|> is called the ket, and in multi-qubit gates there are now 2 numbers inside it giving the states |00>, |01>, |10> and |11>. The first number is the state of the 2nd qubit and the second of the 1st qubit - ie it's back to front. It is easy to remember by thinking that the right most number is always the 1st qubit.
Though there are 2-qubit gates, there are not any higher order gates due to the technical challenges of building them. Higher order operations are built up from 1 and 2 qubit gates.
Any multi-qubit logic gate can be made up from CNOT and single qubit gates
Similar to an if statement. If the control bit (the right most bit) is 1, then flip the other bit else leave it. This gives it the 'Truth table' below :
| Before | After |
|---|---|
| |00> | |00> |
| |10> | |10> |
| |01> | |11> |
| |11> | |01> |
NB in the composer the small circle represents the control qubit, and the large circle with a plus represents the affected qubit
CNOT gates are difficult to make, so their numbers should be kept as low as possible.
These are classical gates that can also be implemented on a quantum computer due to the fact they already have an inverse - the gate itself. A Toffoli gate can be thought of as a CCNOT gate - so a NOT gate with 2 control wires. This is implemented as 3 inputs - a,b,c - and 3 outputs - a,b, c XOR (a AND b), and workign through an example it can be see how the inputs can be recovered from the results (via another pass through the gate).
This principle can also be expanded for example a NAND gate can be implemented when c = 1. (It can be helpful to draw these circuits out to work through examples)
Entanglement is not technically a gate, but can be induced by a combination of gates. Entanglement causes 2 qubits to behave in ways which are individually random, but not independent.
The state (|00> +|01>)/ root 2 is not entangled as the 1st qubit (right most) is in the superposed single qubit state (|0> +|1>)/root 2, and the 2nd qubit can be described similarly. However the state (|01> + |10>)/root 2 is entangled.
When you measure one qubit in an entangled state it behaves randomly, but the result of the measurement allows you to completely predict how the other qubit would behave is measured in the same way. This would not be the case if the qubits were unentangled and so independent of eachother.
These diagrams show Bell states, and the graphs below show that the results are exactly what you would expect them to be.
The superposition between the states where all 3 qubits are in the |0> state and the one where all 3 qubits are in the |1> state is called the GHZ state. This is written (1/root 2)(|000> - |111>).
The combined properties of a 3 qubit system can be predicted - eg what happens if we measure all 3 qubits in the X direction (XXX) - but the individual outcomes per qubit cannot be predicted. When all 3 qubits are measured in the X direction the result is always -1. If 2 are measured in the Y direction and 1 in the X we get 1. The result is calculated by multiplying all the values read together, where a read 0 is mapped to a 1, and a read 1 is mapped to a -1.
A helpful video explaining entanglement and Bell tests.
Einstein described entanglement as spooky action at a distance as it seemed to suggest that the qubits could communicate at speeds faster than the speed of light, which went against his theory of relativity. He theorised that the particles 'agreed' before they were separated and measured which spin they would have, however the Bell test disproves this. Over several Bell tests you can show statistically that the theory of local variables must not be true, and a GHZ circuit can also show this but in one run.