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Error Correction with the Shor Code


Introduction

This project extends your work in Week 21 to implement, simulate, and analyze the first ever error correction code proposed that can handle any single qubit errors (bit-flips, phase-flips, or others) using the Shor code (Sho95).


Description

The Shor code is a natural extension of the Bit- and Phase- flip codes that avoid syndrome measurements where:

  • A bit-flip error means $|0\rangle$ unintentionally turning into $|1\rangle$ or vice versa.
  • A phase-flip error means $|+\rangle$ unintentionally turning into $|-\rangle$ or vice versa.

This code can be viewed as two levels of quantum error correction where:

  1. The logical qubits are first encoded into a 3-qubit phase-flip code.
  2. Each of the resulting physical qubits are encoded into a 3-qubit bit-flip code.

Key Questions

In this project, you will answer the following questions:

  1. How does using the Shor code compare to using no error correction? Are there any trends you notice?

  2. When using the Shor code, how does the average fidelity change with increasing probability of errors? Why is this?

  3. Are there any unexpected patterns in the average fidelity? Why do you think they might be happening if so?


Structure

This project is broken into 3 parts:

Part 1: Defining the Components

Part 1.1: The Phase-Flip Code

Part 1.2: The Bit-Flip Code

Part 2: Implementing the Shor Code

Part 3: Analyzing the Shor Code

Wrapping Up


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