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# 1.2 What can quantum computers do?

Quantum computing offers a new computing approach that is based on the idea that the rules of quantum mechanics can allow shortcuts to solutions for certain tasks. There are quantum algorithms with polynomial run-times that can solve problems where the only known classical algorithms have exponential run-times. Quantum computers make use of the fact that quantum states can be made of linear combinations of individual states and when a measurement is taken, one of the individual states will be measured with a given probability. You will see that entanglement is another fundamental resource for quantum computing.

The integer factorisation problem, which seeks to find the prime number factors of an n-digit integer, is of particular interest because it is the basis of a lot of information security. The run-time of the classical algorithm scales exponentially so as n increases a classical computer takes longer and longer to factorise the integers as n increases. The integer factorisation problem is an example of a a problem that quantum computers can solve much more efficiently. If quantum computing was able to solve the integer factorisation problem with a polynomial run-time then there would be a major problem for information security.

Another example of a problem with an exponential run-time is the travelling salesperson problem, which seeks to find the shortest route between a number of cities subject to visiting each city only once and returning to the starting city at the end of the journey (see Figure 2). This is a typical optimisation problem which can be used to demonstrate the power of quantum computing as well as being a problem which delivery companies would like to be able to solve quickly.

Figure 2 An illustration of the travelling salesperson problem. The image on the left shows 7 cities (in blue) and a trial of all 360 possible routes between them (in red). The image on the right shows the optimum distance after trialling all possible routes.

## Exercise 2

Write a sentence or two to summarise in general terms the context in which quantum computers are considered to be an improvement on classical computers.

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### Answer

Your answer will not be the same but a possible answer is:

Quantum computers may be able solve some problems more quickly than classical computers if problem solving algorithms which have exponential run-times on a classical computer can be written to have polynomial run-times on a quantum computer.

Your answer should include the same conclusions.

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OpenLearn - Introduction to quantum computing
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