Ron Cohen

21 ביוני 2022

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After you convinced yourself that **(QEC) Quantum Error Correction is so important**here and here, you probably thought "Where to begin learning?". First, let's align with the fundamental knowledge that you need.

Generally speaking, I recommend to all of you who are interested in Quantum Computing, to go through the qiskit notebook of IBM, which will give you a great introduction to the field. To those of you who just can't wait to keep reading QEC articles, here is a **pocket guide to the hatching chick.**

##### Qubit

Qubit, just **like a bit**, is a creature that stores a basic unit of 0/1 information. **Unlike a bit**, a qubit can store both 0 and 1 at the same time, where each of the 0 and 1 is stored with some **probability **to be 0 and 1 at the time of measuring the qubit.

We represent those 2 probabilities as 2 numbers in a **vector**, or a sum (a superposition) of |0> state and |1> state at the same time:

Once we **measure **the qubit, it will not be in a superposition of 0 and 1 anymore, but its state will **collapse **into 0 or 1, in a probability that depends on those 2 probabilities (which their sum is 1):

##### Bloch Sphere

A very useful **visualization **of a qubit is the **Bloch Sphere**. It is a sphere, with a radius of 1. Where:

A point on its

**north pole**represents a state that is**pure |0**⟩**.**

A point on its

**south pole**represents a state that is**pure |1**⟩.

**Between the 2 poles**are all the states that are**superpositions**of |0⟩ and |1⟩. The latitude is representing the ratio of probabilities. For example, in the equator, are all the states with probabilities of 50%/50%. Again, you might think that the qubit is in state 0 or 1 and we just do not know yet, but no! The**power**of the qubit, is that it is in**0 and 1 at the same time!**

The

**longitude**(going only around Z-axis) represents the**relative phase**between the |1⟩ term to the |0⟩ term of the superposition. The importance of the phase will be discussed later.

##### 1 Qubit Gate

Once we have a qubit, we can perform basic operations on it, and change it using **qubit gates. **As you can think of a qubit as a 2x1 vector, a gate is a 2x2 matrix, that acts on this vector, and crates another 2x1 vector - the resulted qubit.

I would recommend this tutorial of qiskit, Unless the following is a piece of cake for you:

Now we can think of the mathematical meaning of non-coherent error that was mentioned in the last article:

A

**Bit Flip**noise as an**X gate**that is acting on the qubit.

A

**Phase Flip**noise is a**Z gate**that is acting on the qubit.

##### Eigen States of Operator

Once we know what an operator/gate is, we can learn what is an **eigenstate **of it. Just as in **linear algebra**, a quantum eigenstate of operator A is a state that will not change itself up to a constant multiplication (eigenvalue) when A will be applied on that state. For example, the eigenstates of Z and X:

##### Measure in a different base

Perhaps you thought that once we measure we collapse to a 0 or 1 state. There is a way to measure in a different base - meaning that we **collapse to a state that is neither 0 nor 1**, with probabilities that are different.

The way to do it is **cheating**. The measurement is actually done in a manner of collapse to 0 or 1, but, before measurement and after measurement, we change the state in a way that we choose. For example, if we want to collapse ourselves to an eigenstate of the X operator (|+⟩=|0⟩+|1⟩ or |-⟩=|0⟩-|1⟩), we can use the following trick:

H operator that will change |+⟩ to |0⟩ and |-⟩ to |1⟩

Measure and collapse the state to |0⟩ or |1⟩

Again H operator will change |0⟩ to |1⟩ and |1⟩ to |-⟩

##### Any rotation

By using those 2x2 matrixes, we can rotate our state on the Bloch Sphere, and change it to **any point that we like**.

Now we can think of a small **coherent noise** that was mentioned in the last article, as a** small rotation**, with a small angle on the Bloch sphere. Here are the matrixes that are needed to rotate the state around the x/y/z axis. See how for example, Ry or Rx rotation, will change the probabilities of the 0 and 1 of the state, and Rz will change only the relative phase between them:

##### To Conclude

If you are not familiar with those, I will recommend again to learn and practice this qiskit page.

If you learned anything new, vote 💡and please feel free to ask any question

If it was all a piece of cake for you, but you still want to learn about QEC, vote 🤔

And if you are professional ducks, that liked the explanation, just thumb up 👍

Well done! If you have got so far, **you are ready to step up** to the next fundamentals, which will start to touch QEC concepts.

*Stay tuned...*