Quantum Walks

Quantum walks are the quantum analogue of random walks — they spread quadratically faster and underpin fast algorithms for search and graph problems.

Spreading

ballistic (∝ t) vs diffusive (∝ √t)

Types

discrete-time & continuous-time

Uses

element distinctness, graph search

What it does

Instead of a coin flip driving a classical step, a quantum walk keeps a superposition over positions that interfere, spreading much faster. Search algorithms built on quantum walks match or beat Grover on structured graphs.

Where it's used

Element distinctness, spatial search, and a universal model of quantum computation in its own right.

In code (Qiskit)

# continuous-time: evolve exp(-iAt) with A the graph adjacency/Laplacian