Quantum computing can sound like magic, but it’s really just math. The difference is in how we do the math. And at the center of it all is the humble qubit.

Not Just a Bit: Complex Numbers and Quantum States

In classical computing, a bit can only take the values 0 or 1. But a qubit is different: it is a vector in a two-dimensional complex space. This means it doesn't just take the value 0 or 1. Instead, it's a superposition of both, with complex amplitudes (numbers with a real and imaginary part) that encode the probability of measuring a 0 or a 1.

If that sounds like a lot of information for a single qubit to carry, that’s because it is. A classical system with bits can represent one of possible values at a time. But a quantum system with qubits can simultaneously represent a linear combination of all possibilities. Basically, we’re storing a complex number instead of a boolean. That means exponential storage capacity in the number of qubits, and it’s not because we are storing more values intrinsically, but because we are manipulating probability amplitudes in a vector space.

Let’s make this more concrete: the states |0> and |1> can be represented as the following column matrices:

|0> = \col([1,0 ])  |1> = \col([0,1])

Another important state in Quantum, |+> = 1 / sqrt(2) x |0> + 1 / sqrt(2) x |1>, is represented like this in the |0>, |1> basis:

|+> = \col(1/sqrt(2), 1/sqrt(2))

Where it gets interesting, is that this is also a valid quantum state for a qubit: |ψ⟩ = i / sqrt(2) x |0⟩ + (1 + i ) / 2 x |1⟩. Any |ψ⟩ = α|0⟩ + β|1⟩ state is valid, as long as |α|² + |β|² = 1 because |α|² encodes the probability for the qubit to be measure in the state |0> while |β|² is the probability to measure the qubit in the state |1>. We will discuss this in more detail in a subsequent section.

A Qubit is a Matrix Citizen

In quantum computing, we don’t operate directly on qubit values. Instead, we apply unitary matrices to qubit states. Think of each quantum operation as a rotation in a high-dimensional complex space. These matrices preserve information while transforming it in ways that are reversible (unlike most classical operations).

Example of transformation: the bit flip operation X (or Pauli-X matrix) swap the probabilities of the qubit being measured as |0⟩ and |1⟩.

X = \matrix(0,1,1,0)

We will discuss the other typical transformation (and the quantum gates that they are associated with) in a subsequent section.

To get a better grasp on the concept, imagine rotating a qubit's state vector on the surface of a sphere, which we call the Bloch sphere. Each quantum gate is like a little nudge in some direction. But what’s unique in the quantum world is that these nudges interfere, entangle and collapse based on very different rules from classical logic.

Understanding qubits means understanding that quantum computing is fundamentally about transforming vector spaces instead of flipping bits. In Quantum Computing, we’re not programming logic gates; we’re orchestrating wavefunctions. When people think about Quantum, they often assume it’s primarily about acceleration, but the reality is that the future of computation won’t just be faster: it’ll be weirder, more beautiful, and built on mathematics that doesn’t just predict the future, but describes and models it.

To Remember:

A qubit stores a superposition of states, represented not by simple real numbers, but by complex amplitudes.