arXiv:2608.24668

Scalable Lindblad Noise Learning via Stochastic Tensor-Network Simulation

Alejandro R. Ramos Ramos, Maximilian Fröhlich, Aaron Sander, Robert Wille, Martin Eigel, Patrick Gelß, Sebastian Pokutta

quant-phSubmitted Aug 25, 2026• Announced Aug 26, 2026• 0 citations• Revised Sep 22, 2026 (v3)

arXiv HTML PDF

Abstract

Learning dissipation rates in large-scale open quantum systems is a major obstacle for near-term quantum technologies, as existing Lindblad estimation methods are typically limited to small system sizes due to the computational complexity of repeatedly solving the Lindblad equation during optimization. Here, we propose a scalable noise-learning framework for Lindblad dissipation rates that combines a stochastic simulation method, the Tensor Jump Method (TJM), with gradient-free optimization of a least-squares cost-function defined on time series of local-observable expectation values. We demonstrate the approach on two noise models in the Ising model: a site-resolved (local) model, in which independent dissipation rates are learned for each site up to $N_{\mathrm{site}}=16$, and a spatially homogeneous (global) model with only seven parameters, scaled to $N_{\mathrm{site}}=160$ sites.We complement these numerical results with a series of exact, provable guarantees: the Frobenius variance of the TJM density-matrix estimator is shown to equal $(1-\mathrm{Tr}[ρ^2])/N_{\mathrm{traj}}$, an exact purity-based characterization of the stochastic estimation error; the corresponding purity evolution is proven to be monotonically non-increasing for Hermitian jump operators; and, under a finite covariance distance assumption, the standard deviation of the cost-function is shown to decrease with system size, so that fewer trajectories are needed to reach a fixed target accuracy as the system grows. Together, this combination of scalable numerics and rigorous theoretical guarantees positions TJM-based noise learning as a practical foundation for characterizing dissipation in large quantum devices and for guiding future work on error mitigation and quantum error correction.

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arXiv:2608.24668

Scalable Lindblad Noise Learning via Stochastic Tensor-Network Simulation

Alejandro R. Ramos Ramos, Maximilian Fröhlich, Aaron Sander, Robert Wille, Martin Eigel, Patrick Gelß, Sebastian Pokutta

quant-phSubmitted Aug 25, 2026• Announced Aug 26, 2026• 0 citations• Revised Sep 22, 2026 (v3)

arXiv HTML PDF

Abstract

N_{\mathrm{site}}=16

, and a spatially homogeneous (global) model with only seven parameters, scaled to

N_{\mathrm{site}}=160

sites.We complement these numerical results with a series of exact, provable guarantees: the Frobenius variance of the TJM density-matrix estimator is shown to equal

(1-\mathrm{Tr}[ρ^2])/N_{\mathrm{traj}}

, an exact purity-based characterization of the stochastic estimation error; the corresponding purity evolution is proven to be monotonically non-increasing for Hermitian jump operators; and, under a finite covariance distance assumption, the standard deviation of the cost-function is shown to decrease with system size, so that fewer trajectories are needed to reach a fixed target accuracy as the system grows. Together, this combination of scalable numerics and rigorous theoretical guarantees positions TJM-based noise learning as a practical foundation for characterizing dissipation in large quantum devices and for guiding future work on error mitigation and quantum error correction.

Related papers

Noisy quantum circuit simulation with the tensor jump method

Maximilian Fröhlich, Aaron Sander, Martin Eigel +2 more

Classical simulation of noisy quantum circuits is essential for validating algorithms, bench-marking hardware, and guiding error-mitigation strategies. Yet it remains challenging due to either the exponential memory footprint of density matrix methods or the high variance in trajectories of standard Kraus-insertion approaches. We introduce a variance aware tensor network framework that unifies the tensor jump method (TJM) with a local time-dependent variational principle (TDVP) and a sparse Pauli-Lindblad model (SPLM) of hardware noise. Gates are realized as short variational evolutions on the matrix product state (MPS) manifold, while noise is applied stochastically per circuit window using Pauli-Lindblad jump sets that render the sampling hazard state-independent and the dissipative contraction trivial after renormalization. Strikingly, our method supports arbitrary Lindblad correlated multi-qubit noise channels consistent with hardware connectivity, including long-range noise operators on non-adjacent subsets, enabling direct simulation of crosstalk and other connectivity-induced errors beyond local noise models. Building on prior work that introduced analog unitary unravelings, we adapt this scheme to a framework based on the Lindblad master equation with exact generator matching under discretized angle laws and a parameter choice. This exactly matches the Lindblad generator, while selecting Gaussian or two-point angle laws. In parallel, we introduce an MPS-projector-jump unraveling that yields state-independent hazards and closed-form variance laws. Both schemes are unbiased and inherit the standard 1/√N Monte Carlo convergence, but with substantially smaller constants due to variance reduction. Empirically, projector sampling markedly reduces the required bond dimension per trajectory across many circuit architectures, while analog sampling excels at low noise. We demonstrate accurate, scalable noisy-circuit simulation on a 25-qubit noisy XY quench and IBM’s 127-qubit kicked Ising benchmark with long-range depolarizing noise, achieving substantially reduced Monte Carlo variance and favorable bond dimension growth compared to standard Kraus insertion baselines.Classical simulation of noisy quantum circuits is essential for validating algorithms, bench-marking hardware, and guiding error-mitigation strategies. Yet it remains challenging due to either the exponential memory footprint of density matrix methods or the high variance in trajectories of standard Kraus-insertion approaches. We introduce a variance aware tensor network framework that unifies the tensor jump method (TJM) with a local time-dependent variational principle (TDVP) and a sparse Pauli-Lindblad model (SPLM) of hardware noise. Gates are realized as short variational evolutions on the matrix product state (MPS) manifold, while noise is applied stochastically per circuit window using Pauli-Lindblad jump sets that render the sampling hazard state-independent and the dissipative contraction trivial after renormalization. Strikingly, our method supports arbitrary Lindblad correlated multi-qubit noise channels consistent with hardware connectivity, including long-range noise operators on non-adjacent subsets, enabling direct simulation of crosstalk and other connectivity-induced errors beyond local noise models. Building on prior work that introduced analog unitary unravelings, we adapt this scheme to a framework based on the Lindblad master equation with exact generator matching under discretized angle laws and a parameter choice. This exactly matches the Lindblad generator, while selecting Gaussian or two-point angle laws. In parallel, we introduce an MPS-projector-jump unraveling that yields state-independent hazards and closed-form variance laws. Both schemes are unbiased and inherit the standard 1/√N Monte Carlo convergence, but with substantially smaller constants due to variance reduction. Empirically, projector sampling markedly reduces the required bond dimension per trajectory across many circuit architectures, while analog sampling excels at low noise. We demonstrate accurate, scalable noisy-circuit simulation on a 25-qubit noisy XY quench and IBM’s 127-qubit kicked Ising benchmark with long-range depolarizing noise, achieving substantially reduced Monte Carlo variance and favorable bond dimension growth compared to standard Kraus insertion baselines.

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Dissipation accelerates quantum and classical simulation of open-system dynamics

Armando Angrisani, Ricard Puig, Yanting Teng +1 more

Simulating open quantum systems reveals how environmental coupling shapes relaxation, excitation transport, and the dynamics of quantum correlations. On quantum hardware, dissipative channels add operations and might seem to increase cost. However, we show that a broad class of Pauli noise, including depolarization, can ease quantum and classical simulation by exponentially suppressing high-weight components of Heisenberg-evolved observables. For bounded-degree Lindblad dynamics, this permits system-size-independent Trotter steps when estimating local observables. Combining this compression with Richardson extrapolation, we provide classical and quantum error bounds with respect to the 2-norm of the propagated observable. These give high-probability expectation-value guarantees for random input states or random Hamiltonians acting on a fixed state and control (out-of-time-order) correlators. Concretely, we show that expectation values of local observables at time $t$ can be estimated to accuracy $\varepsilon$ using circuits of maximum depth $O\!\left(\left[1+\left(t/γ\right)^{3/2}\right]\log^2(1/\varepsilon)\right)$ , independent of the system size $n$ . Classically, we show that sparse Pauli propagation runs in time polynomial in $n$ , $t$ and $1/\varepsilon$ for every fixed $γ>0$ , with polynomial degree $O(1/γ)$ in the weak dissipation limit. The gap between our quantum and classical upper bounds leaves room for a substantial polynomial quantum advantage as dissipation weakens, an observation further supported by our numerical results.Simulating open quantum systems reveals how environmental coupling shapes relaxation, excitation transport, and the dynamics of quantum correlations. On quantum hardware, dissipative channels add operations and might seem to increase cost. However, we show that a broad class of Pauli noise, including depolarization, can ease quantum and classical simulation by exponentially suppressing high-weight components of Heisenberg-evolved observables. For bounded-degree Lindblad dynamics, this permits system-size-independent Trotter steps when estimating local observables. Combining this compression with Richardson extrapolation, we provide classical and quantum error bounds with respect to the 2-norm of the propagated observable. These give high-probability expectation-value guarantees for random input states or random Hamiltonians acting on a fixed state and control (out-of-time-order) correlators. Concretely, we show that expectation values of local observables at time $t$ can be estimated to accuracy $\varepsilon$ using circuits of maximum depth $O\!\left(\left[1+\left(t/γ\right)^{3/2}\right]\log^2(1/\varepsilon)\right)$, independent of the system size $n$. Classically, we show that sparse Pauli propagation runs in time polynomial in $n$, $t$ and $1/\varepsilon$ for every fixed $γ>0$, with polynomial degree $O(1/γ)$ in the weak dissipation limit. The gap between our quantum and classical upper bounds leaves room for a substantial polynomial quantum advantage as dissipation weakens, an observation further supported by our numerical results.

quant-pharXiv:2609.40174Oct 1, 20260 citations

N_{\mathrm{site}}=16

, and a spatially homogeneous (global) model with only seven parameters, scaled to

N_{\mathrm{site}}=160

sites.We complement these numerical results with a series of exact, provable guarantees: the Frobenius variance of the TJM density-matrix estimator is shown to equal

(1-\mathrm{Tr}[ρ^2])/N_{\mathrm{traj}}

t

can be estimated to accuracy

\varepsilon

using circuits of maximum depth

O\!\left(\left[1+\left(t/γ\right)^{3/2}\right]\log^2(1/\varepsilon)\right)

, independent of the system size

n

. Classically, we show that sparse Pauli propagation runs in time polynomial in

n

t

and

1/\varepsilon

for every fixed

γ>0

, with polynomial degree

O(1/γ)

in the weak dissipation limit. The gap between our quantum and classical upper bounds leaves room for a substantial polynomial quantum advantage as dissipation weakens, an observation further supported by our numerical results.