Augmenting Density Matrix Renormalization Group with Matchgates and Clifford 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 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.

Figures (7)

• Figure 1: Schematic illustration of MCA-MPS. Left: The structure of the MCA-MPS ansatz, $|\text{MCA-MPS}\rangle = C G |\text{MPS}\rangle$, depicting the sequential application of Matchgate circuits $G$ and Clifford circuits $C$ to the Matrix Product State $|\text{MPS}\rangle$ corresponding to Hamiltonian $H$. Notice that we only deal with $|\text{MCA-MPS}\rangle$ which is less entangled than $|\text{MPS}\rangle$. Right: The equivalent transformation of the Hamiltonian, where the original Hamiltonian $H$ is transformed to a effective Hamiltonian ($H_\text{MCA-MPS} = C G H G^\dagger C^\dagger$) by the successive application of Matchgates and Clifford circuits.
• Figure 2: Ground-state energy error (in the unit of Hartree) for the 1D hydrogen chain H12 (with restricted Hartree-Fock orbital) relative to exact energy, plotted against bond dimension D. Results are shown for both MPS, CAMPS and MCA-MPS. Errors are defined as $|E_{\text{exact}} - E_{\text{MPS}}|$, $|E_{\text{exact}} - E_{\text{CAMPS}}|$ and $|E_{\text{exact}} - E_{\text{MCA-MPS}}|$ respectively. The inset displays the ratio of the error of MPS over MCA-MPS, quantifying the improvement achieved by MCA-MPS over standard MPS with same bond dimension. We can find that MCA-MPS can achieve several orders of magnitude and the improvement becomes more dramatic with the increase of bond dimension.
• Figure 3: Entanglement entropy of the 1D hydrogen chain H12 for MPS, CAMPS and MCA-MPS. The entropy is computed for a bipartition of the system into two equal halves and plotted as a function of bond dimension $D$. We can find that the entanglement entropy is converged with small bond dimension in MCA-MPS. The entanglement entropy in MCA-MPS is reduced by a factor about $4$ to the MPS results, consistent with the significant improvement of ground state energy accuracy in Fig. \ref{['Energy_err']}.
• Figure 4: Similar as Fig. \ref{['Energy_err']}: energy error (in the unit of Hartree) for the 1D hydrogen chain H12 relative to exact energy but for OAO orbitals. For OAO orbitals the Matchgate offers no improvement. So only results for MPS and CAMPS are shown. Errors are defined as $|E_{\text{exact}} - E_{\text{MPS}}|$ and $|E_{\text{exact}} - E_{\text{CAMPS}}|$ respectively. The inset displays the ratio of these errors, quantifying the improvement achieved by CAMPS over standard MPS with same bond dimension. We can find that CAMPS can achieve several orders of magnitude and the improvement becomes more dramatic with the increase of bond dimension.
• Figure S1: 2D OAO results for MPS, MA-MPS and MCA-MPS. Errors (in the unit of Hartree) are defined as $|E_{\text{ref}} - E_{\text{MPS}}|$, $|E_{\text{ref}} - E_{\text{MA-MPS}}|$ and $|E_{\text{ref}} - E_{\text{MCA-MPS}}|$ respectively. The inset displays the ratio of MPS error to MCA-MPS error, quantifying the improvement achieved by MCA-MPS over standard MPS with same bond dimension.