arXiv:2211.11221 · Chaos An Interdisciplinary Journal of Nonlinear Science

Loschmidt echo and scrambling of systematic errors in tomography—A quantum signature of chaos

Abinash Sahu, Naga Dileep Varikuti, Vaibhav Madhok

Published Mar 1, 2026• 0 citations

Abstract

How does quantum chaos lead to rapid scrambling of information as well as systematic errors across a system when one introduces perturbations in the dynamics? What are its consequences for the reliability of quantum simulations and quantum information processing? We employ continuous measurement quantum tomography as a paradigm to study these questions. The measurement record is generated as a sequence of expectation values of a Hermitian observable evolving under repeated application of the Floquet map of the quantum kicked top. We construct a quantity to capture the scrambling of systematic errors, an out-of-time-ordered correlator (OTOC), which serves as a signature of chaos and quantifies the spread of errors. We show that the spread of errors, as quantified by the OTOC, is related to the operator Loschmidt echo, which is defined as the Hilbert-Schmidt inner product of the operators On, and O'n generated from repeated application of the Floquet map for ideal (unperturbed) dynamics and the true (perturbed) dynamics, respectively. This also gives us an operational interpretation of Loschmidt echo (LE) for operators by connecting it to the performance of quantum tomography. We show how our results demonstrate not only a link between LE and scrambling of errors different than previous studies, but also that such a link can have operational consequences in quantum information processing.

DOI: 10.1063/5.0289048

Related papers

Information scrambling and the growth of errors in noisy tomography - a quantum signature of chaos

Vaibhav Madhok, Abinash Sahu, Naga Dileep Varikuti

Abstract How does quantum chaos lead to rapid scrambling of information as well as errors across a system when one introduces perturbations in the dynamics? What are its consequences for the reliability of quantum simulations and quantum information processing? We employ continuous measurement quantum tomography as a paradigm to study these questions. The measurement record is generated as a sequence of expectation values of a Hermitian observable evolving under repeated application of the Floquet map of the quantum kicked top. Interestingly, we find that the reconstruction fidelity initially increases regardless of the degree of chaos or the strength of perturbations in the dynamics. For random states, when the measurement record is obtained from a random initial observable, the subsequent drop in the fidelity obtained is inversely correlated to the degree of chaos in the dynamics. More importantly, this also gives us an operational interpretation of Loschmidt echo for operators by connecting it to the performance of quantum tomography. We define a quantity to capture the scrambling of errors, an out-of-time-ordered correlator (OTOC) between two operators under perturbed and unperturbed system dynamics that serves as a signature of chaos and quantifies the spread of errors. Our results demonstrate not only a fundamental link between Loschmidt echo and scrambling of errors, as captured by OTOCs, but that such a link can have operational consequences in quantum information processing.

Aug 23, 20230 citations

Information Gain, Operator Spreading, and Sensitivity to Perturbations as Quantifiers of Chaos in Quantum Systems

Abinash Sahu

We adopt a continuous weak measurement tomography protocol to explore the signatures of chaos in the quantum system(s). We generate the measurement record as a series of expectation values of an observable evolving under the desired dynamics, which can show a transition from integrability to chaos. We find that the rate of information gain depends on the degree of chaos in the dynamics, the choice of initial observable, and how well the operator is aligned along the density matrix. The amount of operator spreading in the Krylov subspace, as quantified by the fidelity in quantum tomography and various other metrics of information gain, increases with the degree of chaos in the system. We study operator spreading in many-body quantum systems by its potential to generate an informationally complete measurement record. Our quantifiers for operator spreading are more consistent indicators of quantum chaos than Krylov complexity. Our study gives an operational interpretation for operator spreading in terms of fidelity gain in quantum tomography. Continuing in our journey of finding the footprints of chaos in the quantum domain, we explore the growth of errors in noisy tomography. For random states, when the measurement record is obtained from a random operator, the subsequent drop in the fidelity obtained is inversely correlated to the degree of chaos in the dynamics. This gives us an operational interpretation of Loschmidt echo for operators by connecting it to the performance of quantum tomography. We find a quantity to capture the scrambling of errors, an out-of-time-ordered correlator (OTOC) between two operators under perturbed and unperturbed dynamics that serves as a signature of chaos. Our results demonstrate a fundamental link between Loschmidt echo and scrambling of errors, as captured by OTOCs, with operational consequences in quantum information processing.

arXiv:2404.09464Apr 15, 20240 citations

A study of chaos and randomness in quantum systems

Sreeram PG

How classical chaos emerges from the underlying quantum world is a fundamental problem in physics. The origin of this question is in the correspondence principle. Classical chaos arises due to non-linear dynamics, whereas quantum mechanics, driven by unitary evolution, is linear. The question that still remains is - what are the footprints of classical chaos in the quantum world? One can understand the quantum signatures of classical chaos by studying a quantum system whose classical analogue is chaotic. In this thesis, we use the quantum kicked top model of few qubits in the deep quantum regime to investigate signatures that can be considered as a precursor to chaos in the classical limit. In particular, we study out-of-time-ordered correlators (OTOCs) and Loschmidt echo, the two well-known dynamical diagnostics of chaos. We find vestiges of classical chaos even in such a deep quantum regime. Another arena where one can study the effects of chaos and randomness is quantum state tomography. We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators generated by a unitary evolution of an initial observable. The rate of information gain and reconstruction fidelity shows vestiges of chaos. As another contribution of this thesis, we have harnessed the power of randomness inherent in the maximally mixed state to give an efficient quantum algorithm to measure OTOCs. The protocol achieves an exponential speedup over the best known classical algorithm, provided the OTOC operator to be estimated admits an efficient gate decomposition. This protocol also helps benchmark unitary gates, which is important from the quantum computation and control perspective.

arXiv:2402.00287Feb 1, 20240 citations

Effect of chaos on information gain in quantum tomography

Abinash Sahu, Sreeram PG, Vaibhav Madhok

Does chaos in the dynamics enable or impede information gain in quantum tomography? We address this question by considering continuous measurement tomography in which the measurement record is obtained as a sequence of expectation values of a Hermitian observable evolving under the repeated application of the Floquet map of the quantum kicked top. For a given dynamics and Hermitian observables, we observe completely opposite behavior in the tomography of well-localized spin coherent states compared to random states. As the chaos in the dynamics increases, the reconstruction fidelity of spin coherent states decreases. This contrasts with the previous results connecting information gain in tomography of random states with the degree of chaos in the dynamics that drives the system. The rate of information gain and hence the fidelity obtained in tomography depends not only on the degree of chaos in the dynamics and to what extent it causes the initial observable to spread in various directions of the operator space, but, more importantly, how well these directions are aligned with the density matrix to be estimated. Our study also gives an operational interpretation for operator spreading in terms of fidelity gain in an actual quantum information tomography protocol.

arXiv:2203.07692Physical review. EAug 22, 20228 citations

Information Gain in Tomography–A Quantum Signature of Chaos

Vaibhav Madhok, Carlos A. Riofrío, Shohini Ghose +1 more

We find quantum signatures of chaos in various metrics of information gain in quantum tomography. We employ a quantum state estimator based on weak collective measurements of an ensemble of identically prepared systems. The tomographic measurement record consists of a sequence of expectation values of a Hermitian operator that evolves under repeated application of the Floquet map of the quantum kicked top. We find an increase in information gain and, hence, higher fidelities in the reconstruction algorithm when the chaoticity parameter map increases. The results are well predicted by random matrix theory.

arXiv:1301.4296Physical Review LettersJan 6, 201432 citations

Generalized Loschmidt echo and information scrambling in open systems

Yi-Neng Zhou, Chang Liu

Quantum information scrambling, typically explored in closed quantum systems, describes the spread of initially localized information throughout a system and can be quantified by measures such as the Loschmidt echo (LE) and out-of-time-order correlator (OTOC). In this paper, we explore information scrambling in the presence of dissipation by generalizing the concepts of LE and OTOC to open quantum systems governed by Lindblad dynamics. We investigate the universal dynamics of the generalized LE across regimes of weak and strong dissipation. In the weak dissipation regime, we identify a universal structure, while in the strong dissipation regime, we observe a distinctive two-local-minima structure, which we interpret through an analysis of the Lindblad spectrum. Furthermore, we establish connections between the thermal averages of LE and OTOC and prove a general relation between OTOC and Rényi entropy in open systems. Finally, we propose an experimental protocol for measuring OTOC in open systems. These findings provide deeper insights into information scrambling under dissipation and pave the way for experimental studies in open quantum systems.

arXiv:2412.01851SciPost PhysicsMar 6, 20261 citation