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Shor's Error Correction Code

The first quantum code to correct arbitrary single-qubit errors

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Phase Flip Correction

Introduction to Shor's Code

Shor's 9-qubit code was the first quantum error correction code capable of correcting arbitrary single-qubit errors. It combines bit flip and phase flip codes through concatenation.

Code Parameters:
[[9, 1, 3]] code
9 physical qubits → 1 logical qubit
Distance d = 3

The Challenge

✓Bit Flip Code

  • →Corrects X errors
  • ✗Vulnerable to Z errors

✓Phase Flip Code

  • →Corrects Z errors
  • ✗Vulnerable to X errors

Arbitrary Single-Qubit Errors

Any single-qubit error can be decomposed into Pauli operators:

I
No Error
Identity
X
Bit Flip
|0⟩↔|1⟩
Z
Phase Flip
|+⟩↔|-⟩
Y
Both
Y=iXZ

Why 9 Qubits?

3
Bit Flip
Each block uses 3 qubits for bit flip protection
×3
Blocks
Three blocks needed for phase flip protection
=9
Total
Final concatenated code uses 9 physical qubits