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Lobachevskii Journal of Mathematics

Quantum Process Tomography on Cloud-accessible Quantum Computing Platforms

Павел Евгеньевич Ведруков, A. D. Ivlev, Alexey Liniov, Iosif B. Meyerov, Mikhail Ivanchenko

Published Jan 1, 2024• 1 citation
Publisher

Abstract

Special quantum circuits make it possible to collect experimental data to study the dynamics of quantum processor qubits [1]. The harmonic inversion method restores a set of eigenvalues that form a diagram qualitatively similar to the full spectrum of the open quantum system Liouvillian [2]. The Lindblad tomography method [3] evaluates initial state preparation and measurement error (SPAM), Kraus operators, non-Markovian measure, Hamiltonian and Lindblad operators describing the evolution of an open quantum system. We estimated SPAM errors, reconstructed the evolution using Kraus operators for discrete times, and estimated the non-Markovianity of the first qubits of the $$OriginQ \; Wuyuan \;1$$ (Origin Quantum Cloud) and $$ibmq\_belem$$ (IBM Quantum Computing) quantum computers. The obtained results demonstrate the comparability of the platforms parameters and a low degree of non-Markovian behavior. The nearest future challenges are related to describing the observed processes in the form of the time-independent Lindblad equation along with the experiments involving two-qubit interaction.

DOI
10.1134/s1995080224010529

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Special quantum circuits make it possible to collect experimental data to study the dynamics of quantum processor qubits [1]. The harmonic inversion method restores a set of eigenvalues that form a diagram qualitatively similar to the full spectrum of the open quantum system Liouvillian [2]. The Lindblad tomography method [3] evaluates initial state preparation and measurement error (SPAM), Kraus operators, non-Markovian measure, Hamiltonian and Lindblad operators describing the evolution of an open quantum system. We estimated SPAM errors, reconstructed the evolution using Kraus operators for discrete times, and estimated the non-Markovianity of the first qubits of the OriginQ  Wuyuan  1OriginQ \; Wuyuan \;1OriginQWuyuan1 (Origin Quantum Cloud) and ibmq_belemibmq\_belemibmq_belem (IBM Quantum Computing) quantum computers. The obtained results demonstrate the comparability of the platforms parameters and a low degree of non-Markovian behavior. The nearest future challenges are related to describing the observed processes in the form of the time-independent Lindblad equation along with the experiments involving two-qubit interaction.
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.
As progress is made towards the first generation of error-corrected quantum computers, robust characterization and validation protocols are required to assess the noise environments of physical quantum processors. While standard coherence metrics and characterization protocols such as T1{T}_{1}T1​ and T2{T}_{2}T2​, process tomography, and randomized benchmarking are now ubiquitous, these techniques provide only partial information about the dynamic multiqubit loss channels responsible for processor errors, which can be described more fully by a Lindblad operator in the master equation formalism. Here, we introduce and experimentally demonstrate Lindblad tomography, a hardware-agnostic characterization protocol for tomographically reconstructing the Hamiltonian and Lindblad operators of a quantum noise environment from an ensemble of time-domain measurements. Performing Lindblad tomography on a small superconducting quantum processor, we show that this technique naturally builds on standard process tomography and T1/T2{T}_{1}/{T}_{2}T1​/T2​ measurement protocols, characterizes and accounts for state-preparation and measurement errors, and allows one to place bounds on the fit to a Markovian model. Comparing the results of single- and two-qubit measurements on a superconducting quantum processor, we demonstrate that Lindblad tomography can also be used to identify and quantify sources of crosstalk on quantum processors, such as the presence of always-on qubit-qubit interactions.
A central aim of physics is to describe the dynamics of physical sys-tems. Schrödinger’s equation does this for isolated quantum systems. Describing the time evolution of a quantum system that interacts with its environment, in its most general form, has proved to be difficult because the dynamics is dependent on the state of the environment and the corre-lations with it. For discrete processes, such as quantum gates or chemical reactions, quantum process tomography provides the complete descrip-tion of the dynamics, provided that the initial states of the system and the environment are independent of each other. However, many physical systems are correlated with the environment at the beginning of the ex-periment. Here, we give a prescription of quantum process tomography that yields the complete description of the dynamics of the system even when the initial correlations are present. Surprisingly, our method also gives quantitative expressions for the initial correlation. There is a rich history to the studies of decoherence of quantum systems due to the interactions with the surrounding degrees of freedom. When the dynamics of the system (S) is Markovian it can be described by a master equation [1, 2, 3]. Nowadays many researchers are interested in systems that are non-Markovian, as there is mounting evidence that some natural systems of importance may be non-Markovian [4] and such features may allow to manipulate and control quantum systems in desired ways. There is also a great deal of interest in systems that are initially correlated with their environments (E) because non-Markovianity and initial system-environment (SE) correlations are intimately related [5, 6, 7]. 1 ar
Abstract We introduce Lindblad-like quantum tomography (LℓQT) as a quantum characterization technique of time-correlated noise in quantum information processors. This approach enables the estimation of time-local master equations, including their possible negative decay rates, by maximizing a likelihood function subject to dynamical constraints. We discuss LℓQT for the dephasing dynamics of single qubits in detail, which allows for a neat understanding of the importance of including multiple snapshots of the quantum evolution in the likelihood function, and how these need to be distributed in time depending on the noise characteristics. By a detailed comparative study employing both frequentist and Bayesian approaches, we assess the accuracy and precision of LℓQT of a dephasing quantum dynamical map that goes beyond the Lindblad limit, focusing on two different microscopic noise models that can be realised in either trapped-ion or superconducting-circuit architectures. We explore the optimization of the distribution of measurement times to minimize the estimation errors, assessing the superiority of each learning scheme conditioned on the degree of non-Markovinity of the noise, and setting the stage for future experimental designs of non-Markovian quantum tomography.
Abstract Established methods for characterizing quantum information processes do not capture non-Markovian (history-dependent) behaviors that occur in real systems. These methods model a quantum process as a fixed map on the state space of a predefined system of interest. Such a map averages over the system’s environment, which may retain some effect of its past interactions with the system and thus have a history-dependent influence on the system. Although the theory of non-Markovian quantum dynamics is currently an active area of research, a systematic characterization method based on a general representation of non-Markovian dynamics has been lacking. In this article we present a systematic method for experimentally characterizing the dynamics of open quantum systems. Our method, which we call quantum process identification (QPI), is based on a general theoretical framework which relates the (non-Markovian) evolution of a system over an extended period of time to a time-local (Markovian) process involving the system and an effective environment. In practical terms, QPI uses time-resolved tomographic measurements of a quantum system to construct a dynamical model with as many dynamical variables as are necessary to reproduce the evolution of the system. Through numerical simulations, we demonstrate that QPI can be used to characterize qubit operations with non-Markovian errors arising from realistic dynamics including control drift, coherent leakage, and coherent interaction with material impurities.
Abstract Hamiltonian exceptional points (HEPs) are spectral degeneracies of non-Hermitian Hamiltonians describing classical and semiclassical open systems with losses and/or gain. However, this definition overlooks the occurrence of quantum jumps in the evolution of open quantum systems. These quantum effects are properly accounted for by considering quantum Liouvillians and their exceptional points (LEPs). Specifically, an LEP corresponds to the coalescence of two or more eigenvalues and the corresponding eigenmatrices of a given Liouvillian at critical values of external parameters (Minganti et al 2019 Phys. Rev. A 100 062131). Here, we explicitly describe how standard quantum process tomography, which reveals the dynamics of a quantum system, can be readily applied to detect and characterize quantum LEPs of quantum non-Hermitian systems. We conducted experiments on an IBM quantum processor to implement a prototype model with one-, two-, and three qubits simulating the decay of a single qubit through competing channels, resulting in LEPs but not HEPs. Subsequently, we performed tomographic reconstruction of the corresponding experimental Liouvillian and its LEPs using both single- and two-qubit operations. This example underscores the efficacy of process tomography in tuning and observing LEPs even in the absence of HEPs.