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Where the crossover lies, in qubit number, depth and noise rate, is not known.","posed_since":"","precise":"For random circuit sampling with $n$ qubits, depth $D$ and per-gate error $\\epsilon$, Aharonov et al. (2023) give a classical algorithm polynomial in $n$ at constant $\\epsilon$, with cost exponential in $1/\\epsilon$. Determine the best classical cost to reach the experimental linear cross-entropy fidelity $F_{\\mathrm{XEB}} \\sim \\exp(-\\epsilon n D)$ as a function of $(n, D, \\epsilon)$, and locate the easy-hard boundary for parameters of current experiments ($n \\sim 70-100$, $\\epsilon \\sim 10^{-3}$).","problem_ref":null,"references":"","settled_by":"Classical algorithms and complexity bounds that place each published experiment on a quantified side of the boundary.","status_note":"Google Quantum AI (Nature 2024) reported a noise-driven phase transition in random circuit sampling and placed its experiment in the hard phase; classical simulation efforts continue.","title":"Finite-size boundary of classical hardness for noisy random circuit 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