Completely Positive Map for Noisy Driven Quantum Systems Derived by Keldysh Expansion

Ziwen Huang1, Yunwei Lu2, Anna Grassellino1, Alexander Romanenko1, Jens Koch2, and Shaojiang Zhu1

1Superconducting Quantum Materials and Systems Center, Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA
2Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA

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Accurate modeling of decoherence errors in quantum processors is crucial for analyzing and improving gate fidelities. To increase the accuracy beyond that of the Lindblad dynamical map, several generalizations have been proposed, and the exploration of simpler and more systematic frameworks is still ongoing. In this paper, we introduce a decoherence model based on the Keldysh formalism. This formalism allows us to include non-periodic drives and correlated quantum noise in our model. In addition to its wide range of applications, our method is also numerically simple, and yields a CPTP map. These features allow us to integrate the Keldysh map with quantum-optimal-control techniques. We demonstrate that this strategy generates pulses that mitigate correlated quantum noise in qubit state-transfer and gate operations.

Recently, remarkable progress has been made toward achieving practical quantum computing, as both the number and quality of the qubits on quantum-computing chips have increased significantly. However, ensuring accurate gate operations on these qubits remains a challenge, which is primarily due to the occurrence of decoherence errors. To analyze and mitigate these errors, the development of a tool capable of systematically predicting the format and magnitude of the errors is crucial.

In this work, we present a simple decoherence model that can fulfill this task. The model is derived rigorously using the Keldysh formalism and possesses the following important features. First, this model is highly versatile, since it can handle arbitrary drives on quantum processors with classical or quantum noise, as well as Markovian or non-Markovian noise. Second, the dynamical maps predicted by this model are guaranteed to be physical – they are rigorously CPTP. Third, the computational complexity of the model is manageable, enabling straightforward integration with quantum-optimal-control techniques. Through the integration of this model, we numerically demonstrate improved fidelities in both state-transfer and gate operations.

Using our model, we can more conveniently understand qubit decoherence across a variety of scenarios, including those that have rarely been explored. Furthermore, this Keldysh-based model can potentially be extended to higher orders, which would be relevant if the non-Gaussian noise properties becomes important. Finally, by utilizing our model to optimize gates in real quantum processors, we can potentially further reduce inaccuracies and move closer to achieving full quantum error correction.

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Cited by

[1] Ziwen Huang, Taeyoon Kim, Tanay Roy, Yao Lu, Alexander Romanenko, Shaojiang Zhu, and Anna Grassellino, "Fast ZZ-Free Entangling Gates for Superconducting Qubits Assisted by a Driven Resonator", arXiv:2311.01332, (2023).

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