Clifford Circuits Augmented Grassmann Matrix Product States

TL;DR

Addresses the challenge of simulating strongly correlated fermionic systems with entanglement that strains tensor-network representations. Proposes Clifford circuits augmented Grassmann MPS (CAMPS), integrating Clifford-based disentanglers into a locality-preserving Grassmann TN framework to remove classically simulable entanglement. Exploiting fermionic parity conservation $P_f$, the gate set reduces to 12 unique Grassmann Clifford gates, enabling efficient disentangling with maintained accuracy. Benchmarks on the $t$-$V$ model and free-fermion chains show improved energy accuracy at fixed bond dimension $\chi$ and correct universal scaling with central charge $c=1$, indicating faithful low-energy physics and entanglement compression. The approach is scalable to higher dimensions and suggests connections to circuit compression and boundary conformal-field-theory analyses.

Abstract

Recent advances in combining Clifford circuits with tensor-network (TN) methods have shown that classically simulable disentanglers can suppress substantial portions of the entanglement structure, effectively alleviating the bond-dimension bottleneck in TN simulations. In this work, we develop a variational TN framework based on Grassmann tensor networks, which natively encode fermionic statistics while preserving locality. By incorporating locally defined Clifford circuits within the fermionic formalism, we simulate benchmark models including the tight-binding and $t$-$V$ models. Our results show that Clifford disentangling removes the classically simulable component of entanglement, leading to a reduced bond dimension and improved accuracy in ground-state energy estimates. Interestingly, once the natural Grassmann-evenness requirement of the fermionic formulation is taken into account and Clifford gates with identical entanglement action are grouped together, the original set of 11520 two-qubit Clifford gates reduces to only 12 distinct gates. This strong reduction leads to a more efficient disentangling scheme within the fermionic framework. These findings highlight the potential of Clifford-augmented Grassmann TNs as a scalable and accurate tool for studying strongly correlated fermionic systems, particularly in higher dimensions.

Figures (4)

• Figure 1: Overview of the Clifford circuits augmented Grassmann MPS algorithm. (a) As in conventional two-site DMRG, to update the MPS on two adjacent sites $j$ and $j+1$, one constructs the effective Hamiltonian from the given Grassmann MPS and the Hamiltonian represented as Pauli strings, and then solves its eigenproblem to obtain the eigenvector $\ket{\psi_{j,j+1}}$. (b) The state $\ket{\psi_{j,j+1}}$ is disentangled across sites $j$ and $j+1$ using a Clifford circuit. Among all possible Clifford circuits, the optimal one that minimizes the entanglement is then identified and applied, after which the updated Grassmann MPS is obtained via singular value decomposition. (c) The same Clifford circuit is also applied to the Hamiltonian. Due to the properties of Clifford circuits, the Pauli operators on sites $j$ and $j+1$ are finally mapped to other Pauli operators.
• Figure 2: The systematic error of the energy of the $t$--$V$ model (Eq. \ref{['eq:tV_model']}; $t=1$, $V=2$, $L=32$) as a function of bond dimension. The dashed lines are for guiding the eyes.
• Figure 3: The entanglement entropy of the $t$--$V$ model (Eq. \ref{['eq:tV_model']}; $t=1$, $V=2$, $L=50$, $\chi=64$) as a function of the cut positions.
• Figure 4: Volume scaling of the entanglement entropy of the tight-binding model ($t=1$, $V = 0$, $\chi=64$). Fitting results with the fitting function $f(L)=\frac{1}{6}\log L + a + b/L$ are also shown in dashed lines. The data is compared with \ref{['eq:EE_volume_scaling']} and gives the central charge consistent with $c=1$ for both GMPS and CGMPS.