QCQI – Chapter 9: Quantum Error Correction & Noise

Course Slides (MSU)

Big Picture¶

Noise $\Rightarrow$ Pauli basis; K–L conditions; stabilizers $\Rightarrow$ syndromes; distance & decoding; small codes; CSS/Steane; FT glimpse.

K–L Conditions¶

$\langle \bar{i}|E_a^\dagger E_b|\bar{j}\rangle = C_{ab}\delta_{ij}$ ensures no logical info leaks.

Eureka! All damage lives in the syndrome register.

Stabilizers in Practice¶

Pick commuting generators, measure to get bits, map syndrome $\to$ correction.

Eureka! Commutation $\Rightarrow$ simultaneous readout without collapsing the logical state.

Toy Codes¶

Repetition/bit-flip; phase-flip via $H^{\otimes 3}$; Shor $[[9,1,3]]$ idea; CSS/Steane intuition.

Distance $d$ controls detection/correction power.

Syndrome Schematic¶

Hands-on Notebook¶

L13_QCQI_Ch09_Quantum_Error_Correction_and_Noise

Set backend $\in$ {cirq, pennylane, braket, pyquil, qiskit}.

Mini-Lab Ideas¶

• Logical error vs. physical error $p$ (repetition-3).

• Phase-flip variant via $H^{\otimes 3}$.

• (Stretch) Shor encoder/decoder demo.