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Combining tensor networks and Pauli propagation to simulate quantum dynamics

Alessandro Cerati

Published Sep 24, 2026• 0 citations

Abstract

The study of classical simulability is a key part of quantum information theory, providing new tools for fundamental research and the development of quantum computing. An increasingly successful framework for the study of simulability is resource theory. The best-known and best-understood quantum resource is entanglement, but in recent years nonstabilizerness, or magic, has also been gathering attention from the community. It is not yet well understood to what extent different resources play off each other in making systems hard to classically simulate. It has recently been shown that Clifford-augmented matrix product states (CAMPS) allow for efficient circuit simulation in some cases where both entanglement and magic are present. Another emerging method which has shown the ability to simulate systems with both resources is Pauli propagation (PP). Furthermore, the interplay between entanglement and magic at the level of physical phenomena is also poorly understood, in particular in the context of real-time dynamics. In this work, I implement an analytical algorithm proposed by Fux et al. (2025) which allows CAMPS to efficiently simulate essentially the first N non-Clifford Pauli exponential gates in an N-qubit quantum circuit, reproducing results obtained by greedy-search entanglement cooling in the paper. I then develop CAMPS-PP, a method for expectation value dynamics which evolves the state as a CAMPS and the observable in the Heisenberg picture using PP. I show that this method is significantly faster than pure CAMPS for slightly deep random Pauli rotation circuits at the tested system sizes, up to N=100, with improvements of up to two orders of magnitude for N=20. Finally, in order to investigate the physical relevance of magic I apply CAMPS-PP to an XXZ Heisenberg Floquet circuit with Clifford and non-Clifford angles, finding that CAMPS-PP slightly extends the simulable non-Clifford gate density and significantly reduces memory requirements compared to PP.

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Classical simulations of quantum systems are notoriously difficult computational problems, with conventional state vector and tensor network methods restricted to quantum systems that feature only a small number of qudits. The recently introduced Clifford Augmented Matrix Product State (CAMPS) method offer scalability and efficiency by combining both tensor network and stabilizer simulation techniques and leveraging their complementary advantages. This hybrid simulation method has indeed demonstrated significant improvements in simulation performance for qubit circuits. Our work generalises the CAMPS method to higher quantum degrees of freedom -- qudit simulation, resulting in a generalised CAMPS (GCAMPS). Benchmarking this extended simulator on quantum systems with three degrees of freedom, i.e. qutrits, we show that similar to the case of qubits, qutrit systems also benefit from a comparable speedup using these techniques. Indeed, we see a greater improvement with qutrit simulation compared to qubit simulation on the same

T

-doped random Clifford benchmarking circuit as a result of the increased difficulty of conventional qutrit simulation using tensor networks. This extension allows for the classical simulation of problems that were previously intractable without access to a quantum device and will open new avenues to study complex many-body physics and to develop efficient methods for quantum information processing.

Determining the quantum-classical boundary between quantum circuits that can be efficiently simulated classically and those that cannot remains a fundamental question. One approach to classical simulation is to represent the output of a quantum circuit as a Clifford-augmented matrix product state (CAMPS), which, via a disentangling algorithm, decomposes the wave function into Clifford and MPS components, and from which Pauli expectation values can be computed in time polynomial in the MPS bond dimension. In this work, we develop an optimization-free disentangling (OFD) algorithm for Clifford circuits either doped with T gates or, equivalently, preceded by multiqubit gates of the form α I + β P . We give a simple and easily computed algebraic criterion that characterizes the individual quantum circuits for which OFD generates an efficient CAMPS—the bond dimension is exponential in the null space of a Galois binary field GF(2) matrix induced by a tableau of the twisted Pauli strings P . This significantly increases the number of circuits with rigorous polynomial-time classical simulations. We also give evidence that the typical N qubit random Clifford circuit doped with N uniformly distributed T gates of polylogarithmic depth or greater has a CAMPS with polynomial bond dimension. In addition, we compare OFD against disentangling by optimization. We further explore the representability of CAMPS for random Clifford circuits doped with more than N T gates. We also propose algorithms for sampling, probability, and amplitude estimation of bitstrings, and evaluation of entanglement Rényi entropy from CAMPS, which, though still having exponential complexity, are more efficient than standard MPS simulations. This work establishes a versatile framework for understanding classical simulatability of Clifford + T circuits and explores the interplay between quantum entanglement and quantum magic in quantum systems.

Tensor network methods leverage the limited entanglement of quantum states to efficiently simulate many-body systems. Alternatively, Clifford circuits provide a framework for handling highly entangled stabilizer states, which have low magic and are thus also classically tractable. Clifford tensor networks combine the benefits of both approaches, exploiting Clifford circuits to reduce the classical complexity of the tensor network description of states, with promising effects on simulation approaches. We study the disentangling power of Clifford transformations acting on tensor networks, with a particular emphasis on entanglement cooling strategies. We identify regimes where exact or heuristic Clifford disentanglers are effective, explain the link between the two approaches, and characterize their breakdown as non-Clifford resources accumulate. Additionally, we prove that, beyond stabilizer settings, no Clifford operation can universally disentangle even a single qubit from an arbitrary non-Clifford rotation. Our results clarify both the capabilities and fundamental limitations of Clifford-based simulation methods.

Clifford circuits augmented matrix product states (CAMPS) was recently proposed to leverage the advantages of both Clifford circuits and matrix product states (MPS). Clifford circuits can support large entanglement and can be efficiently simulated classically according to the Gottesman-Knill theorem. So in CAMPS, MPS needs only to handle the so-called nonstabilizerness entanglement entropy which significantly improves the simulation accuracy for a given bond dimension. In this work, we generalize CAMPS to study the fermion system by taking advantage of the Jordan-Wigner transformation which can map the studied fermion system to a spin system. We benchmark the method on both the spinless $t\text{\ensuremath{-}}V$ model and the spinful Hubbard model. Our test results show significant improvement of the accuracy of CAMPS over MPS, especially when the interactions are strong. Fermionic CAMPS provides a useful tool for the accurate study of many-body fermion systems in the future and has the potential to help resolve long-standing issues.

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 classical simulation of quantum circuits is of central importance for benchmarking near-term quantum devices. The fact that gates belonging to the Clifford group can be simulated efficiently on classical computers has motivated a range of methods that scale exponentially only in the number of non-Clifford gates. Here, we consider the expectation value problem for circuits composed of Clifford gates and non-Clifford Pauli rotations and introduce a heuristic perturbative approach based on the truncation of the exponentially growing sum of Pauli terms in the Heisenberg picture. Numerical results are shown on a quantum approximate optimization algorithm benchmark for the E3LIN2 problem, and we also demonstrate how this method can be used to quantify coherent and incoherent errors of local observables in Clifford circuits. Our results indicate that this systematically improvable perturbative method offers a viable alternative to exact methods for approximating expectation values of large near-Clifford circuits.