The algorithm atlas

This atlas extends the main algorithms volume with focused explanations of additional families and applications. Each chapter starts from a concrete question and a small worked example, then explains the access model, output, costs and limits. It is designed to help you connect ideas rather than memorise a list of algorithm names.

Follow a connection

• Search and estimation: minimum finding, counting and mean estimation, amplitude-estimation variants, Monte Carlo, walks, collision problems, backtracking and formula evaluation.
• Spectral tools: signal processing, eigenvalue and singular-value transformations, linear algebra, principal components, gradients and Hamiltonian simulation.
• States and learning: Gibbs preparation, Metropolis proposals, learning theory, trainability and generative models.
• Applications: chemistry, differential equations, finance, optimisation and energy systems.
• Evidence and limits: lower bounds, precision, conditioning, fault-tolerant costs and quantum-inspired classical algorithms.

Use the atlas catalogue to choose a chapter. Read its prerequisites first, calculate its tiny example yourself, and attempt the exercises before revealing the answers. A simplified example establishes the mechanism; the chapter’s assumptions explain what changes at a useful problem size.

Keep one question in view

Ask what the algorithm receives and what it returns. Does it assume a coherent oracle, a prepared quantum state, a sparse matrix or a graph-access operation? Does it return one sample, a scalar estimate or a quantum state? These distinctions determine both the correctness statement and the meaningful classical comparison.

Research continues to change this field. Primary references are linked within the chapters, and dated proposals are distinguished from established guarantees. The atlas provides a broad guide to the families in the book’s roadmap; it does not treat a new algorithm name as evidence of practical advantage.