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Whether both the qubit cost and the slowdown can be constant at once is unknown.","posed_since":"2013","precise":"For quantum circuits on $N$ logical qubits of depth $\\operatorname{poly}(N)$ under local stochastic noise below a constant threshold, does a fault-tolerance scheme exist with physical-to-logical qubit ratio $O(1)$ and time overhead $O(1)$? Gottesman (2013) gave constant space with polynomial time overhead; schemes from 2024 reach constant space with polylogarithmic or $O \\sim (\\mathrm{log} N)$ time overhead (arXiv:2411.03632, arXiv:2411.03683). An answer is a construction or a lower bound proving that $\\omega(1)$ time overhead is necessary at constant space overhead.","problem_ref":"a54a341fc4bc03990e8c80a2d8e5a4890eeeeed62b2e46ebc85fc011e1166b14","references":"","settled_by":"An explicit constant-space, constant-time fault-tolerance scheme with a threshold proof, or a matching lower bound on time overhead.","status_note":"2024 constructions reduced the time overhead to about log N at constant space overhead.","title":"Fault tolerance with constant space and constant time 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