Tutorial

#### Superposition

The H gate takes a pure qubit — $\ket{0}$ or $\ket{1}$ — and splits it putting the qubit in a quantum state that now contains both pentagons and triangles — a quantum state that is a superposition of two different qubelets.

Splitting two qubits considerably thickens the plot. Unlike classical computers where operations on one bit doesn't affect others in the system, in quantum computing an action on one influences the states of others. To handle this characteristic of quantum bits, we must think of all possible states of the quantum bits at once instead of the classical way of thinking of one-bit-at-a-time approach. To help us get comfortable with this shift in mindset, consider the following circuit in which two qubits are split as follows: Each H gate splits the $\ket{0}$ qubit. But, in quantum computing, you don't treat these split pentagons and triangles as individual units. Rather, each shape in the top qubit pairs up with each shape in the bottom qubit forming the four combinations as shown below: These combinations, called qubelet combinations, form a mega-qubit.

If we feed the mega-qubit to other gates, then those gates operate on all the combinations in the mega-qubit simultaneously.

When more qubits are split, the mega-qubit automatically holds the combinations formed from the pentagons and triangles from all the split qubits. For example, when 3 qubits are split by the H gate, the resulting mega-qubit is shown below: Each combination is formed by taking a shape from each of the three qubits in turn. This gives $2 \times 2 \times 2 = 2^3 = 8$ combinations in the mega-qubit on the right in the above figure.

In general, as the number of qubits you split becomes large, the number of combinations in the mega-qubit becomes astronomical and well beyond the means of classical computers. But, this gigantic increase in complexity has no impact for quantum computers which is hard-wired to deal with all combinations simultaneously regardless of how many there are.

This ability to simultaneously deal with all combinations or possible quantum states of the qubits offers a spectacular way to routinely solve industrial-scale applications that even today's supercomputers find impossible to tackle: in a quantum computer all Boolean operations are applied to possible states at the same time unlike a classical computer in which only one combination of states at a time are operated on.

One of these combinations, the solution to your problem, is lurking in the mega-qubit. The goal in designing a quantum algorithm is to apply quantum gates to the qubits to tease this solution out from the all the other combinations.

In the next section, you'll learn that the qubelets model explains entanglement, another quantum phenomena that has no classical counterpart.