The hype on quantum computing is very real, one can see the so called “developments” or some crass comments on the idea of quantum computing. So much that so that it shows up a few times a week atleast. How much of the hype is real and how much is not is a story which is long and something I am not going into. What I am indeed interested in is why are so many people interested in it. Many governments have poured in resources to advance the research in quantum computing seeing its far reaching consequences. But, what are these consequences and why is quantum computing so consequential are the questions that I am going to address.

You might ask, what is the need for another article when there are hundreds online already? Most of the resources that I come across are either too full of hype or just plain not so helpful. Some of them which are very good(I have linked some at the end) seem a bit incomplete to me. I will try to give a slightly wholistic picture of quantum computing in my perspective.

Beyond the Paradigm of Classical Computing

The classical computing has been a holy grail to many fields of science and have revolutionized the world. There is no telling as to the implications from banking to medicine which deeply impacted our daily lives. But, there seems to be a limit to the kind of problems that can be addressed with classical computers.

If we analyze the way we go on with our lives, there is one factor which played a great role. The amount of information we possess and communicate. Since the past few decades, the amount of data a person has has expanded exponentially. Maybe from a few Megabytes (assuming that the paperwork in a general household as information) to many gigabytes at present. We exchange a lot of information unknowingly every text we send, every pictue we see or every site we visit is nothing but a transfer of information. This has only been possible with the help of classical computers. They have been instrumental in increasing our ability to handle large amount of data or in a simple way they are good at numbers. This ability helped us understand and solve many problems which were intractable by hand calculations. This does not mean that all the problems are tractable by computers. For some problems, like finding the most efficient schedule for a company like FEDx, classical computers are very slow.

The problems that we have can generally be classified into different kinds - P-Problems (We know how to solve them efficiently), NP-Problems (We do not know how to solve efficiently). An example for the P-Problems is the numerical evaluation of Fast-Fourier-Transform, and for an NP-Hard problem is to find the number of ways a given protein can be folded. This problem is NP-hard as we know how to solve it for small molecules, but if the molecule becomes large which is most generally the case with a few hundred thousand atoms, the problem is almost impossible to solve. No matter how fast the classical computer becomes, the problem is intractable. This is the juncture where we can start to ask the question, is classical computing the only way we can compute?

The idea of Quantum Computation

There are many such NP-Problems which physicists come across. The calculation of electron wave functions in an atom, inter-atomic forces, understanding collisions between subatomic particles or calculations in quantum-chromodynamics and astronomy.

It was Richard Feynman during his famous lecture at a conference organised by IBM and MIT in 1981 that he theorised a new paradigm of computation, saying “Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.”

The understanding of microscopic world in the language of quantum mechanics was not new during the time. It was clear that calculating the behaviour of electrons inside an atom is far more complicated compared to problems in classical mechanics. Feynman thought, why not use atoms to simulate the real world. As atoms are as complex as nature itself, they will be able to show us a better picture of reality than classical computers which are a lot simpler in comparison. But that’s not the end of the story, he also recognised that the same computers can be used to solve a lot of problems in day-to-day human life which seemed impossible to solve with a classical computer.

Before digging into the story of why quantum computers are interesting, taking a short look at the nature of quantum mechanics is useful. It helps us to better appreciate the complexity and potential in quantum computation.

Quantum Quirkiness!

I am not going to bore you with the gory math equations or its history. We need to understand a couple of properties that are essential to get a grasp of why the quantum world is so different, and then we’ll get back to continue the above discussion.

Quantum mechanics, like any other theory, starts with some basic set of rules or postulates. I am only going to talk about the few that we need for this discussion.

Measurement Postulate

First, the postulate of measurement. In our daily lives measurement is a process which does not affect what is being observed. We take a look at our watches and that does not mean the watch is going to change it’s time. This is no longer true for a quantum particle. The moment you make an observation to a particle the particle falls or collapses into one of the allowable states whose value we will measure. After a measurement, the particle stays in the same state whose value was observed and any successive measurements would give the same result. This is a bit muddy, let me clarify it for a bit.

Taking, for an example, an experiment where want to measure the energy of an electron in an atom. Accoding to quantum mechanics, the electron cannot have any energy on the real line, instead it is only allowed to be in states which have a certain and have a very particular energy. (Why is this so, is a question with a long answer. You can leave it to quantum mechanics or visit the Wiki page to get some idea). The quirky part is that we do not know in which state the electron is in untill we make a measurement! In other words, the electron can be in any state or position but it is impossible for us to know. The moment one makes a measurement, the electron randomly picks up a state from the range of allowed states and the energy corresponding to that state is what shows up.

“Superposition”

As remarked earlier, a quantum particle has a certain number of allowed states. This postulate allows the particle to be in any possible superposition of the allowed states. If we denote the $n$ states with symbols $| \psi_i\rangle$, then the particle can be in a state

$$\Phi = \sum_i a_i |\psi_i\rangle$$

Where $a_i$'s are $n$ complex numbers.

Probability of Measurement

Even though the randomness is baked right in, the state that the electron might pick up due to a measurement is not completely random. The laws of quanutm mechanics give us the probabilities of which state the electron might collapse into. These probabilities are calculated using linear algebra. Further, if we are able to get the probabilities of measurement for each of the allowed states we will have solved the problem entirely. So, if the electron in the above example has $n$ allowed states and we denote the probability of measurement for state $i$ with $p_i$, we need n numbers to totally describe the system. But from the basic probability theory we know that sum of all the possible probabiities should be $1$,

$$\sum_i p_i = 1$$

This equality imposes the condition that the complex numbers $a_i$ are normalized according to the equation

$$\sum_i |a_i|^2 = 1$$

This implies that we have $n$ unknowns along with one equation, therefore, we would need $(n-1)$ numbers to describe the system.

If you were able to get a slight gist of what is happening in quantum with the above description, you’re set for the sake of this article. If you want to understand more of it’s wonders, you should definitely watch this lecture by Jim Al-khalili at Royal Society. Now, lets see how this ties up into quantum computing.

Quantum Advantage

To understand the advantage that is gained by changing to quantum lets us compare the to number of values required to describe a classical computer system with $n$ bits. We know $n$ bits can take up $2^n$ different values, but each state can be described with $n$ numbers or bits in this case.

Let us consider a quantum version of a bit - Qubit, that is a quantum system with just two states. Then as we saw earlier, we need exactly $(n-1)$ or one number to describe it. Things get interesting when we consider 2 qubits together. Each of the qubit can take two states. We can denote the states of first qubit by $|\psi^1_{1}\rangle$ and $|\psi^1_{2}\rangle$ and the second qubit by $|\psi^2_{1}\rangle$ and $|\psi^2_{2}\rangle$, where the superscript marks the qubit at hand.

Since each qubit has two states, the system of two qubits as a whole has the following 4 states $|\psi^1_1\psi^2_1\rangle$, $|\psi^1_1\psi^2_2\rangle$, $|\psi^1_2\psi^2_1\rangle$ and $|\psi^1_2\psi^2_2\rangle$. From the above postulates, we can understand that the system can now stay in any superposition of the above four states.

$$\Phi = a_1 |\psi^1_1\psi^2_1\rangle + a_2 |\psi^1_1\psi^2_2\rangle + a_3 |\psi^1_2\psi^2_1\rangle + a_4 |\psi^1_2\psi^2_2\rangle$$

with the same condition of normalization for $a_i$'s. Therefore, we would need $(4-1)$ that is three numbers to describe a system of $n$ qubits. This can be extrapolated to see that for a system of $n$ qubits, we would need $2^n - 1$ numbers to describe it. Whereas for a classical computer with 2 bits, we would only need $2$ numbers. When n keeps on inreasing, the difference increases rapidly. When we increase the number to 300 where the classical analogue would need 300 numbers, quantum system would need $2^{300}-1$ numbers to completely describe it, this is the famous number which reaches the estimate of the total number of atoms in the visible universe. This shows us that the space spanned by a quantum computer is much larger compared to a classical computer which represents its immense potential.

From the above discussion, it is clear to see that the advantage gained in a quantum computer is due to the superpostiion principle which allows the system to be in a state which is a random mixture of the allowed states. This superposition is what gives the huge increase in the space spanned.

Now, let us suppose that we try to come up with a set of operations to perform on a state for any given calculation. Since, in the quantum domain we can also prepare a state which is a superposition of all the states the operation is performed directly on the superposition and hence on all the states in the superposition at once. You might think that the result is also a superpositon and it might not be useful as we need a single state to know what the final answer is. It is possible to come up with operations which will result in the attenuation of all the undesired states, and thus leaving us with the desired result. This gives a huge performance edge as a classical computer can only perform an operation on a single state but not on the superposition. This advantage is very clearly explained in this video by MinutePhysics, which is very instructive.

Thus, a quantum computer would be very powerful in solving some very specific set of problems. As explained in this video by Veritasium, it is not going to replace our classical computers. As I remarked earlier, the classical computers are good at handling large data sets quickly, for example sharing photographs or watching a video online. A quantum computer is not good at that, what it is good at is to perform some type of calculation in a much faster fashion. This capability is invaluable in medicine for protein analysis or vaccine discovery or analyzing stocks and trading. More importantly, they are particularly good at working on optimization problems, for example adjusting the parameters in a food processing facility to maximise the produce per day. This kind of a problem has so many control parameters that it is hard to analyse each case individually. But, a quantum computer can readily analyse all the possible results at the same time thereby reducing the computation time required by a few orders of magnitude.

A few other sources which I found to be useful: