A distance-3 error-correcting code has a correction circuit (encoding
PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Open quantum systems, decoherence and error correction · Open quantum systems, decoherence and error correction
Quantum Information and Technology, Ch Quantum error correction
Problem
A distance-3 error-correcting code has a correction circuit (encoding, syndrome extraction and recovery) with N=300 distinct locations per code cycle at which a single fault (a faulty gate or a faulty measurement) can occur. A single fault on its own is always caught and corrected, but a fault at two of these locations occurring in the same cycle sometimes defeats the code. Suppose a fraction f=0.22 of all pairs of fault locations are 'harmful': if faults strike both members of such a pair in the same cycle, a logical error results.
((a)) Estimate the number of harmful fault pairs c.
((b)) Taking the logical error rate to be pL≈cp2 for physical error probability p per location, estimate the pseudo-threshold p∗, the value of p below which this code reduces the error rate (that is, where pL(p∗)=p∗).