Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2602.15942 · Physical review. B./Physical review. B

Limits of Clifford Disentangling in Tensor Network States

Sergi Masot-Llima, Piotr Sierant, Paolo Stornati, Artur Garcia-Saez

quant-phSubmitted Feb 17, 2026• Announced Feb 19, 2026• 0 citations• Revised Feb 22, 2026 (v2)
v1 · Feb 17, 2026v2 · Feb 22, 2026
arXivHTMLPDF

Abstract

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.

DOI
10.1103/5g5b-5rrz
Comments
17 pages, 11 figures

Related papers

Stabilizer Tensor Networks: Universal Quantum Simulator on a Basis of Stabilizer States

Sergi Masot-Llima, Artur García-Sáez

Efficient simulation of quantum computers relies on understanding and exploiting the properties of quantum states. This is the case for methods such as tensor networks, based on entanglement, and the tableau formalism, which represents stabilizer states. In this Letter, we integrate these two approaches to present a generalization of the tableau formalism used for Clifford circuit simulation. We explicitly prove how to update our formalism with Clifford gates, non-Clifford gates, and measurements, enabling universal circuit simulation. We also discuss how the framework allows for efficient simulation of more states, raising some interesting questions on the representation power of tensor networks and the quantum properties of resources such as entanglement and magic, and support our claims with simulations.

arXiv:2403.08724•Physical Review Letters•Dec 3, 2024•39 citations

Clifford disentanglers for entanglement reduction in molecular electronic structure simulations

Longfei Chang, Zibo Wu, Yunzhi Li +5 more

Entanglement is a key bottleneck limiting the efficiency of tensor-network and quantum simulations of molecular electronic structures. Here, we systematically assess and extend Clifford disentanglers as a structure-preserving approach to entanglement reduction: they can modify the entanglement structure of qubit wavefunctions while retaining the Pauli-string form of qubit Hamiltonians. To enable a practical search over Clifford transformations, we classify Clifford operators by their action on the Schmidt spectrum across a bipartition, reducing the two- and four-qubit search spaces to 20 and 91392 representatives, respectively. Embedded in an iterative Clifford-augmented matrix product state framework, these transformations reduce the energy errors at fixed bond dimension for the molecular test cases studied and mitigate the dependence on orbital orderings and fermion-to-qubit mappings. We further show that Clifford disentanglers can also benefit quantum simulations such as the shallow-circuit variational quantum eigensolver calculations. Together, these results establish Clifford disentanglers as a useful structure-preserving entanglement-engineering tool for tensor-network and quantum simulations of molecular electronic structure, while also clarifying their correlation dependence and motivating future developments.

quant-pharXiv:2606.12056•Journal of Chemical Theory and Computation•Jun 11, 2026•0 citations

Augmenting Finite Temperature Tensor Network with Clifford Circuits

Xiangjian Qian, Jiale Huang, Mingpu Qin

Recent studies have highlighted the combination of tensor network methods and the stabilizer formalism as a very effective framework for simulating quantum many-body systems, encompassing areas from ground state to time evolution simulations. In these approaches, the entanglement associated with stabilizers is transferred to Clifford circuits, which can be efficiently managed due to the Gottesman-Knill theorem. Consequently, only the non-stabilizerness entanglement needs to be handled, thereby reducing the computational resources required for accurate simulations of quantum many-body systems in tensor network related methods. In this work, we adapt this paradigm for finite temperature simulations in the framework of Time-Dependent Variational Principle, in which imaginary time evolution is performed using the purification scheme. Our numerical results on the one-dimensional Heisenberg model and the two-dimensional $J_1-J_2$ Heisenberg model demonstrate that Clifford circuits can significantly improve the efficiency and accuracy of finite temperature simulations for quantum many-body systems. This improvement not only provides a useful tool for calculating finite temperature properties of quantum many-body systems, but also paves the way for further advancements in boosting the finite temperature tensor network calculations with Clifford circuits and other quantum circuits.

arXiv:2410.15709•Oct 21, 2024•1 citation

Stabilizer Tensor Networks with Magic State Injection

Azar C. Nakhl, Ben Harper, Maxwell T. West +4 more

This Letter augments the recently introduced stabilizer tensor network (STN) protocol with magic state injection, reporting a new framework with significantly enhanced ability to simulate circuits with an extensive number of non-Clifford operations. Specifically, for random T-doped N-qubit Clifford circuits the computational cost of circuits prepared with magic state injection scales as O[poly(N)] when the circuit has t≲N T gates compared to an exponential scaling for the STN approach, which is demonstrated in systems of up to 200 qubits. In the case of the hidden bit shift circuit, a paradigmatic benchmarking system for extended stabilizer methods with a tunable amount of magic, we report that our magic state injected STN framework can efficiently simulate 4000 qubits and 320T gates. These findings provide a promising outlook for the use of this protocol in the classical modeling of quantum circuits that are conventionally difficult to simulate efficiently.

arXiv:2411.12482•Physical Review Letters•May 15, 2025•16 citations

Combining tensor networks and Pauli propagation to simulate quantum dynamics

Alessandro Cerati

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.

•Sep 24, 2026•0 citations

Efficient Learning of Clifford-Scrambled Product States

Tobias Haug

Clifford circuits acting on product magic states provide a compact ansatz exhibiting extensive magic, volume-law entanglement, and even classically hard sampling under standard complexity assumptions. Here, we efficiently recover its hidden product subsystems from two-copy Bell sampling. Quadratic relations between Bell samples determine the irreducible blocks after removing Pauli stabilizers, and binary linear algebra constructs a Clifford disentangler. For logarithmic-size blocks, approximate learning of the full state is efficient whenever the state remains inverse-polynomially separated from acquiring additional Pauli stabilizers. We efficiently learn $n$-qubit states prepared by random $T$-doped Clifford circuits with $T$-gate density below one via $O(n^2)$ Bell samples, and disentangle hidden product blocks in Clifford-augmented matrix product states. For unitary learning, we exactly learn $T$-depth-one circuits using $O(n^2)$ queries. Finally, we rule out pseudorandom states and unitaries with a product bipartition hidden by Clifford circuits.

quant-pharXiv:2609.27128•Sep 24, 2026•0 citations
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.
Efficient simulation of quantum computers relies on understanding and exploiting the properties of quantum states. This is the case for methods such as tensor networks, based on entanglement, and the tableau formalism, which represents stabilizer states. In this Letter, we integrate these two approaches to present a generalization of the tableau formalism used for Clifford circuit simulation. We explicitly prove how to update our formalism with Clifford gates, non-Clifford gates, and measurements, enabling universal circuit simulation. We also discuss how the framework allows for efficient simulation of more states, raising some interesting questions on the representation power of tensor networks and the quantum properties of resources such as entanglement and magic, and support our claims with simulations.
Entanglement is a key bottleneck limiting the efficiency of tensor-network and quantum simulations of molecular electronic structures. Here, we systematically assess and extend Clifford disentanglers as a structure-preserving approach to entanglement reduction: they can modify the entanglement structure of qubit wavefunctions while retaining the Pauli-string form of qubit Hamiltonians. To enable a practical search over Clifford transformations, we classify Clifford operators by their action on the Schmidt spectrum across a bipartition, reducing the two- and four-qubit search spaces to 20 and 91392 representatives, respectively. Embedded in an iterative Clifford-augmented matrix product state framework, these transformations reduce the energy errors at fixed bond dimension for the molecular test cases studied and mitigate the dependence on orbital orderings and fermion-to-qubit mappings. We further show that Clifford disentanglers can also benefit quantum simulations such as the shallow-circuit variational quantum eigensolver calculations. Together, these results establish Clifford disentanglers as a useful structure-preserving entanglement-engineering tool for tensor-network and quantum simulations of molecular electronic structure, while also clarifying their correlation dependence and motivating future developments.
Recent studies have highlighted the combination of tensor network methods and the stabilizer formalism as a very effective framework for simulating quantum many-body systems, encompassing areas from ground state to time evolution simulations. In these approaches, the entanglement associated with stabilizers is transferred to Clifford circuits, which can be efficiently managed due to the Gottesman-Knill theorem. Consequently, only the non-stabilizerness entanglement needs to be handled, thereby reducing the computational resources required for accurate simulations of quantum many-body systems in tensor network related methods. In this work, we adapt this paradigm for finite temperature simulations in the framework of Time-Dependent Variational Principle, in which imaginary time evolution is performed using the purification scheme. Our numerical results on the one-dimensional Heisenberg model and the two-dimensional J1−J2J_1-J_2J1​−J2​ Heisenberg model demonstrate that Clifford circuits can significantly improve the efficiency and accuracy of finite temperature simulations for quantum many-body systems. This improvement not only provides a useful tool for calculating finite temperature properties of quantum many-body systems, but also paves the way for further advancements in boosting the finite temperature tensor network calculations with Clifford circuits and other quantum circuits.
This Letter augments the recently introduced stabilizer tensor network (STN) protocol with magic state injection, reporting a new framework with significantly enhanced ability to simulate circuits with an extensive number of non-Clifford operations. Specifically, for random T-doped N-qubit Clifford circuits the computational cost of circuits prepared with magic state injection scales as O[poly(N)] when the circuit has t≲N T gates compared to an exponential scaling for the STN approach, which is demonstrated in systems of up to 200 qubits. In the case of the hidden bit shift circuit, a paradigmatic benchmarking system for extended stabilizer methods with a tunable amount of magic, we report that our magic state injected STN framework can efficiently simulate 4000 qubits and 320T gates. These findings provide a promising outlook for the use of this protocol in the classical modeling of quantum circuits that are conventionally difficult to simulate efficiently.
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.
Clifford circuits acting on product magic states provide a compact ansatz exhibiting extensive magic, volume-law entanglement, and even classically hard sampling under standard complexity assumptions. Here, we efficiently recover its hidden product subsystems from two-copy Bell sampling. Quadratic relations between Bell samples determine the irreducible blocks after removing Pauli stabilizers, and binary linear algebra constructs a Clifford disentangler. For logarithmic-size blocks, approximate learning of the full state is efficient whenever the state remains inverse-polynomially separated from acquiring additional Pauli stabilizers. We efficiently learn nnn-qubit states prepared by random TTT-doped Clifford circuits with TTT-gate density below one via O(n2)O(n^2)O(n2) Bell samples, and disentangle hidden product blocks in Clifford-augmented matrix product states. For unitary learning, we exactly learn TTT-depth-one circuits using O(n2)O(n^2)O(n2) queries. Finally, we rule out pseudorandom states and unitaries with a product bipartition hidden by Clifford circuits.