From Bits to Qubits: Where the Rules Change

A classical bit is the most boring object in information theory. It is either 0 or it is 1. You can read it without changing it. You can copy it without limit. You can send it anywhere. You can store it forever. It has no secrets. It does exactly what you expect.

The qubit is none of these things. And the way it differs from a classical bit is not a small extension or a minor upgrade. The rules change so completely that the two objects share almost nothing beyond the fact that both produce binary outcomes when measured. Starting from the bit and trying to reach the qubit by adding features is like starting from a bicycle and trying to reach a submarine by adding parts. At some point you have to abandon the frame entirely.

Let us start with what we are abandoning.

The classical world: definite states, deterministic operations

A classical bit has a state: 0 or 1. You can perform operations on it: flip it (NOT gate), leave it alone (identity), combine it with other bits (AND, OR, XOR). At every moment, the bit has a definite value. After every operation, the bit has a definite value. The history of a classical computation is a sequence of definite states connected by deterministic operations.

This is so natural that it is hard to see it as a choice. But it is a choice. The assumption that information carriers always have definite states is not a logical necessity. It is a feature of the physics we grew up with. Nature chose different rules for very small things.

Probability amplitudes: the new currency

Here is where the rules change. A qubit has a state, but that state is not simply 0 or 1. It is described by two numbers, typically labeled alpha and beta. These numbers are the probability amplitudes for the two measurement outcomes.

The word “amplitude” matters. In everyday life, probability is a number between 0 and 1 that measures how likely something is. Amplitudes are different. They are complex numbers, meaning each one has both a magnitude and a phase (you can think of phase as a direction, like an arrow on a clock face). The magnitude tells you something about likelihood. The phase tells you something about how this amplitude will interact with other amplitudes.

The relationship between amplitudes and probabilities is given by the Born rule: the probability of getting outcome 0 equals the squared magnitude of alpha, and the probability of getting outcome 1 equals the squared magnitude of beta. The total probability must equal 1, so the squared magnitudes of alpha and beta always sum to 1.

This might seem like an unnecessary complication. Why not just work with probabilities directly? Because probabilities cannot do what amplitudes can do. Probabilities are always positive. Amplitudes can be negative. Amplitudes can be complex. And this means amplitudes can interfere.

Interference: what amplitudes can do that probabilities cannot

Interference is the phenomenon that makes quantum computing possible. It is also the concept most often glossed over in popular explanations, which is a problem because without interference, quantum computing has no advantage over classical computing.

Consider two waves arriving at the same point. If both waves have peaks at that point, they add up (constructive interference) and the combined wave is larger. If one wave has a peak and the other has a trough, they cancel (destructive interference) and the combined wave is smaller or even zero.

Probability amplitudes behave the same way. When a quantum computation has multiple paths leading to the same outcome, the amplitudes for those paths add up. If they have the same phase, they reinforce. If they have opposite phases, they cancel. The total probability of the outcome depends on whether the amplitudes interfere constructively or destructively.

This is the mechanism behind every quantum speedup. A quantum algorithm arranges things so that the amplitudes for wrong answers interfere destructively (cancel each other out) while the amplitudes for correct answers interfere constructively (reinforce each other). The result is that when you measure at the end, you are much more likely to get the right answer.

A classical computer cannot do this. Classical probabilities are always positive. They can only add up. There is no classical mechanism for the probability of a wrong answer to decrease because two contributions to that probability happened to cancel. Interference is a purely quantum resource.

Superposition: what it actually means

Now we can say what superposition actually is. A qubit is in superposition when its state has nonzero amplitudes for both measurement outcomes. That is it. The qubit is described by a specific, well-defined quantum state. It is not “in two states at once.” It is in one state that happens to have nontrivial amplitudes for two outcomes.

The phrase “in two states at once” comes from mapping the quantum situation onto classical language. In classical physics, the only way a bit could give you 0 sometimes and 1 other times would be if you did not know which state it was in. A coin under a cup is either heads or tails, and if you do not know which, you assign probabilities. But the coin has a definite state. Your uncertainty is about your knowledge, not about the coin.

A qubit in superposition is fundamentally different. It is not that you lack knowledge of the qubit’s “real” state. The qubit does not have a “real” classical state. The quantum state (the pair of amplitudes) is the complete description. There is nothing else to know. This is exactly the first assumption from Chapter 1 doing its damage: your brain insists that the qubit must “really” be 0 or 1, and superposition is just your ignorance. That insistence is wrong.

Why does this matter practically? Because if superposition were just ignorance, interference would be impossible. If the qubit were secretly 0, it would behave like 0. If it were secretly 1, it would behave like 1. In neither case would you get the constructive and destructive interference that quantum computation relies on. The fact that interference happens is experimental proof that superposition is not ignorance. The qubit genuinely has no definite classical value.

The Bloch sphere: a map of qubit states

A useful way to visualize a single qubit’s state is the Bloch sphere. Imagine a sphere. The north pole represents the state 0. The south pole represents the state 1. Every other point on the surface represents a superposition state.

The equator of the Bloch sphere holds states with equal probabilities of measuring 0 or 1, but different phases. A state on the equator pointing “east” and a state pointing “west” both give 50/50 outcomes, but they differ in phase, and that phase difference will produce different interference effects in a computation.

The Bloch sphere is not a physical location where the qubit lives. It is a mathematical map of all possible states of a single qubit. Points on the surface represent pure quantum states (states with no classical uncertainty mixed in). Points inside the sphere represent mixed states, but we will set those aside for now.

Two things to notice about the Bloch sphere. First, it is a sphere, which means the space of qubit states is continuous. A qubit can be in any state on the surface, not just 0 and 1. There are infinitely many possible states. This is different from a classical bit, which has exactly two. Second, opposite points on the sphere (like 0 and 1, or “east” and “west”) are states that are perfectly distinguishable by measurement. Points that are close together on the sphere are states that are hard to distinguish. This geometry will matter when we discuss gates.

Measurement: where quantum meets definite

Measurement is where superposition ends and a definite outcome appears. When you measure a qubit in the standard basis, two things happen:

First, you get a result: either 0 or 1. The probability of each result is determined by the Born rule applied to the qubit’s amplitudes.

Second, the qubit’s state changes. After measurement, the qubit is in the state corresponding to the result you got. If you measured 0, the qubit is now definitely in state 0. If you measured 1, the qubit is now in state 1. The superposition is gone.

This is irreversible. The amplitudes that described the qubit before measurement are not recoverable from the measurement result. You got one bit of information (0 or 1) from a state that contained a continuous amount of information (the full pair of amplitudes). Information was lost.

Measurement in quantum mechanics is not like reading a file from a hard drive. Reading a file does not change the file. Quantum measurement changes the qubit. You can measure a qubit only once. After that, the qubit is in whatever state the measurement left it in, and the original state is gone.

This is the second assumption from Chapter 1 in action. Your brain models measurement as passive observation: looking at something and reading off what was already there. In quantum mechanics, measurement is an active process that creates a definite result from an indefinite state. The result did not pre-exist the measurement.

Single-qubit gates: rotations on the Bloch sphere

In classical computing, the only nontrivial single-bit operation is the NOT gate: flip 0 to 1 and 1 to 0. For qubits, operations are far richer because the state space is a continuous sphere, not a pair of points.

A single-qubit gate is a rotation on the Bloch sphere. The qubit starts at some point on the sphere, and the gate moves it to another point. Because rotations are continuous, there are infinitely many possible single-qubit gates. Three are particularly important.

The Hadamard gate takes the state 0 (north pole) and rotates it to a specific point on the equator: an equal superposition of 0 and 1 with positive amplitudes. It takes the state 1 (south pole) and rotates it to the opposite point on the equator: an equal superposition but with amplitudes of opposite sign. The Hadamard gate is the standard way to create superposition from a definite state. It appears in almost every quantum algorithm.

Phase gates leave the probabilities unchanged but rotate the qubit around the vertical axis of the Bloch sphere. They change the phase of the amplitudes without changing their magnitudes. This sounds trivial, since measurement outcomes are determined by magnitudes, not phases. But phases determine interference effects, so changing the phase can completely alter what happens later in a computation. Phase gates are how quantum algorithms steer interference.

Rotation gates allow continuous rotation around any axis of the Bloch sphere by any angle. These give you fine-grained control over the qubit’s state. Together with the Hadamard and phase gates, they let you move a qubit to any point on the Bloch sphere.

All single-qubit gates share a property: they are reversible. Every gate has an inverse that undoes it. In classical computing, the NOT gate is its own inverse (flip twice and you are back to the start). For qubit gates, the inverse is simply the opposite rotation. This reversibility is a fundamental feature of quantum mechanics: the time evolution of a closed quantum system is always reversible. Measurement is the only irreversible operation.

Why this matters for computation

Let us put the pieces together. A qubit carries amplitudes. Those amplitudes can interfere. Gates manipulate those amplitudes with surgical precision, rotating the qubit’s state on the Bloch sphere. The game of quantum computing is to arrange a sequence of gates so that the amplitudes for correct answers interfere constructively and the amplitudes for wrong answers interfere destructively. When you measure at the end, the correct answer appears with high probability.

This is not “trying all answers at once.” A qubit in superposition does not evaluate all possibilities simultaneously in any useful sense. What it does is carry amplitudes that can be manipulated through interference. The quantum advantage comes from the interference, not from the parallelism.

Getting this distinction right matters more than any formula. If you think quantum computing works by trying all answers at once, you will overestimate what quantum computers can do (since “trying all answers” sounds like it should solve everything) and misunderstand why quantum algorithms are hard to design (since “just try everything” sounds easy). The correct picture, interference-based computation, explains both why quantum computers are powerful for specific problems and why designing quantum algorithms requires carefully engineering the interference pattern.

One qubit is interesting but not useful for computation. The real power emerges when multiple qubits interact, which brings us to the subject of the next chapter: entanglement.