Preparing Projected Entangled Pair States on a Quantum Computer

We present a quantum algorithm to prepare injective projected entangled pair states (PEPS) on a quantum computer, a class of open tensor networks representing quantum states. The run time of our algorithm scales polynomially with the inverse of the minimum condition number of the PEPS projectors and, essentially, with the inverse of the spectral gap of the PEPS’s parent Hamiltonian.

Figure 1

Algorithm constructing injective PEPS.

Figure 2

In each step the algorithm processes one vertex $v$: $H_t$ is grown into $H_{t+1}$ by removing all existing terms referring to $v$ before the $2k$-local parent Hamiltonian terms are added implementing PEPS projector $A^{(v)}$ at $v$. Note that terms around $v$ connecting to an open virtual index $v^{\prime }$ (bonds with dotted border) are only temporary and are removed in later steps of the algorithm. All terms constraining a vertex $v$ only restrict the physical subspace $P_{\mathrm{phy}}^{(v)}$, while degrees of freedom in the orthogonal subspace ($\mathbb{1}-P_{\mathrm{phy}}^{(v)}$) are eliminated with an additional penalty term $H_{\mathrm{phy}}^{(v)}$ that is added per vertex.