  Demonstrations of quantum computational advantage and benchmarks of quantum
processors via quantum random circuit sampling are based on evaluating the
linear cross-entropy benchmark (XEB). A key question in the theory of XEB is
whether it approximates the fidelity of the quantum state preparation. Previous
works have shown that the XEB generically approximates the fidelity in a regime
where the noise rate per qudit $\varepsilon$ satisfies $\varepsilon N \ll 1$
for a system of $N$ qudits and that this approximation breaks down at large
noise rates. Here, we show that the breakdown of XEB as a fidelity proxy occurs
as a sharp phase transition at a critical value of $\varepsilon N$ that depends
on the circuit architecture and properties of the two-qubit gates, including in
particular their entangling power. We study the phase transition using a
mapping of average two-copy quantities to statistical mechanics models in
random quantum circuit architectures with full or one-dimensional connectivity.
We explain the phase transition behavior in terms of spectral properties of the
transfer matrix of the statistical mechanics model and identify two-qubit gate
sets that exhibit the largest noise robustness.
