Fault-tolerance thresholds and logical error scaling

4 problems

• Does logical error suppression continue exponentially to large code distance?

  Open

  Recent chips showed that enlarging an error-correcting code cuts the error rate by a constant factor at each step. It is unknown whether this continues to the much larger codes that useful computations need, or whether rare correlated events set a floor.

• Threshold theorem for noise with slowly decaying spatial correlations

  Open

  Error correction is proven to work when errors on distant qubits are nearly independent. It is unknown whether it still works when each qubit feels a fixed amount of noise but a shared environment links the noise on far-apart qubits with correlations that fall off slowly with distance.

• Does the optimal toric-code depolarizing threshold equal the hashing bound?

  Open

  With a perfect decoder (the classical routine that infers which errors occurred), the toric code tolerates a certain depolarizing error rate, and numerics put it at about 18.9%. Whether this equals exactly the hashing bound, a limit from information theory with the same value to three digits, is unknown.

• Largest noise rate allowing fault-tolerant universal quantum computation

  Open

  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.