Follow us
Ron Cohen
21 ביוני 2022
.
After you convinced yourself that (QEC) Quantum Error Correction is so importanthere 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...
