3. Linear Algebra: Eigenvalues

Eigenvalues and Eigenvectors

In Quantum Computing, Eigenvalues and Eigenvectors are the language that translates a quantum operation into a physical, observable result.

1. 🔑 The Core Equation

An Eigenvector is a special vector $|\psi\rangle$ that, when acted upon by a matrix $U$, only gets scaled, not rotated.

The amount it is scaled by is the Eigenvalue $\lambda$:

$$U|\psi\rangle = \lambda|\psi\rangle$$

🧠 QC Interpretation: Observables and Outcomes

• The Matrix $U$ is the Observable: A matrix that represents a physical property we can measure, like energy or spin. In our case, this is the measurement operator (usually the Pauli-Z gate).
• The Eigenvectors are the Measurable States: These special states ($\{|\psi\rangle\}$) are the only states the qubit can collapse into upon measurement. For a single qubit measured in the standard basis, the eigenvectors are simply $|0\rangle$ and $|1\rangle$.
• The Eigenvalues are the Measurement Results: The values ($\lambda$) tell us what the result of the measurement is. In quantum mechanics, these are real numbers. We map them to the outcomes 0 and 1.

🚪 Introducing the Pauli-Z Gate ($Z$)

The Pauli-Z gate is the standard observable for measurement in the computational basis. Its matrix is:

$$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

Your Task:

Apply the Pauli-Z matrix ($Z$) to our basis states, $|0\rangle$ and $|1\rangle$.

• Calculate $Z|0\rangle$. What is the resulting vector and its corresponding eigenvalue ($\lambda_0$)?
• Calculate $Z|1\rangle$. What is the resulting vector and its corresponding eigenvalue ($\lambda_1$)?

This will reveal why $|0\rangle$ and $|1\rangle$ are the special states we measure.