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arXiv:2609.40174

Dissipation accelerates quantum and classical simulation of open-system dynamics

Armando Angrisani, Ricard Puig, Yanting Teng, Zoë Holmes

quant-phcond-mat.stat-mechSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

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.

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10 + 64 pages, 2 figures

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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 ttt can be estimated to accuracy ε\varepsilonε using circuits of maximum depth O ⁣([1+(t/γ)3/2]log⁡2(1/ε))O\!\left(\left[1+\left(t/γ\right)^{3/2}\right]\log^2(1/\varepsilon)\right)O([1+(t/γ)3/2]log2(1/ε)), independent of the system size nnn. Classically, we show that sparse Pauli propagation runs in time polynomial in nnn, ttt and 1/ε1/\varepsilon1/ε for every fixed γ>0γ>0γ>0, with polynomial degree O(1/γ)O(1/γ)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.
Abstract The fundamental difference between closed and open quantum dynamics lies in their environmental interaction: closed systems are perfectly isolated and evolve reversibly under unitary Hamiltonian dynamics, whereas open systems continuously couple to an external bath, resulting in irreversible dissipation and information loss. In this work, we show internal Hamiltonian dynamics can be "faked`` via external pure dissipation, i.e., Lindbladians without a coherent Hamiltonian part. More concretely, we show that, in a GKSL representation with zero explicit Hamiltonian, bounded-norm dissipative generators can approximate Hamiltonian dynamics within ϵ\epsilonϵ error in diamond norm using O(t2/ϵ)\mathcal{O}(t^2/\epsilon)O(t2/ϵ) evolution time. We further prove that for time-independent dynamics this O(t2/ϵ)\mathcal{O}(t^2/\epsilon)O(t2/ϵ) scaling is in the worst case, necessary and optimal from a geometric perspective, which captures the fundamental decoherence cost for catching up with the speed of Hamiltonian dynamics. For practical considerations, we establish an autonomous QDRIFT protocol and show how to exponentially improve the dependence on ϵ\epsilonϵ for expectation value estimation via extrapolations. Our construction leads to various implications, including the BQP-completeness of purely dissipative dynamics even before reaching approximate equilibrium, a Zeno-adjacent state-independent freezing effect, the no super-quadratic fast-forwarding theorem of a class of purely dissipative dynamics, and reducing Lindbladian simulation cost via gauge changing.
Open-system dynamics govern decoherence, transport, and engineered dissipation in quantum many-body systems. We propose a quantum algorithm for simulating time-independent, finite-range Lindblad dynamics with bounded local strength on a constant-dimensional lattice of nnn sites, allowing noncommuting jump operators. For time T≥1T\ge1T≥1 and diamond-norm error at most εεε, it has depth O(Tpolylog⁡(nT/ε))O(T\operatorname{polylog}(nT/ε))O(Tpolylog(nT/ε)) and gate count O(nTpolylog⁡(nT/ε))O(nT\operatorname{polylog}(nT/ε))O(nTpolylog(nT/ε)). These results extend the HHKL spatial decomposition to open systems. The main obstacle is that the reversible overlap corrections used for unitary dynamics have no direct analogue for dissipative channels. We overcome this by retaining the environment in a shared unitary dilation, keeping information lost from the system coherently available and making the spatial corrections reversible. We prove exponential occupation bounds for the environmental states generated by the decomposition, allowing compact representation and coherent simulation of local factors on the inputs that arise. The construction reduces each short-time evolution to a constant number of parallel layers on regions of logarithmic diameter, using nearest-neighbour gates and polylogarithmic auxiliary density. We also prove unconditional worst-case lower bounds using time-independent, purely dissipative processes that transport quantum information. For circuits with local quantum and classical communication, the lower bounds are Ω(nT)Ω(nT)Ω(nT) for the expected gate count and Ω(T)Ω(T)Ω(T) for depth on time scales up to the lattice diameter. They match the upper bounds' leading time and volume dependence up to polylogarithmic factors. At fixed time and sufficiently small fixed error, we further prove a depth lower bound of Ω(log⁡n/log⁡log⁡n)Ω(\log n/\log\log n)Ω(logn/loglogn), showing that global accuracy can require increasing depth as the lattice grows.
We present a polynomial-time classical algorithm for estimating expectation values of arbitrary observables on typical quantum circuits under any incoherent local noise, including non-unital or dephasing. Although previous research demonstrated that some carefully designed quantum circuits affected by non-unital noise cannot be efficiently simulated, we show that this does not apply to average-case circuits, as these can be efficiently simulated using Pauli-path methods. Specifically, we prove that, with high probability over the circuit gates’ choice, Pauli propagation algorithms with tailored truncation strategies achieve an inversely polynomially small simulation error. This result holds for arbitrary circuit topologies and for any local noise, under the assumption that the distribution of each circuit layer is invariant under single-qubit random gates. Under the same minimal assumptions, we also prove that most noisy circuits can be truncated to an effective logarithmic depth for the task of estimating expectation values of observables, thus generalizing prior results to a significantly broader class of circuit ensembles. We further numerically validate our algorithm with simulations on a 6 × 6 lattice of qubits under the effects of amplitude damping and dephasing noise, as well as real-time dynamics on an 11 × 11 lattice of qubits affected by amplitude damping.
Entanglement is not only the origin of exotic quantum phenomena, but also widely regarded as the fundamental barrier to classical simulation of quantum dynamics. We overturn this intuition: for predicting local observables under Hamiltonian evolution, entanglement in the quantum state actually heals the classical simulation error in the Heisenberg picture. Classical algorithms that propagate observables in the Pauli basis with proper truncation have shown remarkable empirical success in simulating noiseless quantum dynamics, yet all prior rigorous guarantees required noise or randomness---leaving the physically relevant noiseless regime without theoretical foundation. To close this gap, we prove that the truncation error of Low-weight Pauli Dynamics (LPD) admits an average-case bound without assuming randomness, provided the state is sufficiently entangled. Since tensor-network methods efficiently simulate low-entanglement states while LPD thrives precisely in the complementary regime, together they extend rigorous classical simulation to longer times, sharpening the boundary between classically simulable and genuinely quantum dynamics.
As quantum devices grow in size while remaining affected by noise, a key challenge is to determine when and how they can surpass classical computation on practical tasks. This requires the development of highly efficient classical simulation algorithms. Pauli backpropagation algorithms are among the most promising, having shown efficiency in simulating certain classes of parameterized quantum circuits, strong candidates for near‑term quantum advantage, under random circuit assumptions and depolarizing noise. However, their efficiency had not been demonstrated for more realistic nonunital noise models, such as amplitude damping, which better capture the behaviour of current hardware. We close this gap by adapting Pauli backpropagation to most single‑qubit noise channels, including nonunital noise, and proving that it remains efficient under these conditions. Our algorithm is parallelizable, memory‑efficient, and requires no assumptions on the locality of the observable, the geometry of the hardware, or the depth of the circuit. While hardware noise can strongly affect computational power in highly structured, fault‑tolerant circuits, our findings indicate little difference for most near‑term variational architectures. This establishes Pauli backpropagation as an efficient method for simulating noisy quantum devices under realistic noise assumptions.While these advances improve our ability to reproduce noisy expectation values of observables, many practical applications, including combinatorial optimization, require direct access to the bitstrings output by quantum circuits, as these encode candidate solutions. Classical replication of such noisy samples has so far been limited beyond the random circuit setting. To bridge this gap, we first focus on problems whose objective functions depend only on two‑body correlations, such as Max‑Cut, and apply Gaussian randomized rounding to the circuit's two‑qubit marginals. This produces classical samples whose expected cost is provably close to that of the noisy quantum device. Large‑scale simulations and experiments on IBMQ hardware confirm that the rounded samples reproduce the full energy distribution, with similar behaviour under various noise models. Coupled with Pauli backpropagation for efficiently estimating noisy marginals, this yields an end‑to‑end classical sampler operating in polynomial time for a broad class of variational circuits. Our results therefore show that sufficiently noisy quantum circuits for combinatorial optimization can be sampled from classically, clarifying where near‑term hardware may or may not outperform classical algorithms.Finally , we go beyond one and two‑body marginals and present an algorithm that samples valid bitstrings directly from a maximum‑entropy distribution matching the given expectation values of the quantum circuit. This method does not rely on the locality or sparsity of the problem's Hamiltonian and performs well across a variety of quantum circuits, including deep circuits on many qubits. When combined with Pauli propagation or error mitigation, it provides a practical pipeline for obtaining samples from quantum circuits in the near term, where full error correction might not yet be available.
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 Nsite=16N_{\mathrm{site}}=16Nsite​=16, and a spatially homogeneous (global) model with only seven parameters, scaled to Nsite=160N_{\mathrm{site}}=160Nsite​=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−Tr[ρ2])/Ntraj(1-\mathrm{Tr}[ρ^2])/N_{\mathrm{traj}}(1−Tr[ρ2])/Ntraj​, 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.