Fault tolerance with constant space and constant time overhead
In plain words
Theorists can now run a quantum computation reliably with only a constant factor more qubits, but at the price of a slowdown that grows slowly with problem size. Whether both the qubit cost and the slowdown can be constant at once is unknown.
Precise statement
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~(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.
What would settle it
An explicit constant-space, constant-time fault-tolerance scheme with a threshold proof, or a matching lower bound on time overhead.
Status in the literature
Unverified note
2024 constructions reduced the time overhead to about log N at constant space overhead.