**Qubit** (**Quantum Bit**) is the **fundamental unit of information** in **quantum computing** as like bit which is the basic unit of information in classical computing. However, unlike bit in the classical computing which can exist in one of two states such as 0 and 1 at any given point of time, the qubit can be found to exist in states 0, 1 or the **superposition state** which, simply speaking, can be said to be both 0 and 1.

In this post, you will learn some of the following:

- Qubit and Superposition State
- Superposition state explained with examples

### Qubit and Superposition State

The following are some of the important points about the state of Qubits:

- The qubit can be found to exist in the state of 0, 1 or the superposition state which represents the state of both 0 and 1.
- However,
**when measured, Qubit would found to be in only one of the two states 0 and 1 and not the superposition state**. This is very important to understand.

The state in Quantum theory is represented using **|** and **>**. However, for the sake of understanding, we will use vector notation to represent the 0 and 1 state. Thus, the following holds true:

- The state of 0 can be represented using [latex]\vec{0}[/latex]
- The state of 1 can be represented using [latex]\vec{1}[/latex]
- The superposition state can be represented as the following:

[latex]\vec{\gamma}[/latex] = [latex]\alpha\vec{0} + \beta\vec{1}[/latex]where [latex]\alpha[/latex] and [latex]\beta[/latex] are complex numbers and [latex]\vec{\gamma}[/latex] is the superposition state of state [latex]\alpha\vec{0}[/latex] and [latex]\alpha\vec{1}[/latex]. Recall that the complex numbers are of the form,*c = a + ib*, where i = [latex]\sqrt{-1}[/latex].

As per the quantum mechanics, the **modulus squared** of [latex]\alpha[/latex] and [latex]\beta[/latex] represents the probability of finding the qubit in state [latex]\vec{0}[/latex] and [latex]\vec{1}[/latex] respectively.

The fact that probability must sum upto 1 leads to following equation which could be used for determining value of [latex]\alpha[/latex] and [latex]\beta[/latex].

[latex]|{\alpha}|^2 + |{\beta}|^2 = 1[/latex]

One needs to note that if [latex]{\alpha}, {\beta}[/latex] are real complex numbers with imaginary coefficients such as (x + iy), the way to find the modulus squared is to multiple the complex number with its conjugate. This esssentially means that modulus squared for complex number [latex]x + iy[/latex] is following:

[latex](x + iy)\times(x – iy)[/latex]

### Superposition states explained with examples

**Example 1:** **Let’s determine if the below can be said to represent a valid superposition state of a qubit:**

[latex]\frac{1}{2}\vec{0} + \frac{\sqrt{3}}{2}\vec{1} [/latex]

The coefficient of [latex]\vec{0}[/latex] is the complex number, [latex]\frac{1}{2}[/latex]. The modulus squared of the coefficient, [latex]\frac{1}{4}[/latex], is the probability that state 0 will happen.

The coefficient of [latex]\vec{1}[/latex] is the complex number, [latex]\frac{\sqrt{3}}{2}[/latex]. The modulus squared of the coefficient, [latex]\frac{3}{4}[/latex] is the probability that state 0 will happen.

Going by above, the following gets evaluated:

[latex](\frac{1}{2})^2 + (\frac{\sqrt{3}}{2})^2[/latex]

[latex] = \frac{1}{4} + \frac{3}{4} = 1[/latex]

Thus, it could be said that [latex]\frac{1}{\sqrt{2}}\vec{0} + \frac{1}{\sqrt{2}}\vec{1} [/latex] is a valid superposition state.

**Example 2**: **Let’s consider another example and check whether it is a valid superposition state:**

[latex]\frac{1}{\sqrt{2}}\vec{0} + \frac{1}{2}\vec{1}[/latex]

The sum of modulus squared of coefficients (probability) of states should be equal to 1. Let’s check this out.

[latex](\frac{1}{\sqrt{2}})^2 + (\frac{1}{2})^2[/latex]

[latex] = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \ne 1[/latex]

Thus, the superposition state represented by above is not a valid superposition state.

### Further Reading / References

- Quantum Superposition
- Book – Quantum Computing Explained (David McMahon). It is a great book for learning quantum computing. If you are planning to get deeper understanding of the book, you may not want to miss this one.

### Summary

In this post, you learned about some of the following:

- Introduction to Superposition states of Qubit
- Superposition state explained with examples

Did you find this article useful? Do you have any questions about this article or **understanding superposition states of Qubit in Quantum computing**? Leave a comment and ask your questions and I shall do my best to address your queries.

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