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The reality about a quantum computer is that it can do some problems like prime factorization very fast, but it can’t perform as well as a normal computer in solving many other problems.

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Let’s try to understand the matter in more detail. In order to solve a problem, the computer has to perform some calculations in many steps which we call algorithm. The lower the number of steps in the algorithm, the less calculations are required and the faster the computer can solve the problem. Now the number of steps depends on the size of the input. If the input is of size n and takes n^2 steps then that algorithm is called O(n2) algorithm (read order of n square algorithm). If the size of input is 100 in O(n2) algorithm, then the problem can be solved in maximum 10000 steps. Just like that can be an O(n), O(logn) algorithm. These are called the time complexity of the algorithm, which tells us which algorithm will work faster.

Now some questions, if you have two algorithms, one O(n^2) and one O(n3), which one will solve the problem faster? What is the maximum number of book titles you should read if you have to find one book out of n books? How much is the complexity of finding books? Now a question to think about, if there are 100 words in the dictionary, then how many steps can you find a specific word if you search wisely? If there are n words in place of 100?

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Now n2, n3, nk are all polynomial complexity. There is another type of complexity called exponential complexity, which is of size kn, ie 2n, 3n. As the value of n increases, the number of steps in the algorithm increases, the number of steps in the exponential algorithm increases much faster than the number of steps in the polynomial algorithm. Do you remember a story where the little boy asked his mother for 1 rupee the first day, 2 rupees the next day, 4 rupees, 8 rupees, 16 rupees in the following days for 1 year? That means 2n has to be paid on the nth day. See the table below how big the number becomes after 50 steps if we increase it like this:

The step number for the 2n algorithm becomes a 16-digit number when the input size is only 50.

Unfortunately, there are many real-life problems for which we don’t know anything better than exponential algorithms. Scientists call these np or non-deterministic-polynomial category problems. If one could find a polynomial solution to a problem in this category with 100% certainty, the face of the world would change at that moment, the “holy grail” of computer science. More interesting thing is that if you can solve even 1 np problem then all np problems will be solved. Currently, in this type of problem, the best one is selected by looking at all possible outcomes and the steps are reduced by imposing various conditions.

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Now a supercomputer might be a few thousand times faster than your average computer, but they’re also 2100 times faster.

Putting in a stepwise algorithm will end the galaxy lifespan but not solve the problem. Supercomputers can therefore work faster than ordinary computers, but cannot reduce the complexity of algorithms. We need a computer that can minimize the number of steps in the algorithm. Only then we can solve the problem of np category quickly.

Now the question is whether a quantum computer can solve the np category problem? Sadly, the answer is no until now. So even quantum computers cannot do better than ordinary computers in these problems.

So what problem would a quantum computer be good at solving? A qubit can be either 0 or 1 just as a bit in a normal computer can be 0 or 1. But the interesting thing about a qubit is that it can simultaneously exist in a state of 0-1 combinations called superposition. If we have 1000 qubits, it is possible to hold 21000 numbers simultaneously which is more than the number of molecules in the visible universe. Now if we had an algorithm that would do an operation on the qubits at the same time to make all the numbers into one possible answer, we could find the correct answer very quickly. But the problem is that when we try to see which state the qubits are in at the end of the algorithm, we will only get 1 state, according to the rules of quantum mechanics, we will not be able to read the rest at all. [1]

But there is something called interference that we can use to some advantage. First, let’s look at a picture of the amplitude of the wave:

Now let’s look at interference[6]:

In the upper right image positive and negative amplitudes combine to cancel each other out, while in the left image similar amplitudes combine to increase the amplitude. If we can create an algorithm that cancels out wrong answers through destructive interference and increases the amplitude of correct answers, the probability of getting the correct answer in the final state will greatly increase. [1]

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There is an algorithm for analyzing prime factorization or prime product using this property which is Shor’s algorithm. This algorithm is O(n3)

Step n

K can be written as the product of some prime numbers, which cannot be done in polynomial time on classical computers (but this is not a problem of the np category). There are many protocols in cryptography, and on the principle that ordinary computers cannot quickly prime factorize large numbers, quantum computers will be able to break these protocols very quickly. [4]

At the beginning I asked a question that n

How many book titles should be read to find out 1 book from the book? The answer is very simple, the book can be at the end of all so only n book titles need to be read, complexity is O(n). Until now, we have assumed that finding the data will take O(n) time if the data is not sorted in a particular order (eg, from smallest to largest). A quantum algorithm called Grover Search has been shown to have O(squareroot(n))

Or the square root of n steps to find data from any database! [3]

But breaking cryptography is not the only task of quantum computers, there are other great possibilities. With quantum computers we can simulate chemical reactions, we can figure out how an atom reacts with whom. Since nanotechnology is dependent on quantum mechanics, quantum simulation will play a very important role there as well. [2] At that time, we may be able to simulate the effectiveness of new drugs on computers without testing them on animals. However, according to MIT’s Scott Aronson, research into quantum computing, even if it turns out that building a quantum computer is not possible, we will gain new insights into how the universe works. [1]

Where is the problem in making a quantum computer? The main problem is quantum decoherence [1][5]. The qubits interact with the environment to destroy the state they were in, causing what physicists call “wave function collapse”. We know that qubits can exist in multiple states of superposition at the same time, causing decoherence to “collapse” into a single state. And once “collapsed” it cannot be undone. This is one of the biggest hurdles in building a quantum computer.

[1] Limitations of Quantum Computer – Scott Aaronson, Associate Professor at MIT (Published in 2008 SCIENTIFIC AMERICAN)

[2] Quantum Theory – Tanvirul Islam, free-spirited blogger and quantum researcher

[3] Grover search

[4] Shors algorithm

[5] Decoherence

[6] Interference

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