MODULE 1 · LESSON 2
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Sign in to track progress / enrolWhat a Qubit Is
A classical bit is 0 or 1. A qubit is usually introduced as being 0 and 1 at the same time, which sounds profound and explains nothing. Here is a more useful description.
Amplitudes
A qubit's state is described by two numbers, one associated with the outcome 0 and one associated with the outcome 1. These numbers are called amplitudes.
When you measure the qubit you get either 0 or 1, never anything in between. The probability of each outcome is the square of its amplitude.
So a qubit with amplitude 1 for outcome 0 and amplitude 0 for outcome 1 always yields 0. A qubit with amplitudes of about 0.707 each yields 0 half the time and 1 half the time, because 0.707 squared is roughly 0.5.
That is superposition. Not the qubit being both values, but the qubit carrying a pair of numbers that determine what you are likely to see when you look.
The feature that matters: amplitudes can be negative
Here is the part that ordinary probability does not have, and it is the single most important idea in this course.
A probability cannot be negative. There is no such thing as a minus 30 percent chance of rain.
An amplitude can be negative. A qubit can have amplitude 0.707 for outcome 0 and minus 0.707 for outcome 1. Square them both and you still get 50 percent each, so a single measurement cannot tell this qubit apart from the previous one.
But the two are genuinely different states, and they behave differently when you operate on them. Positive and negative amplitudes can meet and cancel.
That cancellation is the entire mechanism by which quantum computers do anything useful. Everything in Module 2 builds on it. If you remember one sentence from this lesson, remember that the useful strangeness is not that a qubit can be in two states, it is that the numbers describing it can be negative and therefore cancel.
A quantum computer is not a machine that tries everything at once. It is a machine that arranges for wrong answers to cancel out.
Why not just a random bit?
A fair question. If measuring a qubit in superposition gives 0 or 1 with some probability, why is that different from flipping a coin?
The difference is that a coin's outcome is already determined and merely unknown to you. Your 50 percent describes your ignorance, not the coin.
A qubit in superposition is not hiding a value. Before measurement there is no fact about which outcome it will produce, and the amplitudes are physically real in the sense that they interfere. Two paths leading to the same outcome can cancel, which cannot happen with probabilities, because probabilities only ever add.
This is testable rather than philosophical. Experiments distinguish the two cases, which is why we can say the qubit is not simply a coin whose result we have not seen yet.
What 2 to the n really means
Now the number that generates most of the hype.
One qubit needs 2 amplitudes. Two qubits need 4. Three need 8. In general n qubits require 2 to the power n amplitudes to describe fully.
Three hundred qubits would require more amplitudes than there are atoms in the observable universe. This is a real and remarkable fact, and it is the reason simulating quantum systems on classical computers becomes impossible so quickly.
It is also the origin of the misconception. The reasoning goes: the machine holds an astronomical number of amplitudes, therefore it is doing an astronomical number of calculations, therefore it can try every answer at once.
The first part is right and the conclusion does not follow, for a reason covered fully in the next lesson: you cannot read the amplitudes. Measuring gives you one outcome, chosen at random according to those amplitudes, and the rest are gone.
So the challenge of quantum algorithm design is not generating a superposition over many possibilities, which is easy. It is arranging the operations so that when you finally measure, the answer you want is overwhelmingly likely to be the one you get. Very few problems permit such an arrangement, which is exactly why the useful algorithm list is short.
Popular explanations reach for a spinning coin, a sphere, or a maze explored down every path simultaneously. Each captures something and misleads about something more important.
The spinning coin. Suggests the qubit is rapidly alternating between 0 and 1. It is not, and nothing is oscillating between two values. Worse, a spinning coin has no analogue of a negative amplitude, so it cannot represent cancellation, which is the only part that matters.
Exploring every path of a maze at once. The most damaging, because it implies you can then look at the paths and pick the one that worked. You cannot. You get one path back, at random, weighted by amplitudes. If the algorithm has not been constructed so that the good path is overwhelmingly likely, a random path is exactly what you receive.
The Bloch sphere. Genuinely used by practitioners: a qubit's state is a point on a sphere, with the poles as 0 and 1 and everything else a superposition. Accurate and useful for single qubits, but it does not extend to multiple qubits, which is where the interesting behaviour lives. Fine as a picture, not as a foundation.
The analogy that holds up best is waves, because waves have the property that matters. Two waves meeting crest to crest reinforce. Crest to trough, they cancel and you get nothing, which is why noise cancelling headphones work and why light passing through two slits produces dark bands where light plainly arrives.
Amplitudes behave like wave heights. They add, they subtract, and they cancel. A quantum algorithm is a carefully engineered interference pattern, arranged so that paths leading to wrong answers cancel and paths leading to the right answer reinforce.
That is not a metaphor bolted on afterwards. It is what the mathematics says, and it is why the next lesson treats interference as the engine of the whole field.
What is the most important difference between a qubit in superposition and a classical bit whose value you happen not to know?
Amplitude
Click to flipA number associated with each possible measurement outcome. Its square gives the probability of that outcome, and crucially it may be negative.
Click to flip backA qubit is described by amplitudes, numbers whose squares give measurement probabilities. The essential feature is not that a qubit can be in two states but that its amplitudes can be negative and therefore cancel, which ordinary probability can never do. The 2 to the n scaling is genuine and explains why classical simulation fails quickly, but it does not mean the machine tries every answer, because the amplitudes cannot be read out. Designing a quantum algorithm means engineering an interference pattern in which wrong answers cancel.