Preparing topological projected entangled pair states on a quantum computer

Simulating exotic phases of matter that are not amenable to classical techniques is one of the most important potential applications of quantum information processing. We present an efficient algorithm for preparing a large class of topological quantum states, the $G$-injective projected entangled pair states (PEPS), on a quantum computer. Important examples include the resonant valence bond states, conjectured to be topological spin liquids. The runtime of the algorithm scales polynomially with the condition number of the PEPS projectors and inverse polynomially in the spectral gap of the PEPS parent Hamiltonian.

Figure 1

(a) Illustrates the decomposition of the original tensor in the tensors $A$. We mark in white the bonds in which we have $U_g$ and in black those in which we have ${\overline{U}}_g$. (b) Illustrates the new way of grouping the tensors to get a $G$-isometric PEPS, called $C$. The bonds of this new tensor are numbered clockwise as in the figure.

Figure 2

Preparing a $G$-injective PEPS. $H_t$ ($t=1...N$) is the parent Hamiltonian for the $G$-injective PEPS $|A^1...A^t\rangle$, $P_t$ the projector onto its ground-state subspace. Note that by specifying $A^v$, we are implicitly selecting a particular semiregular represent-ation of $G$.

Figure 3

The sequence of outcomes of the binary measurements $\{P_t,P_t^{\perp }\}$ and $\{P_{t+1},P_{t+1}^{\perp }\}$. We start in an eigenstate $|{\psi }_t\rangle$ of $P_t$, and want to transition to a state in the subspace $P_{t+1}$. With nonzero probability, the first measurement succeeds with outcome $P_{t+1}$. If it fails, we have prepared a state in the $P_{t+1}^{\perp }$ subspace. We “unwind” the measurement by measuring $\{P_t,P_t^{\perp }\}$ again. Upon repeating the $\{P_{t+1},P_{t+1}^{\perp }\}$ measurement, we again have a nonzero probability of successfully obtaining the $P_{t+1}$ outcome. This is repeated until success.