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arXiv:2510.22311 · Communications Physics

Characterizing Pauli propagation via operator complexity

Yuguo Shao, Song Cheng, Zhengwei Liu

Published Jul 27, 2026• 0 citations
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Abstract

Pauli-propagation simulation represents observables in the Pauli basis and evolves their coefficients in the Heisenberg picture. In this work, we bridge operator complexity and the resource cost of Pauli-propagation methods by proving that the truncation error is governed by the Operator Stabilizer Rényi entropy (OSE) $${{{\mathcal{S}}}}^{\alpha }(O)$$. Our bounds quantify how OSE controls the compressibility of the evolving operator and give explicit prescriptions for the Top-K budget required to achieve a target accuracy. As an analytic test case, we prove that for the 1D Heisenberg model at Jz = 0, the number of non-zero Pauli coefficients generated from a local operator grows at most quadratically with the number of Trotter steps. We then benchmark the Top-K Pauli propagation on XXZ Heisenberg chains. The numerical results show high accuracy with a small truncation number K in the free regime (Jz = 0) and competitive performance against tensor-network methods, such as TDVP, in the interacting case (Jz = 0.5). These results position OSE as a resource measure for Pauli-propagation methods. Classical simulations of quantum dynamics are often limited by rapidly growing entanglement, motivating observable-based methods. Here, the authors show that operator stabilizer entropy gives a priori error bounds for Pauli propagation and determines the Pauli-term budget needed to reach a target accuracy.

DOI
10.1038/s42005-026-02789-z

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Pauli-propagation simulation represents observables in the Pauli basis and evolves their coefficients in the Heisenberg picture. In this work, we bridge operator complexity and the resource cost of Pauli-propagation methods by proving that the truncation error is governed by the Operator Stabilizer Rényi entropy (OSE) Sα(O){{{\mathcal{S}}}}^{\alpha }(O)Sα(O). Our bounds quantify how OSE controls the compressibility of the evolving operator and give explicit prescriptions for the Top-K budget required to achieve a target accuracy. As an analytic test case, we prove that for the 1D Heisenberg model at Jz = 0, the number of non-zero Pauli coefficients generated from a local operator grows at most quadratically with the number of Trotter steps. We then benchmark the Top-K Pauli propagation on XXZ Heisenberg chains. The numerical results show high accuracy with a small truncation number K in the free regime (Jz = 0) and competitive performance against tensor-network methods, such as TDVP, in the interacting case (Jz = 0.5). These results position OSE as a resource measure for Pauli-propagation methods. Classical simulations of quantum dynamics are often limited by rapidly growing entanglement, motivating observable-based methods. Here, the authors show that operator stabilizer entropy gives a priori error bounds for Pauli propagation and determines the Pauli-term budget needed to reach a target accuracy.
We study a nonstabilizerness resource theory for operators, which is dual to that describing states. We identify that the stabilizer Rényi entropy analog in operator space is a good magic monotone satisfying the usual conditions while inheriting efficient computability properties and providing a tight lower bound to the minimum number of non-Clifford gates in a circuit. Operationally, this measure quantifies how well an operator can be approximated by one with only a few Pauli strings-analogous to how entanglement entropy relates to tensor-network truncation. A notable advantage of operator stabilizer entropies is their inherent locality, as captured by a Lieb-Robinson bound. This feature makes them particularly suited for studying local dynamical magic resource generation in many-body systems. We compute this quantity analytically in two distinct regimes. First, we show that under random evolution, operator magic typically reaches near-maximal value for all Rényi indices, and we evaluate the Page correction. Second, harnessing both dual unitarity and ZX graphical calculus, we solve the operator stabilizer entropy for interacting integrable XXZ circuit, finding that it quickly saturates to a constant value. Overall, this measure sheds light on the structural properties of many-body nonstabilizerness generation and can inspire Clifford-assisted tensor network methods.
The complexity of simulating quantum many-body dynamics, or quantum computations, in the Heisenberg picture is governed by the scrambling of initially simple operators into superpositions of exponentially many Pauli strings. The corresponding expansion coefficients define the Pauli spectrum, whose structure controls the performance of classical algorithms based on truncating Pauli expansions. Here we determine the finite-depth Pauli spectrum of random quantum circuits, both in the noiseless case and in the presence of local noise, through its moments, given by the operator stabilizer Rényi entropies. In noiseless circuits, we uncover a hierarchy in the approach to the fully scrambled regime: low moments equilibrate at relatively short depths, while higher moments, which are sensitive to rare, large-amplitude Pauli coefficients, require parametrically larger depths. In noisy circuits, scrambling competes with an effective suppression of operator spreading. Above a critical error per cycle γcN=O(1)γ_c N=\mathcal{O}(1)γc​N=O(1), the operator fails to reach the fully scrambled distribution and remains supported on an atypically sparse subset of Pauli strings. Conversely, below this scale, we rigorously show that classical simulation remains exponentially hard, demonstrating that finite noise does not automatically imply classical simulability. The resulting noise-induced transition in operator complexity therefore delineates the boundary between intrinsically hard quantum dynamics and those that remain classically accessible.
The pursuit of quantum advantage is driving the co-evolution of quantum processors and classical simulation methods. Despite advances in scale and quality, the accuracy of quantum simulation is ultimately limited by error rates and sampling overheads. Similarly, while classical simulation methods such as Pauli propagation have made remarkable progress, their accuracy is ultimately limited by the exponential growth of operator paths and the truncations needed to control memory and runtime. Here we show that these complementary limitations can be mitigated by embedding Pauli propagation within a hybrid error-mitigation framework that reduces quantum sampling overhead while achieving lower truncation errors with fewer classical resources than traditional Pauli propagation alone. In this framework, a target observable is classically propagated through noise-canceling inverse channels, producing a modified observable that is measured directly on a quantum processor. We prototype two implementations and benchmark their performance numerically on canonical models that challenge traditional Pauli propagation. We also perform experiments on a quantum processor using 56 superconducting qubits, revealing the tradeoffs of their respective truncation strategies. These results illustrate how classical and quantum resources can be orchestrated to extend observable estimation beyond the limits of either approach alone, providing a foundation for quantum-centric supercomputing and future demonstrations of quantum advantage.
Local-operator entanglement (LOE) dictates the complexity of simulating Heisenberg evolution using tensor network methods, and it bears witness to many-body chaos for local dynamics. We show that LOE is also sensitive to how non-Clifford a unitary is-its magic resources. In particular, we prove that LOE is always upper-bound by three distinct magic monotones: T-count, unitary nullity, and operator-stabilizer Rényi entropy. Moreover, in the average case for large, random circuits, LOE and magic monotones approximately coincide. Our results imply that an operator evolution that is expensive to simulate using tensor network methods must also be inefficient using both stabilizer and Pauli truncation methods. In terms of a previous conjecture on the characteristic scaling of LOE, our results imply that nonintegrable spin chains cannot be simulated classically. Entanglement in operator space therefore measures a unified picture of nonclassical resources, in stark contrast to the Schrödinger picture.
We extend the Pauli Propagation framework to imaginary-time evolution and introduce imaginary-time Pauli Propagation (ITPP), an operator-based algorithm for approximating thermal and ground-state properties directly in the Pauli basis. We derive explicit imaginary-time propagation rules for Pauli strings and analyze the main approximation errors arising from Trotterization and Pauli-space truncation, including bounds that account for the non-unitary nature of the evolution. Benchmarking ITPP on the one-dimensional transverse field Ising model, we find that the method performs best in the high-temperature regime, where thermal states admit a compressed Pauli-basis approximation, while the required number of Pauli terms grows rapidly as the system approaches the ground-state limit. We further compare with tensor network simulations and study finite-temperature scalability through the number of retained Pauli terms required to reach a fixed target accuracy. These results establish ITPP as a complementary framework for imaginary-time simulation and suggest that combining imaginary-time and real-time Pauli Propagation could provide a pathway toward simulating more general non-unitary and open quantum system dynamics within a unified Pauli-based framework.
We present a method for simulating operator dynamics in out-of-equilibrium quantum systems. Due to the rapid growth of complexity in these systems, this is typically speaking an intractable task. However, exceptional progress has been made in recent years to sidestep this barrier, with the introduction of a number of methods that make use of a truncation of the simulation to low-weight (the number of non-trivial terms in a Pauli string basis expansion) observables, which turns out to be a good approximation for many dynamical quantities of interest, e.g., two-point infinite-temperature correlation functions between local operators. In this work, we extend this idea to a leaner truncation protocol, truncating operators based on their diameter - that is, the size of the region on the lattice on which they are non-trivially supported. Using existing analysis for generic circuits we argue that this kind of truncation protocol is physically well-motivated, and show via extensive numerical simulations for a number of systems of interest (here, the kicked Ising model and the Heisenberg XXZ model) that it is effective, and allows us to efficiently and accurately extract local correlation functions and transport properties.