebras
  5. Generalized Markovian embedding and maximum likelihood invariant algebra estimation
  6. Validation and application
  7. Invariant structure "guessing" tests and embeddings hierarchy
  8. Repetitive measurement -- preparation tradeoff
  9. Inferring effective physics from restricted observations
  10. Genuinely hidden structures in waveguide quantum electrodynamics
  11. Conclusion

Key Result

Theorem 1

Any matrix $*$-algebra $\mathcal{A} \subseteq \mathbb{C}^{n \times n}$ can be represented as where $n_0$ is a non-negative number (by $n_0=0$ we mean that the block $0 I_{n_0}$ is absent in the sum above), and $n_k, m_k, k=1, \ldots, K$ are positive integer numbers, such that $U$ is a unitary matrix, $\mathbb{C}^{n_k \times n_k}$ are *-algebras of all complex $n_k \times n_k$ matrices, and $I_d$

Figures (5)

  • Figure 1: Decoherence-free algebras and corresponding models embeddings hierarchy, arranged from unitary dynamics (the largest algebra) to free open GKSL dynamics (the smallest algebra). That is, the diagram shows transitions from the simplest possible model (top) to the most complex one (bottom). The colors indicate the structures $\nu_{\mathbf{E}}$ being used in the conducted experiments: blue for the Table \ref{['tab:res1A']} experiments, red --- Table \ref{['tab:res1B']}, green --- Table \ref{['tab:res1C']}.
  • Figure 2: Dynamics of the functional values on the test dataset for the models considered in the the numerical experiments from Table \ref{['tab:res1A']}. Each cluster of the models is presented with one model. Plots are shown up to the best epoch. Best achieved objective values are marked with the crosses at the final epoch.
  • Figure 3: Test functional evolutions for the structure investigation experiments from Table \ref{['tab:res2']}: (A) for $\nu_{\mathbf{E}} = (\left\{2, \, 3\right\})$ data; (B) $\nu_{\mathbf{E}} = (\left\{3, \, 2\right\})$ data; (C) $\nu_{\mathbf{E}} = (\left\{1, \, 5\right\}, \left\{1, \, 1\right\} )$ data; same hyperparameters $\nu$ are used for the trained models, and the correct structures are successfully found.
  • Figure 4: The embedding hierarchy of decoherence-free algebra structures presented in Table \ref{['tab:res5']} for the three-qubit model. Each node corresponds to a structure $\nu$, and arrows indicate embeddings between the corresponding algebras. The structures are arranged from simpler (top) to more complex (bottom) models in terms of the associated class of GKSL generators. The length of an arrow reflects the number of intermediate structures in the hierarchy that are not explicitly shown here.
  • Figure 5: The bar chart of the test functional values $F/N$ presented in Table \ref{['tab:res5']}. Columns which are related to structures located in different branches of the embeddings hierarchy (see Figure \ref{['diag:embedn8']}) are grouped together.

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Theorem 2