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arXiv:1007.4537 · Physical Review A

Reconstruction of time-dependent coefficients: A check of approximation schemes for non-Markovian convolutionless dissipative generators

Bruno Bellomo, Antonella De Pasquale, Giulia Gualdi, Ugo Marzolino

Published Dec 7, 2010• 20 citations
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Abstract

We propose a procedure to fully reconstruct the time-dependent coefficients of convolutionless non-Markovian dissipative generators via a finite number of experimental measurements. By combining a tomography-based approach with a proper data sampling, our proposal allows to relate the time-dependent coefficients governing the dissipative evolution of a quantum system to experimentally accessible quantities. The proposed scheme not only provides a way to retrieve the full information about potentially unknown dissipative coefficients, but also, most valuably, can be employed as a reliable consistency test for the approximations involved in the theoretical derivation of a given non-Markovian convolutionless master equation.

DOI
10.1103/physreva.82.062104

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We propose a procedure to fully reconstruct the time-dependent coefficients of convolutionless non-Markovian dissipative generators via a finite number of experimental measurements. By combining a tomography-based approach with a proper data sampling, our proposal allows to relate the time-dependent coefficients governing the dissipative evolution of a quantum system to experimentally accessible quantities. The proposed scheme not only provides a way to retrieve the full information about potentially unknown dissipative coefficients, but also, most valuably, can be employed as a reliable consistency test for the approximations involved in the theoretical derivation of a given non-Markovian convolutionless master equation.
We propose a procedure based on symplectic tomography for reconstructing the unknown parameters of a convolutionless non-Markovian Gaussian noisy evolution. Whenever the time-dependent master equation coefficients are given as a function of some unknown time-independent parameters, we show that these parameters can be reconstructed by means of a finite number of tomograms. Two different approaches towards reconstruction, integral and differential, are presented and applied to a benchmark model made up of a harmonic oscillator coupled to a bosonic bath. For this model the number of tomograms needed to retrieve the unknown parameters is explicitly computed.
In open quantum systems, phenomenological master equations with unknown parameters are often introduced. Here we propose a time-independent procedure based on quantum tomography to reconstruct the potentially unknown parameters of a wide class of Markovian master equations. According to our scheme, the system under investigation is initially prepared in a Gaussian state. At an arbitrary time ttt, in order to retrieve the unknown coefficients one needs to measure only a finite number (ten at maximum) of points along three time-independent tomograms. Due to the limited amount of measurements required, we expect our proposal to be especially suitable for experimental implementations.
Characterization of near-term quantum computing platforms requires the ability to capture and quantify dissipative effects. This is an inherently challenging task, as these effects are multifaceted, spanning a broad spectrum from Markovian to strongly non-Markovian dynamics. We introduce Quantum Liouvillian Tomography (QLT), a protocol to capture and quantify non-Markovian effects in time-continuous quantum dynamics. The protocol leverages gradient-based quantum process tomography to reconstruct dynamical maps and utilizes regression over the derivatives of Pauli string probability distributions to extract the Liouvillian governing the dynamics. We benchmark the protocol using synthetic data and quantify its accuracy in recovering Hamiltonians, jump operators, and dissipation rates for two-qubit systems. Finally, we apply QLT to analyze the evolution of an idling two-qubit system implemented on a superconducting quantum platform to extract characteristics of Hamiltonian and dissipative components and, as a result, detect inherently non-Markovian dynamics. Our work introduces the first protocol capable of retrieving generators of generic open quantum evolution from experimental data, thus enabling more precise characterization of many-body non-Markovian effects in near-term quantum computing platforms.
The time-convolutionless master equation provides a general framework to model non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to the perturbative time-convolutionless quantum master equations. We further introduce a nonperturbative approach that approximates the time-convolutionless generator as a nested time-ordered exponential function.
We investigate what a snapshot of a quantum evolution--a quantum channel reflecting open system dynamics--reveals about the underlying continuous time evolution. Remarkably, from such a snapshot, and without imposing additional assumptions, it can be decided whether or not a channel is consistent with a time (in)dependent Markovian evolution, for which we provide computable necessary and sufficient criteria. Based on these, a computable measure of "Markovianity" is introduced. We discuss how the consistency with Markovian dynamics can be checked in quantum process tomography. The results also clarify the geometry of the set of quantum channels with respect to being solutions of time (in)dependent master equations.
Characterizing non-Markovian quantum dynamics is essential for accurately modeling open quantum systems, particularly in near-term quantum technologies. In this work, we develop a structure-preserving approach to characterizing non-Markovian evolution using the time-convolutionless (TCL) master equation, considering both linear and nonlinear formulations. To parameterize the master equation, we explore two distinct techniques: the Karhunen-Loeve (KL) expansion, which provides an optimal basis representation of the dynamics, and neural networks, which offer a data-driven approach to learning system-environment interactions. We demonstrate our methodology using experimental data from a superconducting qubit at the Quantum Device Integration Testbed (QuDIT) at Lawrence Livermore National Laboratory (LLNL). Our results show that while neural networks can capture complex dependencies, the KL expansion yields the most accurate predictions of the qubit's non-Markovian dynamics, highlighting its effectiveness in structure-preserving quantum system characterization. These findings provide valuable insights into efficient modeling strategies for open quantum systems, with implications for quantum control and error mitigation in near-term quantum processors.