## Introduction to Perfect Quantum State Transfer

Many aspects of the world of our everyday experience, which we call the classical world, are just so familiar that the way they work seems self-evident. It seems almost impossible to conceive of how things could function any differently. This is a trap that the pioneers of computing science fell into. Modern computer science is founded on bits - entities which take on the values 0 or 1. You can look at them, make copies of them, and act based on their values in order to develop interesting algorithms.

However, the apparently self-evident basis of computer science predicts that certain operations are impossible. For instance, think of the NOT gate. It looks at the value of an incoming bit, and outputs the opposite value. One can ask "is there such a think as the square root of NOT gate", by which we mean a gate which takes an input, and produces an output which, when fed back as the input to another identical gate gives the same output as NOT would have given. Such a thing does not exist, and cannot exist, in a modern computer. However, by exploring different extremes of the physical world, where one has to modify the classical laws of physics to explain the results of experiments, then one can find a device that implements the square root of NOT operation. This extreme is the regime of Quantum Mechanics.

Suddenly, when one has a new computational gate, you need to rewrite the theory of computation, and this gives us Quantum Computers. Rather than bits, these operate on qubits, which take on not only the values 0 and 1, but potentially some superposition of the two. Quantum Mechanics is very strange, and behaves very differently from the world of our intuition. For instance, it is impossible to perfectly copy qubits. The problem is that quantum mechanical effects are very fragile, and it is extremely easy to get things just a little bit wrong, and return to the classical regime. This is a major challenge for experimentalists who are trying to build a quantum computer.

There is now well established theory relating to sufficient requirements for being able to implement a quantum computation. In particular, if one can operate on single quantum bits (qubits), and between arbitrary pairs of qubits, then arbitrary quantum computations can be realised. Unfortunately, most experimental designs don't allow for direct interaction between arbitrary pairs of qubits, but only between those that are close to each other. The ideal situation is conveyed in the "No Restrictions" choice in the animation below, where we can just take one qubit, and move it wherever we want.