Question

We analyzed the Shor algorithm for error correction that uses three physical qubits to produce one corrected logical qubit. Assume that the probability for a single qubit error during qubit a machine cycle for operating a gate (during which some qubits can sit idle, but still can be affected by error) to be p. The Shor algorithm corrects bit flip and phase errors and it ignores the chances that two errors occur on two different qubits simultaneously.

The probability that a single qubit stays good is 1-p.

The Shor probability that a Shor corrected logical qubit of 3 qubits stays good is (1-p)^3 + 3p(1-p)^2.

Prove that the Shor corrected probability for remaining good is higher than the uncorrected probability when p<1/2.

Answer 1