VECTORS: ARROWS THAT DESCRIBE QUBIT STATES
Building vectors from scratch, and writing qubit states as columns of numbers for the first time.
In Module 1, we described a qubit purely with pictures: an arrow pointing somewhere on the Bloch sphere, located by two angles, theta and phi. That picture is complete for intuition, but it is not something you can hand to a computer, plug into a formula, or compute a probability from directly. To do any of that, we need to write a qubit's state as actual numbers — and the tool that holds those numbers is called a vector.
This lesson builds vectors from scratch, assuming nothing beyond basic arithmetic. By the end of this 9-lesson module, you will be able to write down the exact numerical recipe behind everything Module 1 showed you as pictures.
What a Vector Actually Is
Forget any physics baggage attached to the word "vector." For our purposes:
A vector is simply an ordered list of numbers. If you have ever used an array in code — [3, 7]
or [0.5, -2, 9] — you have already used the core idea. The only new ingredient is that we are
going to interpret this list geometrically, as an arrow starting at the origin.
From List to Arrow
Take the vector [3, 2]. Draw a standard 2D graph with a horizontal axis and a vertical axis. Starting at the origin (0, 0), draw an arrow that ends at the point 3 units right and 2 units up. That arrow is the vector [3, 2]. The first number tells you how far to go along the horizontal axis; the second tells you how far along the vertical axis.
Writing Vectors the Way Quantum Computing Does
Quantum computing almost always writes vectors vertically (as a column) rather than horizontally (as a row). This is purely a notational convention — a "column vector":
This is the exact same vector as [3, 2] above — same arrow, same meaning — just written top-to-bottom instead of left-to-right. We use this column convention because, starting in Lesson 3, we will be multiplying vectors by matrices, and the column layout is what makes that multiplication work correctly.
Qubits as 2-Dimensional Vectors
Here is the payoff, right away: a single qubit's state is a vector with exactly two entries, because there are exactly two classical outcomes (0 and 1) it can resolve into. Recall from Module 1, Lesson 4, that |0⟩ and |1⟩ are the two poles of the Bloch sphere. In vector form:
Take a moment to notice the pattern: the top entry of the vector is associated with the 0 outcome, and the bottom entry is associated with the 1 outcome. |0⟩ has a "full" top entry (1) and "empty" bottom entry (0) — matching the fact that measuring |0⟩ always gives 0. |1⟩ is the mirror image.
These two vectors are so fundamental that they get a name: the standard basis vectors for a qubit — we will fully define what "basis" means mathematically by the end of this module.
Vector Addition and Scalar Multiplication
We need exactly two operations on vectors for everything ahead in this course.
Scalar multiplication — multiplying every entry of a vector by the same number:
Geometrically, this stretches the arrow to twice its length, without changing its direction (multiplying by a negative number also flips its direction).
Vector addition — adding two vectors entry-by-entry:
Worked Example: Building a Superposition-Shaped Vector
Let's combine both operations to build something that will look extremely familiar once we reach Lesson 9. Take |0⟩ and |1⟩, each scaled by the same number, 0.7071 (an approximation of , a number that will appear constantly throughout this course):
Hold onto this result — in Lesson 9, we will show this exact vector is the numerical form of |+⟩, the equator state from Module 1, Lesson 5.
The Length (Norm) of a Vector
One more tool we need: the length of a vector, also called its norm. For a 2D vector [a, b], the length is found using the Pythagorean theorem you already know from geometry:
Worked Example: Length of |0⟩
Worked Example: Length of Our Superposition Vector
Notice both vectors have length exactly 1. This is not a coincidence — it is a strict requirement for any valid qubit state vector, called normalization. We will explain precisely why this must always be true when we connect vector length to probability in Lesson 5, and formalize it fully in Lesson 9.
The term comes from the same root as "normal," meaning standard or regulated. A normalized vector has been scaled to have length exactly 1 — the standard, agreed-upon length for representing a valid probability-carrying quantum state. You will see this requirement checked constantly once we reach Qiskit in Module 11; if you ever build a qubit state vector whose length is not 1, that is a red flag that something in your math is wrong.
Practice Questions
Test your understanding
KEY TAKEAWAYS
Remember these points
A vector is simply an ordered list of numbers, interpreted geometrically as an arrow starting at the origin
Quantum computing writes vectors vertically ("column vectors"), purely as a convention that makes later matrix multiplication work cleanly
A single qubit's state is a 2-entry vector: |0⟩ = [1, 0] and |1⟩ = [0, 1] as columns — top entry linked to outcome 0, bottom entry linked to outcome 1
Vector addition (entry-by-entry) and scalar multiplication (scale every entry) are the two basic operations used throughout the rest of this module
Every valid qubit state vector must have length exactly 1 — a property called normalization, whose deeper meaning is unpacked fully in Lessons 5 and 9