Largest noise rate allowing fault-tolerant universal quantum computation
In plain words
Proven guarantees say error correction works for error rates up to roughly a hundredth of a percent, simulations reach about one percent, and impossibility proofs only start near one third. The true maximum tolerable error rate is unknown.
Precise statement
Consider circuits of unitary two-qubit gates whose input wires each suffer depolarizing noise of strength $p$, with noise-free one-qubit unitaries and fresh ancilla qubits, as in Kempe, Regev, Unger and de Wolf (arXiv:0802.1464, 2008), who prove that above $p=35.7\%$ ($29.3\%$ if the only two-qubit gate is CNOT) the output of any deep enough circuit is independent of its input. Find $p_{\mathrm{th}}$, the supremum of $p$ at which arbitrarily long computations remain reliable with polylogarithmic overhead. Constructive schemes reach of order $1\%$ in simulation and rigorous lower bounds are of order $10^{-5}$ to $10^{-4}$ (approximate).
What would settle it
Matching or near-matching upper and lower bounds on $p_{\mathrm{th}}$ for the stated noise model.