Learning the structure of open quantum systems

Abstract: We design an algorithm for learning the coefficients of an $n$-qubit constant-local Lindbladian to $\varepsilon$ error with $O(g d^2 \log(n) / \varepsilon^2)$ total evolution time, where $g$ is the single-site energy and $d$ is the (approximate) degree of the interaction graph. Though Lindbladians present new challenges not present in the special case of Hamiltonians, our algorithm achieves the suite of desiderata attained by state-of-the-art Hamiltonian learning algorithms: (1) it uses non-adaptive, ancilla-free randomized Pauli measurement circuits with a time resolution of only $Θ(1/g)$; (2) it works without knowledge of the structure of the unknown Lindbladian; (3) it depends on a smooth form of degree, thereby supporting the learning of quasi-local and power-law Lindbladians. Our algorithm is a simple iterative method, where the objective function consists of Fourier coefficients of the Lindbladian restricted to few-site regions. Its analysis identifies the difficulty unique to open systems, which we call "confusing" terms. For settings where the "confusion" is limited, the performance of the algorithm improves. We demonstrate this for the case of structure learning of Hamiltonians from access to real-time evolution, where we obtain a new algorithm that is significantly simpler than previous work. In addition, using the same iterative method, we design the first efficient algorithm for structure learning Hamiltonians from high-temperature Gibbs states.

• The paper introduces a quantum algorithm that learns Lindbladian coefficients and support from simple, non-adaptive randomized Pauli experiments.
• It leverages localized Fourier analysis and iterative convex optimization to accurately map dissipative dynamics in noisy, open quantum systems.
• The method is robust to various locality regimes, providing practical insights for validating engineered dissipation and advanced quantum simulation.

Learning the Structure of Open Quantum Systems: An Expert Overview

Problem Context and Significance

The task of Hamiltonian learning—estimating the parameters of an unknown quantum many-body Hamiltonian from samples of its dynamics—has seen considerable progress, especially for systems with local interactions. However, real-world quantum systems rarely evolve in isolation; open quantum system dynamics, modeled by Lindbladian superoperators, accommodate the irreversible and dissipative processes arising from the system's interaction with an environment. Learning the microscopic structure (the coefficients and the support of local terms) of such general Lindbladians is substantially more challenging than Hamiltonian learning due to their non-unitary, noisy, and potentially highly structured dissipative parts.

Recent interest in engineered Lindbladians—for applications including quantum thermodynamics and rapid thermalization through dissipative channels—calls for efficient, rigorous methods for inferring Lindbladian structure from feasible quantum experiments. Prior works either focused on strongly restricted classes of Lindbladians (e.g., dissipators acting only on single qubits) or suffered from poor or unquantified dependence on crucial parameters such as system size, degree, and resolution.

Main Contributions and Algorithmic Results

This work introduces and rigorously analyzes a quantum algorithm for structure learning of $n$-qubit, constant-locality Lindbladians in the black-box real-time evolution setting. The proposed algorithm achieves the following key features:

• Query Complexity: The total evolution time required to identify the Lindbladian coefficients to additive error $\epsilon$ scales as $\mathcal{O}(g d^2 \log n / \epsilon^2)$, where $g$ is the site-local one-norm (single-spin energy) and $d$ is a robust notion of approximate local degree—the minimal degree after neglecting terms with total weight $<\eta$ for appropriate $\eta$.
• Experimental Requirements: Only non-adaptive, ancilla-free quantum experiments are required; each involves initialization in a random Pauli eigenstate, time evolution under the unknown Lindbladian, and measurement in a random Pauli basis. The time resolution is $\Theta(1/g)$.
• Structure Agnostic: No a priori structural information (e.g., graph connectivity or term supports) is needed. The algorithm discovers not only coefficients but also the structure—i.e., which terms are present—with near-optimal quantum resources.
• Robust to Quasi-locality and Long-Range Decay: The method is robust to broad classes of physically relevant locality, including geometrically local, quasi-local (exponential decay), and algebraically decaying (power-law) Lindbladians.
• Algorithmic Simplicity: The core algorithm is an iterative convex optimization (akin to a damped Richardson/Newton method) in coefficient-space, using locally supported randomized Fourier coefficients of the Lindbladian channel as the objective function. Rounding and thresholding steps ensure that only "significant" terms are retained at each stage, mitigating the issue of term "confusion" inherent to non-orthogonality in local Fourier estimation.

Summary of Numerical and Theoretical Guarantees

General Lindbladian Learning

• Precision $\epsilon$, failure probability $\delta$:
• Total evolution time: $\epsilon$0
• Classical runtime: Quasipolynomial in $\epsilon$1 for arbitrary $\epsilon$2 (with a path to polynomial via increased quantum resources)
• Success: With high probability, all Lindbladian coefficients are learned to additive error $\epsilon$3 in the appropriate norm.

Applications and Regimes

Setting	Quantum Cost	Scaling with $\epsilon$4 and $\epsilon$5
Strict $\epsilon$6-local, bounded degree	$\epsilon$7	(Optimal in $\epsilon$8)
$\epsilon$9-local, no geometric constraint	$\mathcal{O}(g d^2 \log n / \epsilon^2)$0	(Polynomial in $\mathcal{O}(g d^2 \log n / \epsilon^2)$1 for constant $\mathcal{O}(g d^2 \log n / \epsilon^2)$2)
Quasi-local (exp. decay, $\mathcal{O}(g d^2 \log n / \epsilon^2)$3-dim lattice)	$\mathcal{O}(g d^2 \log n / \epsilon^2)$4	(Mild dependence on $\mathcal{O}(g d^2 \log n / \epsilon^2)$5 for fixed $\mathcal{O}(g d^2 \log n / \epsilon^2)$6)
Power-law ($\mathcal{O}(g d^2 \log n / \epsilon^2)$7)	$\mathcal{O}(g d^2 \log n / \epsilon^2)$8, $\mathcal{O}(g d^2 \log n / \epsilon^2)$9	(Optimal for fast decay)

Performance in Hamiltonian Limits

• When specialized to learning $g$0-local Hamiltonians, this approach yields provably simpler and more efficient structure learning algorithms in both the time-evolution and high-temperature Gibbs state access models. For time-evolution, the total evolution time is independent of local degree—a significant refinement over prior work.

Algorithmic and Analytical Techniques

Fourier-Based Channel Estimation

Unlike Hamiltonian learning, where time-evolved expectation values can directly single out coefficients via carefully chosen observables, Lindbladian evolution’s non-unitarity results in "mixed" contributions among coefficients when estimating via local observable statistics. The authors address this by leveraging a form of randomized Pauli channel tomography and local Fourier analysis. For each few-site region, the algorithm estimates a set of local Fourier coefficients of the channel $g$1, and relates their linear responses to the underlying Lindbladian coefficients. These relationships are captured by a structured linear map $g$2, the properties and invertibility of which are analyzed in detail.

Iterative Convex Estimation with Rounding

To overcome the "confusion" due to non-orthogonality of local measurements (i.e., contributions to an estimated coefficient from several true terms), the algorithm employs an iterative convex optimization with periodic aggressive rounding/thresholding of small values, ensuring only terms of significant magnitude are estimated with high precision. This mechanism controls error inflation due to the possibly ill-conditioned $g$3 and enables support (structure) discovery.

Series Expansion and Cluster Techniques

Rigorous error bounds are developed via cluster expansion methods for Lindbladian channels, bounding the higher-order terms in the local Fourier coefficient response. The sharply controlled locality expansion ensures that for evolution times $g$4 up to $g$5, the linear term in the exponent dominates and higher-order error terms are strictly bounded. Consequently, the overall estimation procedure is robust and achieves high-precision reconstruction.

Relation to Prior and Concurrent Work

Previous rigorous approaches to Lindbladian learning have either not handled arbitrary local dissipators (e.g., restricting to single-site terms), required ancilla-assisted or adaptive protocols, or suffered from quantum resource or classical complexity that scale poorly with system size or local degree. The analysis here overcomes these through a combination of robust locality-aware tomography, iterative variable selection, and new forms of effective sparsity (approximate degree). Notably, several recently concurrent works have tackled related questions, often with higher quantum resource and parameter dependence, or without handling power-law or quasi-local interactions.

Implications and Future Research Directions

Practical Implications

• Quantum Experiment Design: The described protocol, requiring only randomized Pauli preparations and measurements, is implementable on current-day quantum hardware without the need for adaptive control or ancilla.
• Engineered Dissipation and Simulation: Enables validation and fine-tuning of engineered open-system dynamics for quantum simulation, quantum memories, and thermalization processes.
• Foundation for Efficient Long-Range Lindbladian Learning: Empowers efficient diagnostics of models with long-range or fast power-law decay, supporting cutting-edge developments in quantum statistical mechanics.

Theoretical Directions

• Tightness of Local Degree Dependence: The quadratic dependence on approximate degree in Lindbladian learning (as opposed to the linear scaling in Hamiltonian learning) raises the question of whether further algorithmic or analytical innovations can close this gap or if it is fundamentally necessary.
• Low-Temperature State Learning: Extending the regime of tractable structure learning from high-temperature Gibbs states to arbitrary temperatures, potentially leveraging improved cluster expansions or alternative measurement strategies.
• Classical Algorithmic Improvements: The current algorithm attains quasipolynomial classical runtime in unfavorable parameter regimes; further refinements for polynomial-time postprocessing, perhaps by more aggressive variable elimination or parallelization, remain a promising avenue.

Conclusion

This paper provides a rigorous, efficient, and general algorithmic framework for learning the structure of open quantum systems from simple quantum experiments. The analysis successfully extends and improves upon state-of-the-art Hamiltonian learning techniques to the more complex Lindbladian setting by creatively addressing the issues arising from dissipation, locality, and confusion. The results advance both the conceptual and practical toolkit for open system inference, and form a benchmark for quantum learning theory in the presence of realistic physical noise.