A quantum computer is a linear computer, meaning that all of its underlying operations are based on linear algebra. This powerful branch of mathematics provides the tools we need to describe and manipulate the quantum world with precision.
We use it to represent everything from qubits to the quantum operations that transform them. At the core of this math are two key ideas: vectors and a special notation called the braket.
🧮 Understanding Vectors and Complex Numbers
Forget about the simple arrows you might be used to from high school physics. In quantum computing, a vector is simply an ordered list of numbers. Since qubits are represented by complex numbers, we use complex vectors, or lists of complex numbers.
These vectors can be added together or multiplied by a single number (a scalar). When we manipulate qubits, we are effectively manipulating these vectors. While this might seem abstract, this mathematical formalism is the only way we know to accurately and completely describe what is happening in the quantum world.
✍️ The Elegant Bra-Ket Notation
To keep things clean and easy to read, quantum computing uses a special shorthand invented by physicist Paul Dirac called bra-ket notation. In this notation, a quantum state is a vector called a ket, written as $|\psi\rangle$. When we want to perform a quantum operation, such as finding the inner product of two states, we use its corresponding partner, a bra, written as $\langle\psi|$.
Combining a bra and a ket creates a braket, such as $\langle\phi|\psi\rangle$, which is a compact way of writing a full inner product calculation. This notation allows us to express complex mathematical operations on vectors in a clear and concise way, making our algorithms much easier to understand and write.
💪 The Power of Adjoints and Orthonormal Bases
To make the math work, we often use the adjoint of a vector, which is a combination of taking its transpose (flipping its rows to columns) and its complex conjugate. This operation is what turns a ket into its corresponding bra.
This careful system of notation is particularly useful when working with the computational basis, a set of vectors like $|0\rangle$ and $|1\rangle$ that are both normalized (have a magnitude of 1) and orthogonal (perpendicular to each other). We can express any quantum state as a combination of these basis vectors, allowing us to perform powerful operations and analyze the results.
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Glassner, Andrew. Quantum Computing: From Concepts to Code. No Starch Press, 2025.
More Topics
- The Quantum Threat: How Shor’s Algorithm Puts Modern Encryption at Risk
- Cracking the Code: How Quantum Parallelism Solves Deutsch’s Problem in One Step
- The Deutsch-Jozsa Algorithm: An Exponential Leap in Quantum Problem Solving
- The Bernstein-Vazirani Algorithm: Unmasking a Secret With a Single Query
- Simon’s Algorithm: The First Quantum Speedup That Left Classical Computers in the Dust
- The Quantum Enigma: Why Measurement is the Most Unpredictable Step
- Can You Teleport a Qubit? The Amazing Alice & Bob Experiment
