Your Guide to Quantum Algorithms - Deepstash

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## What is Quantum Computing?

Quantum Computing is a form of computing that takes advantage of quantum mechanics to process information exponentially faster than classical computers.

Classical computers use classical bits which can consist of either a 0 or a 1 to encode information.Quantum computers,on the other hand,qubits, like photons,atoms, ions, etc. to encode information into 2 distinguishable quantum states .

Now you may be thinking, what’s the benefit of using qubits over classical bits? The answer lies in the ability of qubits to behave quantumly, specifically the quantum phenomena of superposition and entanglement.

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## Superposition

When a quantum system is in superposition, it can be a 0 or a 1 or a combination of both, at the same time. This allows a quantum computer to process information at a significantly higher rate than classical computers.

For example, 4 regular bits can only represent 1 of the total 16 combinations at one time. 4 qubits in superposition, however, can be all 16 combinations at once!

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## Entanglement

Entanglement is when a pair or group of quantum systems are strongly correlated, giving them the ability to be perfectly in unison, no matter how far apart they are. This means that quantum computers only need to measure 1 qubit and to figure out the value of the other qubit in the pair instantaneously because they are a part of the same entangled system.

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## Quantum Algorithms

An algorithm is essentially a series of steps to solve a problem. However, these steps are limited by the hardware on which the algorithm is being run on.

For example,let's say we have written down the steps needed to find the derivative of a polynomial function. If we gave these directions to a mathematician they would easily be able to arrive at the correct answer, since they know calculus. If we gave these directions to a student that is currently learning calculus, they would also be able to arrive at the correct answer, but it would probably take a lot more time since they just barely have

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## ...

the experience/knowledge needed. But, if we gave these instructions to a kindergartner, they would be very lost and wouldn’t arrive at an answer. There would probably just be scribbles on the page.

In this example, the steps to calculate a derivative represents an algorithm, and the various people using them represent varying degrees of computing software. Just like how as the person became less and less knowledgeable on physics the efficiency of the steps to be carried decrease in the above analogy, so do computer algorithms.

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## ...

Certain algorithms take much longer on classical computers that obey the laws of macro physics because their hardware may not be able to carry out some of the steps efficiently.

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## 1. Simon’s Algorithm

Simon’s problem was one of the first computational problems to prove that a quantum algorithm could solve a problem exponentially faster than a classical algorithm.

This algorithm, although not providing much practical value on its own, inspired the Quantum Fourier Transforms in Shor’s algorithm, one of the most famous quantum algorithms of all time!

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## 2. Shor’s Algorithm

Shor’s algorithm is by far one of the most famous quantum algorithms of all time, as it can factor integers in polynomial time. It was invented in 1994 by Peter Shor to solve the problem of finding the prime factors of a given number, N .

Shor’s algorithm even has the potential to break modern public-key cryptography, like the widespread RSA cryptosystem, on an ideal quantum computer. RSA relies on the impossible nature of factoring the product of two prime numbers for a large enough number.

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## ...

However, this scenario is unlikely to happen in the near future because we still have a lot more progress to make in lowering quantum noise and quantum decoherence in current quantum computers.

Still, Shor’s algorithm is an extremely efficient project in giving you a hands-on experience with using a quantum Fourier transform especially for those that are interesting in the intersection of QC and Cryptography!

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## 3. Deutsch-Jozsa Algorithm

The Deutsch-Jozsa Algorithm was to showcase how quantum algorithms can be exponentially faster than any possible deterministic classical algorithm.

The algorithm itself doesn’t provide much practical use besides being specifically designed to be easy for a quantum algorithm and hard for any deterministic classical algorithm.

The D-J problem consists of an oracle that implements a hidden boolean function, which takes as input a string of bits, and returns either 0 or 1.The D-J quantum algorithm produces an answer that is always correct with a single evaluation of the function.

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## 4. Grover’s Algorithm

Lov Grover created this algorithm to solve the problem of an unstructured search. It can find the unique input to a black box function that produces a particular output value, using just O(sqrt N) evaluation of the function, N being the function’s domain.

In other words, let's say we had a shuffled deck of cards and were tasked with finding 1 specific card. The classical algorithm would solve this by going through all the cards in the deck one by one, or N evaluations. On the other hand, Grover’s algorithm would only require O(sqrt N) evaluation to do the same job.

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## Continuing

Unlike the previous quantum algorithms mentioned, Grover’s algorithm only provides a quadratic speedup in evaluation time for unstructured searches, compared to their exponential speedup. Still, the amplitude amplification trick employed in Grover’s algorithm is extremely useful when trying to obtain quadratic run time improvements for a variety of other algorithms.

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## 5. Bernstein-Vazirani Algorithm

The Bernstein-Vazirani Algorithm was invented by Ethan Bernstein and Umesh Vazirani in 1992. It is a restricted version of the Deutsch–Jozsa algorithm.

The algorithm was created to solve a is

So, let’s just say that we are given a box . Hidden in the box is a secret number. This “secret number” is represented by six bits made up of 0’s and 1’s. We need to figure out what the “secret number” is.

Classically, a computer would find it most efficient to calculate the “secret number” by evaluating the function n times, where x = 2^i and i is the summation of 0, 1, … n-1.

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## Continuing

Now imagine if you could find out what the secret number is in one try, no matter its size. That’s exactly what running the Bernstein-Vazirani algorithm on a quantum computer allows you to do.

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