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.
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\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.
Clifford circuits can be utilized to disentangle quantum states with polynomial cost, thanks to the Gottesman-Knill theorem. Based on this idea, the Clifford Circuits Augmented Matrix Product States (CAMPS) method, which is a seamless integration of Clifford circuits within the density-matrix renormalization group algorithm, was proposed recently and was shown to be able to reduce entanglement in various quantum systems. In this work, we further explore the power of the CAMPS method in critical spin chains described by conformal field theories (CFTs) in the scaling limit. We find that the optimized disentanglers correspond to duality transformations, which significantly reduce the entanglement entropy in the ground state. For the critical quantum Ising spin chain governed by the Ising CFT with self-duality, the Clifford circuits found by CAMPS coincide with the duality transformation, i.e., the Kramer-Wannier self-duality in the critical Ising chain. It reduces the entanglement entropy by mapping the free conformal boundary condition to the fixed one. In the more general case of the XXZ chain, the CAMPS gives rise to a duality transformation mapping the model to the quantum Ashkin-Teller spin chain. Our results highlight the potential of the framework as a versatile tool for uncovering hidden dualities and simplifying the entanglement structure of critical quantum systems.
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.
In this work, we investigate the Kitaev honeycomb model employing the recently developed Clifford Circuits Augmented Matrix Product States (CAMPS) method. While the model in the gapped phase is known to reduce to the toric code model - whose ground state is entirely constructible from Clifford circuits - we demonstrate that the very different gapless quantum spin liquid (QSL) phase can also be significantly disentangled with Clifford circuits. Specifically, CAMPS simulations reveal that approximately two-thirds of the entanglement entropy in the isotropic point arises from Clifford-circuit contributions, enabling dramatically more efficient computations compared to conventional matrix product state (MPS) methods. Crucially, this finding implies that the Kitaev QSL state retains significant Clifford-simulatable structure, even in the gapless phase with non-abelian anyon excitations when time reversal symmetry is broken. This property not only enhances classical simulation efficiency significantly but also suggests substantial resource reduction for preparing such states on quantum devices. As an application, we leverage CAMPS to study the Kitaev-Heisenberg model and determine the most accurate phase boundary between the anti-ferromagnetic phase and the Kitaev QSL phase in the model. Our results highlight how Clifford circuits can effectively disentangle the intricate entanglement of Kitaev QSLs, opening avenues for efficiently simulating related and similar strongly correlated models.
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.
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.