Eigenvalue and Eigenvector: Quantum Computing Glossary

For a linear operator A, an eigenvector |v⟩ satisfies A|v⟩ = λ|v⟩ where λ is the eigenvalue, representing states that are unchanged in direction by the operator, crucial in quantum mechanics where observables are Hermitian operators whose eigenvalues are measurement outcomes.

Eigenvalues and eigenvectors are the backbone of quantum measurement theory. Every physical quantity that can be measured (position, momentum, energy, spin) corresponds to a Hermitian operator, and the only values that can appear as measurement results are the eigenvalues of that operator.

Definition

For a square matrix (or linear operator) $A$ , a nonzero vector $|v\rangle$ is an eigenvector with eigenvalue $\lambda$ if:

$A|v\rangle = \lambda|v\rangle$

Applying $A$ to $|v\rangle$ stretches or shrinks it by $\lambda$ but does not rotate it. The set of all eigenvalues is the spectrum of $A$ .

Hermitian operators and real eigenvalues

Observables are represented by Hermitian operators: $A^\dagger = A$ . Their eigenvalues are always real (since $\lambda\langle v|v\rangle = \langle v|A|v\rangle = \langle v|A^\dagger|v\rangle^* = \lambda^*\langle v|v\rangle$ forces $\lambda = \lambda^*$ ). Eigenvectors of distinct eigenvalues are orthogonal, and the full set forms an orthonormal basis (the spectral theorem).

The measurement postulate

When measuring observable $A$ on state $|\psi\rangle$ :

Express $|\psi\rangle$ in the eigenbasis of $A$ : $|\psi\rangle = \sum_i c_i |v_i\rangle$ where $A|v_i\rangle = \lambda_i|v_i\rangle$ .
The probability of obtaining outcome $\lambda_i$ is $|c_i|^2 = |\langle v_i|\psi\rangle|^2$ .
After the measurement, the state collapses to $|v_i\rangle$ .

The expected value of the measurement is:

$\langle A \rangle = \langle\psi|A|\psi\rangle = \sum_i \lambda_i |c_i|^2$

Examples from quantum gates

Pauli Z gate: eigenvalues $+1$ and $-1$ with eigenvectors $|0\rangle$ and $|1\rangle$ :

$Z|0\rangle = +1\cdot|0\rangle, \qquad Z|1\rangle = -1\cdot|1\rangle$

Measuring in the $Z$ basis is the standard computational-basis measurement.

Pauli X gate: eigenvalues $+1$ (eigenvector $|+\rangle$ ) and $-1$ (eigenvector $|-\rangle$ ), where $|{\pm}\rangle = \frac{1}{\sqrt{2}}(|0\rangle \pm |1\rangle)$ . To measure in the $X$ basis, apply $H$ before measuring in the $Z$ basis.

Eigenvalues in quantum algorithms

Variational Quantum Eigensolver (VQE): finds the ground-state energy of a molecule by minimizing $\langle\psi(\theta)|H|\psi(\theta)\rangle$ over circuit parameters $\theta$ . The ground state energy is the lowest eigenvalue of the molecular Hamiltonian $H$ .

Quantum Phase Estimation (QPE): given a unitary $U$ and eigenvector $|u\rangle$ with $U|u\rangle = e^{2\pi i \phi}|u\rangle$ , QPE estimates the phase $\phi$ in the eigenvalue. Shor’s algorithm uses QPE as a subroutine to find the period of modular exponentiation.

Quantum Fourier Transform: the QFT diagonalizes the shift operator, exposing its eigenstructure. The speedup in many quantum algorithms comes from efficient access to spectral information.

Code example

import numpy as np

# Pauli Z matrix
Z = np.array([[1, 0], [0, -1]], dtype=complex)
eigenvalues, eigenvectors = np.linalg.eigh(Z)
print("Z eigenvalues:", eigenvalues)           # [-1., 1.]
print("Z eigenvectors (columns):\n", eigenvectors)

# Simple 2-qubit Hamiltonian H = Z tensor Z
Z_tensor_Z = np.kron(Z, Z)
evals, evecs = np.linalg.eigh(Z_tensor_Z)
print("\nZ⊗Z eigenvalues:", evals)             # [-1, -1, -1, 1] -- check ordering
print("Ground state energy:", evals[0])

# Verify: A|v> = lambda|v> for the lowest eigenvalue
v0 = evecs[:, 0]
residual = np.linalg.norm(Z_tensor_Z @ v0 - evals[0] * v0)
print("Residual ||A|v> - lambda|v>||:", residual)  # should be ~0

Eigenvalue and Eigenvector