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:2407.03202 · Physical Review Letters

Clifford Circuits Augmented Time-Dependent Variational Principle

Xiangjian Qian, Jiale Huang, Mingpu Qin

Published Apr 15, 2025• 22 citations
arXivHTMLPDF

Abstract

The recently proposed Clifford circuits augmented matrix product states (CA-MPSs) [X. Qian, J. Huang, and M. Qin, Phys. Rev. Lett. 133, 190402 (2024)PRLTAO0031-900710.1103/PhysRevLett.133.190402] seamlessly augment density matrix renormalization groups with Clifford circuits. In CA-MPSs, the entanglement from stabilizers is transferred to the Clifford circuits, which can be easily handled according to the Gottesman-Knill theorem. As a result, an MPS needs only to deal with the nonstabilizer entanglement, which largely reduces the bond dimension and the resource required for the accurate simulation of many-body systems. In this Letter, we generalize CA-MPSs to the framework of the time-dependent variational principle (TDVP) for time evolution simulations. In this method, we apply Clifford circuits to the resulting MPS in each TDVP step with a two-site sweeping process similar as in density matrix renormalization groups, aiming at reducing the entanglement entropy in the MPS, and the Hamiltonian is transformed accordingly using the chosen Clifford circuits. Similar as in CA-MPSs, the Clifford circuits does not increase the number of terms in the Hamiltonian, which makes the overhead very small in the new method. We test this method in XXZ chain, 2D Heisenberg model, and Kitaev honeycomb model. The results show that the Clifford circuits augmented TDVP method can reduce the entanglement entropy in the time evolution process and hence makes the simulation reliable for a longer time. The Clifford circuits augmented time-dependent variational principle provides a useful tool for the simulation of the time evolution process of many-body systems in the future.

DOI
10.1103/physrevlett.134.150404

Related papers

Clifford circuits augmented matrix product states for fermion systems

Jiale Huang, Xiangjian Qian, Mingpu Qin

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.

arXiv:2501.00413•Physical review. B./Physical review. B•Oct 14, 2025•4 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

Augmenting Density Matrix Renormalization Group with Matchgates and Clifford Circuits

Jiale Huang, Xiangjian Qian, Zhendong Li +1 more

Abstract Matchgates and Clifford circuits are two types of quantum circuits which can be efficiently simulated classically, though the underlying reasons are quite different. Matchgates are essentially the single particle basis transformations in the Majorana fermion representation, which can be easily handled classically, while the Clifford circuits can be efficiently simulated using the tableau method according to the Gottesman–Knill theorem. In this work, we propose a new wave-function ansatz in which matrix product states (MPS) are augmented with the combination of Matchgates and Clifford circuits (dubbed MCA-MPS) to take advantage of the representing power of all of them. Moreover, the optimization of MCA-MPS can be efficiently implemented within the density matrix renormalization group method. Our benchmark results on one-dimensional hydrogen chain show that MCA-MPS can improve the accuracy of the ground-state calculation by several orders of magnitude over MPS with the same bond dimension. This new method provides us a useful approach to study quantum many-body systems. The MCA-MPS ansatz also expands our understanding of classically simulatable quantum many-body states.

Chinese Physics Letters•Mar 3, 2026•1 citation

Clifford-dressed variational principles for precise Loschmidt echoes

Antonio Francesco Mello, Alessandro Santini, Mario Collura

We extend the recently introduced Clifford-dressed time-dependent variational principle (TDVP) to efficiently compute many-body wave-function amplitudes in the computational basis. This advancement enhances the study of Loschmidt echoes, which generally require accurate calculations of the overlap between the evolved state and the initial wave function. By incorporating Clifford-disentangling gates during TDVP evolution, our method effectively controls entanglement growth while keeping the computation of these amplitudes accessible. Specifically, it reduces the problem to evaluating the overlap between a matrix product state (MPS) and a stabilizer state, a task that remains computationally feasible within the proposed framework. To demonstrate the effectiveness of this approach, we first benchmark it on the one-dimensional transverse-field Ising model. We then apply it to more challenging scenarios, including a nonintegrable next-to-nearest-neighbor Ising chain and a two-dimensional Ising model. Our results highlight the versatility and efficiency of the Clifford-augmented MPS, showcasing its capability to go beyond the evaluation of simple expectation values. This makes it a powerful tool for exploring various aspects of many-body quantum dynamics.

arXiv:2502.01872•Physical Review A•May 2, 2025•2 citations

Clifford Dressed Time-Dependent Variational Principle

Antonio Francesco Mello, Alessandro Santini, Guglielmo Lami +2 more

We propose an enhanced time-dependent variational principle (TDVP) algorithm for matrix product states that integrates Clifford disentangling techniques to efficiently manage entanglement growth. By leveraging the Clifford group, which maps Pauli strings to other Pauli strings while maintaining low computational complexity, we introduce a Clifford dressed single-site 1-TDVP scheme. During the TDVP integration, we apply a global Clifford transformation as needed to reduce entanglement by iteratively sweeping over two-qubit Clifford unitaries that connect neighboring sites in a checkerboard pattern. We validate the new algorithm numerically using various quantum many-body models, including both integrable and nonintegrable systems. Our results demonstrate that the Clifford dressed TDVP significantly improves entanglement management and computational efficiency, achieving higher accuracy, extended simulation times, and enhanced precision in computed observables compared to standard TDVP approaches. Additionally, we propose incorporating Clifford gates directly within the two-site 2-TDVP scheme.

arXiv:2407.01692•Physical Review Letters•Apr 15, 2025•20 citations

Clifford augmented density matrix renormalization group for ab initio quantum chemistry

Lizhong Fu, Honghui Shang, Jinlong Yang +1 more

The recently proposed Clifford augmented density matrix renormalization group (CA-DMRG) method seamlessly integrates Clifford circuits with matrix product states and takes advantage of the expression power from both. CA-DMRG has been shown to be able to achieve higher accuracy than standard DMRG on commonly used lattice models, with only moderate computational overhead compared to the latter. In this work, we propose an efficient scheme in CA-DMRG to deal with ab initio quantum chemistry Hamiltonians and apply it to study several molecular systems. Our numerical results show that CA-DMRG can reach higher accuracy than DMRG using the same bond dimension, pointing out a promising route to push the boundary of solving ab initio quantum chemistry with strong static correlations.

Physical review. B./Physical review. B•Nov 12, 2025•2 citations
The recently proposed Clifford circuits augmented matrix product states (CA-MPSs) [X. Qian, J. Huang, and M. Qin, Phys. Rev. Lett. 133, 190402 (2024)PRLTAO0031-900710.1103/PhysRevLett.133.190402] seamlessly augment density matrix renormalization groups with Clifford circuits. In CA-MPSs, the entanglement from stabilizers is transferred to the Clifford circuits, which can be easily handled according to the Gottesman-Knill theorem. As a result, an MPS needs only to deal with the nonstabilizer entanglement, which largely reduces the bond dimension and the resource required for the accurate simulation of many-body systems. In this Letter, we generalize CA-MPSs to the framework of the time-dependent variational principle (TDVP) for time evolution simulations. In this method, we apply Clifford circuits to the resulting MPS in each TDVP step with a two-site sweeping process similar as in density matrix renormalization groups, aiming at reducing the entanglement entropy in the MPS, and the Hamiltonian is transformed accordingly using the chosen Clifford circuits. Similar as in CA-MPSs, the Clifford circuits does not increase the number of terms in the Hamiltonian, which makes the overhead very small in the new method. We test this method in XXZ chain, 2D Heisenberg model, and Kitaev honeycomb model. The results show that the Clifford circuits augmented TDVP method can reduce the entanglement entropy in the time evolution process and hence makes the simulation reliable for a longer time. The Clifford circuits augmented time-dependent variational principle provides a useful tool for the simulation of the time evolution process of many-body systems in the future.
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.
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.
Abstract Matchgates and Clifford circuits are two types of quantum circuits which can be efficiently simulated classically, though the underlying reasons are quite different. Matchgates are essentially the single particle basis transformations in the Majorana fermion representation, which can be easily handled classically, while the Clifford circuits can be efficiently simulated using the tableau method according to the Gottesman–Knill theorem. In this work, we propose a new wave-function ansatz in which matrix product states (MPS) are augmented with the combination of Matchgates and Clifford circuits (dubbed MCA-MPS) to take advantage of the representing power of all of them. Moreover, the optimization of MCA-MPS can be efficiently implemented within the density matrix renormalization group method. Our benchmark results on one-dimensional hydrogen chain show that MCA-MPS can improve the accuracy of the ground-state calculation by several orders of magnitude over MPS with the same bond dimension. This new method provides us a useful approach to study quantum many-body systems. The MCA-MPS ansatz also expands our understanding of classically simulatable quantum many-body states.
We extend the recently introduced Clifford-dressed time-dependent variational principle (TDVP) to efficiently compute many-body wave-function amplitudes in the computational basis. This advancement enhances the study of Loschmidt echoes, which generally require accurate calculations of the overlap between the evolved state and the initial wave function. By incorporating Clifford-disentangling gates during TDVP evolution, our method effectively controls entanglement growth while keeping the computation of these amplitudes accessible. Specifically, it reduces the problem to evaluating the overlap between a matrix product state (MPS) and a stabilizer state, a task that remains computationally feasible within the proposed framework. To demonstrate the effectiveness of this approach, we first benchmark it on the one-dimensional transverse-field Ising model. We then apply it to more challenging scenarios, including a nonintegrable next-to-nearest-neighbor Ising chain and a two-dimensional Ising model. Our results highlight the versatility and efficiency of the Clifford-augmented MPS, showcasing its capability to go beyond the evaluation of simple expectation values. This makes it a powerful tool for exploring various aspects of many-body quantum dynamics.
We propose an enhanced time-dependent variational principle (TDVP) algorithm for matrix product states that integrates Clifford disentangling techniques to efficiently manage entanglement growth. By leveraging the Clifford group, which maps Pauli strings to other Pauli strings while maintaining low computational complexity, we introduce a Clifford dressed single-site 1-TDVP scheme. During the TDVP integration, we apply a global Clifford transformation as needed to reduce entanglement by iteratively sweeping over two-qubit Clifford unitaries that connect neighboring sites in a checkerboard pattern. We validate the new algorithm numerically using various quantum many-body models, including both integrable and nonintegrable systems. Our results demonstrate that the Clifford dressed TDVP significantly improves entanglement management and computational efficiency, achieving higher accuracy, extended simulation times, and enhanced precision in computed observables compared to standard TDVP approaches. Additionally, we propose incorporating Clifford gates directly within the two-site 2-TDVP scheme.
The recently proposed Clifford augmented density matrix renormalization group (CA-DMRG) method seamlessly integrates Clifford circuits with matrix product states and takes advantage of the expression power from both. CA-DMRG has been shown to be able to achieve higher accuracy than standard DMRG on commonly used lattice models, with only moderate computational overhead compared to the latter. In this work, we propose an efficient scheme in CA-DMRG to deal with ab initio quantum chemistry Hamiltonians and apply it to study several molecular systems. Our numerical results show that CA-DMRG can reach higher accuracy than DMRG using the same bond dimension, pointing out a promising route to push the boundary of solving ab initio quantum chemistry with strong static correlations.