Vectors

Quantum Computing • Unit 2

Vectors are the language of quantum states. From basic 2D vectors to complex Hilbert spaces, this unit builds the mathematical framework for representing qubits and multi-qubit systems.

Core Concepts

Vector fundamentals – understanding vectors in R² and Rⁿ as ordered lists of numbers
Vector operations – addition, subtraction, and scalar multiplication
Basis vectors – spanning sets and the standard basis in n dimensions
Dot product – measuring similarity and computing angles between vectors
Complex vectors (Cⁿ) – extending to complex entries for quantum mechanics
Hilbert spaces – the complete inner product spaces where quantum states live
Orthonormal bases – the preferred bases for quantum measurement

Key Equations

$\vec{v} = (v_1, v_2, ..., v_n)$

$\vec{u} \cdot \vec{v} = \sum_i u_i v_i$

$\langle u | v \rangle = \sum_i u_i^* v_i$

$||\vec{v}|| = \sqrt{\langle v | v \rangle}$

Real-World Applications

→Representing qubit states as unit vectors in C²
→Computing transition amplitudes via inner products
→Understanding measurement probabilities through projections
→Building multi-qubit state spaces via tensor products

Topic Map

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