Quantum Mindset Blog #1


Quantum Computing Intro



While the concept of quantum mechanics has been around for the last 100+ years, quantum computing is a more recent phenomenon. It originated in the 1980s, when famous physicists, including Richard Feynman and David Deutsch, proposed using quantum systems to simulate classical processes and perform computations with incredible speed.

Currently, quantum computing is being touted as the next frontier in information processing, with the potential to enhance the problem-solving capabilities of modern computers. However, understanding quantum computing can be considerably challenging because it combines high-level computing concepts with considerable amounts of linear algebra.

Rest assured, though, as we will use clear examples throughout this blog series to make complex, math-heavy ideas easier to picture. We will begin with the most fundamental building block of all: the concept of a set.

Sets are convenient and shorthand ways of writing out a list of objects. You can think of them as the mathematical analogue to a basket of items. For example, if we have a basket with the following items — an apple, an orange, and a mango — then we can define a set as:

$$ A = \{apple, mango, orange \}$$

Each member of set \(A\), or any set in general, is also called an element. While our example used fruits, elements can also represent numbers, symbols, or any other object.

Before we can meaningfully explore the quantum world, we must understand how sets and information are used to conceptualize the classical world.

In classical computing, information is commonly expressed in binary, which is a number system consisting entirely of \(0\) and \(1\)s. Therefore, in a similar vein to our fruit basket analogy, we can define the standard classical set as:

where \(\sigma\) is a commonly used Greek letter to represent sets. Each element of any set describing classical phenomena is called a state.

Now that we have clearly defined sets in relation to classical computing, we are ready to tackle the concepts of classical systems.

A classical system is any physical or digital system consistent with the principles of classical physics — think of a traditional pendulum. One of the governing properties of classical systems is the exclusivity of classical states — meaning that the classical system can only be in one state at any given moment. For example, the set \(\sigma\) has two possible states \(0\) and \(1\),but the corresponding system \(\psi\) will be in exactly one of them. This state determinism is what separates classical systems from quantum systems.

Classical systems can be characterized as certain or uncertain, an important distinction that will solidify later once we compare them to quantum systems.

  • A certain classical system is one whose state is fixed and unchanging.

  • Uncertain classical systems never have constant states and are instead described in terms of probabilities until they are measured.

Yay! You’ve made it to the end of my very first quantum blog. Today, we have made major inroads towards understanding quantum computing by covering the basic concepts of sets and connecting them to classical systems.

As we move forward, we will be using vectors, matrices, and complex numbers to ground quantum states and sets in mathematical rigor. If you are unfamiliar with linear algebra, I highly recommend checking out 3Blue1Brown’s Essence of Linear Algebra series. He enhances the abstract concepts of linear algebra using dynamic animations and engaging visuals. For now, focus on the first two videos in the series.

You can start watching here: Essence of Linear Algebra.

To learn more about complex numbers, check out 3Blue1Brown’s excellent introduction, Complex Number Fundamentals. The first free videos should be enough to follow along with my future posts.

Get ready for a quantum leap into my next post!

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