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arXiv:1905.10941 · Physical Review Applied

Non-Markovian Noise Characterization with the Transfer Tensor Method

Yuqin Chen, Kaili Ma, Yi-Cong Zheng, Jonathan Allcock, Shengyu Zhang, Chang‐Yu Hsieh

Published Mar 18, 2020• 37 citations
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Abstract

We propose simple protocols for performing quantum noise spectroscopy based on the method of transfer tensor maps (TTMs) [Cerrillo and Cao, Phys. Rev. Lett. 112, 110401 (2014)]. The TTM approach is a systematic way to deduce the memory kernel of a time-nonlocal quantum master equation via quantum process tomography. With access to the memory kernel, it is possible to (1) assess the non-Markovianity of a quantum process, (2) reconstruct the noise spectral density beyond pure-dephasing models, and (3) investigate collective decoherence in multiqubit devices. We illustrate the usefulness of TTM spectroscopy on the IBM Quantum Experience platform, and demonstrate that the qubits in the IBM device are subject to mild non-Markovian dissipation with spatial correlations.

DOI
10.1103/physrevapplied.13.034045

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We propose simple protocols for performing quantum noise spectroscopy based on the method of transfer tensor maps (TTMs) [Cerrillo and Cao, Phys. Rev. Lett. 112, 110401 (2014)]. The TTM approach is a systematic way to deduce the memory kernel of a time-nonlocal quantum master equation via quantum process tomography. With access to the memory kernel, it is possible to (1) assess the non-Markovianity of a quantum process, (2) reconstruct the noise spectral density beyond pure-dephasing models, and (3) investigate collective decoherence in multiqubit devices. We illustrate the usefulness of TTM spectroscopy on the IBM Quantum Experience platform, and demonstrate that the qubits in the IBM device are subject to mild non-Markovian dissipation with spatial correlations.
The initial stages of the evolution of an open quantum system encode the key information of its underlying dynamical correlations, which in turn can predict the trajectory at later stages. We propose a general approach based on non-Markovian dynamical maps to extract this information from the initial trajectories and compress it into non-Markovian transfer tensors. Assuming time-translational invariance, the tensors can be used to accurately and efficiently propagate the state of the system to arbitrarily long time scales. The non-Markovian transfer tensor method (TTM) demonstrates the coherent-to-incoherent transition as a function of the strength of quantum dissipation and predicts the noncanonical equilibrium distribution due to the system-bath entanglement. TTM is equivalent to solving the Nakajima-Zwanzig equation and, therefore, can be used to reconstruct the dynamical operators (the system Hamiltonian and memory kernel) from quantum trajectories obtained in simulations or experiments. The concept underlying the approach can be generalized to physical observables with the goal of learning and manipulating the trajectories of an open quantum system.
With continuing improvements on the quality of fabricated quantum devices, it becomes increasingly crucial to analyze noisy quantum process in greater details such as characterizing the non-Markovianity in a quantitative manner. In this work, we propose an experimental protocol, termed spectral transfer tensor maps (STTMs), to accurately predict the Rivas-Huelga-Plenio non-Markovian measure of any Pauli channels without state preparation and measurement errors. In fact, for Pauli channels, STTM even allows the reconstruction of a highly precise noise power spectrum for qubits. At last, we also discuss how STTM can be useful to approximately characterize non-Markovianity of non-Pauli channels via Pauli twirling in an optimal basis.
With continuing improvements on the quality of fabricated quantum devices, it becomes increasingly crucial to analyze noisy quantum process in greater details such as characterizing the non-Markovianity in a quantitative manner. In this work, we propose an experimental protocol, termed Spectral Transfer Tensor Maps (SpecTTM), to accurately predict the RHP non-Markovian measure of any Pauli channels without state-preparation and measurement (SPAM) errors. In fact, for Pauli channels, SpecTTM even allows the reconstruction of highly-precised noise power spectrum for qubits. At last, we also discuss how SpecTTM can be useful to approximately characterize non-Markovianity of non-Pauli channels via Pauli twirling in an optimal basis.
The transfer tensor method is a versatile tool for analyzing and propagating general open quantum systems. It captures in a compact manner all memory effects in a non-Markovian system through a straightforward transformation of a set of dynamical maps. Transfer tensors provide the exact convolutional propagator associated with a given time discretization over the past evolution of an open quantum system. Here we show that, for any finite time discretization, the memory kernel of the Nakajima Zwanzig equation deviates from the exact transfer tensors, although both converge in the continuous-time limit, as expected. We examine this behaviour in the context of an analytically solvable model: a two level atom resonant with a lossy cavity in the Jaynes Cummings limit. The atomic dynamics separate into two decoupled degrees of freedom -- the coherence and the population inversion. We derive exact expressions for the dynamical map, the transfer tensors and the memory kernel governing the coherence, and we relate them to their counterparts for the population inversion. As a function of the ratio between the cavity loss rate and the atom-cavity coupling strength, we identify regions of enhanced non-Markovianity in which the system can be described as fully Markovian for certain time-step choices.
Accurate simulations of open quantum system dynamics is a long standing issue in the field of chemical physics. Exact methods exist, but are costly, while perturbative methods are limited in their applicability. Recently a new black-box type method, called transfer tensor method (TTM), was proposed [J. Cerrillo and J. Cao, Phys. Rev. Lett. 112, 110401 (2014)]. It allows one to accurately simulate long time dynamics with a numerical cost of solving a time-convolution master equation, provided many initial system evolution trajectories are obtained from some exact method beforehand. The possible time-savings thus strongly depend on the ratio of total versus initial evolution lengths. In this work, we investigate the parameter regimes where an application of TTM would be most beneficial in terms of computational time. We identify several promising parameter regimes. Although some of them correspond to cases when perturbative theories could be expected to perform well, we find that the accuracy of such approaches depends on system parameters in a more complex way than it is commonly thought. We propose that the TTM should be applied whenever system evolution is expected to be long and accuracy of perturbative methods cannot be ensured or in cases when the system under consideration does not correspond to any single perturbative regime.
Accurate noise characterization is essential for scaling quantum processors toward fault-tolerant operation. Although reduced qubit dynamics are often modeled with Markovian master equations, present-day devices can exhibit memory effects generated by residual qubit-qubit couplings, structured environments, and finite bath correlation times. Here we develop a channel-resolved, Post-Markovian Master Equation model for non-Markovian noise and test it in superconducting qubits. Using idle-evolution tomography on IBM Quantum processors, we identify complementary operational signatures of non-Markovianity, including violations of CP-divisibility and revivals of distinguishability-based information-backflow measures. We further derive a closed-form spectator-ZZZZZZ model with local dissipation and show that it captures the observed transverse Bloch-vector revivals while leaving the longitudinal relaxation mode Markovian within the model. The fitted closed-form dynamics enable an analytical reconstruction of the transverse memory kernel, whose damped oscillatory structure captures the non-Markovian correction beyond the fitted Markovian baseline. Two-qubit tomography shows buildup and revivals of quantum mutual information on comparable timescales, supporting spectator-induced crosstalk as an important contributor to the observed memory effects. Our results connect operational non-Markovianity diagnostics, microscopic crosstalk modeling, and reduced memory-kernel reconstruction in a single experimental framework for superconducting quantum hardware.