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Model Order Reduction for Open Quantum Systems Based on Measurement-Adapted Time-Coarse Graining

Wentao Fan1 and Hakan E. Türeci2

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Phys. Rev. X 16, 021015 – Published 16 April, 2026

DOI: https://doi.org/10.1103/bysj-hjfl

Abstract

Model order reduction encompasses mathematical techniques aimed at reducing the complexity of mathematical models in simulations while retaining the essential characteristics and behaviors of the original model. This is particularly useful in the context of large-scale dynamical systems, which can be computationally expensive to analyze and simulate. Here, we present a model order reduction technique to reduce the time complexity of open quantum systems, grounded in the principle of measurement-adapted coarse graining. This method, governed by a coarse-graining timescale τ and the spectral band center ω0 of the measurement channel, organizes corrections to the lowest-order model which aligns with the rotating-wave approximation Hamiltonian in certain limits and rigorously justifies the resulting effective quantum master equation (EQME). The focus on calculating to a high degree of accuracy what can only be resolved by the measurement introduces a principled regularization procedure to address singularities and generates low-stiffness models suitable for efficient long-time integration. The closed-form EQME parameters greatly enhance interpretability, while the predicted EQME structure provides a foundation for deriving stochastic master equations under continuous measurement and for constructing efficient gray-box models for system identification and control. As a demonstration, we derive the fourth-order EQME for a challenging problem related to the dynamics of a superconducting qubit under high-power dispersive readout in the presence of a continuum of dissipative waveguide modes. This derivation shows that the lowest-order terms align with previous results, while higher-order corrections suggest new phenomena.

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Popular Summary

Efficiently modeling open quantum systems is a primary challenge in quantum science due to the complex multiscale dynamics that arise when nonlinear systems are driven by external sources. We introduce measurement-adapted time-coarse graining, a model order reduction technique that systematically derives effective quantum master equations based on the finite time resolution of a measurement apparatus. By organizing corrections through a coarse-graining timescale and the spectral band center of the measurement channel, we rigorously justify the resulting effective dynamics and provide closed-form formulas for superoperators at arbitrary orders. Our derivation for a superconducting qubit under high-power dispersive readout reveals new phenomena, such as measurement-induced spin energy jumps, that are not captured by standard rotating-wave approximations or effective Hamiltonian methods. This framework provides a self-consistent prescription for deriving stochastic master equations and low-stiffness models suitable for efficient system identification and control.

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Phys. Rev. X 16, 021015– Published 16 April, 2026
  • Received 30 October 2024
  • Revised 10 July 2025
  • Accepted 12 March 2026
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