## QuBits & QuBobs

QuBobs help us explain the basic concepts of quantum computing.

### What is a qubit?

A qubit is the unit of information of a quantum computer. A collection of qubits comprises the memory of the computer. A classical bit can be 0 or 1, which we then can use to encode data in binary, that is, as a sequence of 1s or 0s.

We represent each qubit as a colored disk. The color blue represents 0, and orange represents a 1.

#### Superposition

A qubit has two elementary states that correspond to classical values 0 and 1. A qubit can be in a superposition of these states, that is, it is “simultaneously” 0 and 1. This is similar to a coin that can land on heads or tails. As long as we have not looked at the outcome, our state of knowledge is that it is “both” heads and tails. Just like a biased coin, a qubit can have arbitrary proportions of 0 and 1. We say that the qubit is in superposition of the basis states. Just like randomness, superposition allows us to explore the space of solutions when systematic exploration would be too costly, but where favorable configurations are sufficiently likely for randomness to help us find one.

The proportion of 0 and 1 in the superposition is represented by the proportion of each color on the disk. A handcrank allows us to turn the “qubit” to illustrate the fact that it is in superposition of both of the basis states.

#### Amplitude

The amplitude of the basis state 0 is a coefficient that determines the proportion of the qubit that is in state 0, similarly for the amplitude of 1. Amplitudes can be positive or negative, If there were no signs then we would be describing classical randomized bits, which have nothing quantum about them.

The proportion of blue represented isthe square of the amplitudeof the basis state 0 . This proportion is always between 0 and 100%. The sum of the proportions of 0 and of 1 is 100%.

#### Observation

In order to obtain the result of a quantum computation, it is necessary to observe the system to read off the value that was computed. It is not possible to recover the precise amplitudes of a qubit, just as it it not possible to know the exact bias of a biased coin. We may, however, make an observation of the qubit, that is, obtain 0 or 1 with a probability that is equal to the square of the corresponding amplitude. Unlike a biased coin which has the same bias after each trial and observation, the qubit is much more fragile, Once the observation is made, the amplitudes are lost and the qubit becomes 100% the value that was observed.

To illustrate observation, we turn the hand crank and stop at a random time and look at the color that is under the observation window. This results in a sample of one of the two values with a probability equal to its proportion.

#### Sign and interference

Just as any number has two possible square roots, one negative and one positive, an amplitude can be positive or negative. However, this sign is invisible to the observer making a direct observation of a qubit. Yet the sign is crucial in quantum computation, as it is taken into consideration in all the operations that are made on the qubits during a computation. Each basis state in a superposition can have a positive (+) or negative (-) sign. The sign is also called *phase*. Two basis states can cancel as a result of opposite signs. This phenomenon is called interference. It is used in quantum computing to help useful configurations emerge, and to attenuate unwanted configurations.

The sign of a basis state is represented on a second wheel. The sign visible in the round window indicates the sign of the bases state visible in the qubit's observation window.

#### Normalisation

Just as probabilities sum to 1, the squares of the amplitudes of the basis states 0 and 1 sum to 1. What is important is the proportion of each color. Bringing the sum to 1 is called normalisation.

Since we only consider the proportion of each color, normalisation consists of assuming that the total area of the disk is 100%.

### Computation on several qubits

A quantum computation requires the use of several qubits. There is no hope of beating a classical (randomised) computer unless the qubits get *entangled* in the course of the computation.

#### Two qubits

If each qubit has two basis states, 0 and 1, a pair of qubits has four basis states. 00, 01, 11, 10. A pair of qubits is therefore described by giving four coefficients (amplitudes) that determine the proportion of each of the four configurations.

We represent a pair of qubits by two disks. The way we arrange the disks and make them rotate will determine the proportion of the four configurations.

#### Separable qubits

We say two qubits are separable if they behave as two independent systems. This is analogous to throwing two biased coins. The result of the first coin does not have any influence on the result of the second coin (and vice versa).

Two separable qubits are represented by two disks that are activated by independent hand cranks. The observation of one qubits does not affect the observation of the second qubit.

#### Entangled qubits

If two qubit are not separable, they are called entangled. Without entanglement, no quantum advantage is possible.

Two entangled qubits are activated by a single hand crank and a pulley belt at the back of the machine that joins them together. When the hand crank stops, we can observe the first qubit. Assume we have observed a 0. This allows us to re-evaluate the proportions for the second qubit. This phenomenon is entirely similar to classical probabilities and conditional probabilities.

#### The signs of a pair of qubits

As for the single qubit, the sign of a pair of (entangled) qubits is essential to obtain a faithful representation of a pair of qubits. Without the signs, we would be representing a probabilitic system which requires no quantum technology. Each pair of values 0 0, 01, 11, 10 has a sign (+ or -).

The upshot: the two fundamental characteristics that give quantum computation its power are entanglement and signs.

### More to come : How does a quantum computation work ?

We'll explain the following applications of quantum computation – all without any equations

- Quantum circuits
- Quantum gates
- X gate (classical negation)
- Z gate (sign flip)
- CNOT gate (introduces entanglement)
- Hadamard gate (introduces superposition and signs)

- Quantum teleportation of a qubit
- Deutsch's and Deutsch-Jossa's algorithms to illustrate interference
- Grover's unordered search algorithm
- Quantum non-locality

Also forthcoming : the mathematics behind our representation, and why it works.