Error Correction Capstone

Error Correction Capstone

In this capstone you run a complete quantum error correction cycle, end to end: encode a logical qubit across three physical qubits, let a phase flip error strike, diagnose exactly which qubit was hit without ever looking at the data, undo the damage, and prove the logical state survived. This is the full loop that every fault-tolerant quantum computer must execute millions of times per second.

The code you build protects against Z errors (phase flips). That choice is deliberate: on real hardware, dephasing is typically the dominant noise process, the T2 time that limits superconducting qubits is exactly a phase flip rate. The bit flip code from earlier in this track cannot see a Z error at all; this one is built for it.

The 3-Qubit Phase Flip Code

The phase flip code is the bit flip code viewed through Hadamard glasses. The encoding maps

and the circuit is the familiar repetition spread followed by a basis change: cx(0,1), cx(0,2), then H on all three data qubits. The H layer is what re-aims the protection: since HZH = X, a Z error striking the codeword acts exactly like an X error in the rotated frame, and bit flips are what repetition codes catch.

Diagnosing Without Looking

Measuring a data qubit directly would collapse the encoded superposition. Instead, the code measures parities: ancilla q3 records whether qubits 0 and 1 agree in the X basis, and ancilla q4 does the same for qubits 1 and 2. Each check is an H sandwich: rotate the pair into the computational basis, copy the parity onto a fresh ancilla with two CNOTs, rotate back. The data qubits are touched only by gates, never by measurement.