Scalable projected entangled-pair state representation of random quantum circuit states

Classical simulation of a programmable quantum processor is crucial in identifying the threshold of a quantum advantage. We demonstrate the simple update of projected entangled-pair states (PEPSs) in the Vidal gauge that represent random quantum circuit states, which center around recent quantum advantage claims. Applied to square lattices of qubits akin to state-of-the-art superconducting processors, the PEPS representation is exact for circuit depths less than $\mathcal{D}_\mathrm{tr}$ = $\beta\log_2\chi$, where $\chi$ is the maximum bond dimension and $2 \lesssim \beta \lesssim 4$ depends on the choice of two-qubit gates, independent of the qubit number $n$. We find the universal scaling behaviors of the state fidelity by treating large-scale circuits of $n \leq 10^{4}$, using $\chi \leq 128$ on a conventional CPU. Our method has a polynomial scaling of computational costs with $n$ for circuit depth $\mathcal{D}=O(\log n)$ and is more advantageous than matrix product state approaches if $n$ is large. This work underscores PEPSs as a scalable tool for benchmarking quantum algorithms with future potential for sampling applications using advanced contraction techniques.