Michael A. Nielsen Books

We begin by building a neural network that can recognize handwritten digits — a seemingly simple task with unexpectedly powerful lessons. The MNIST dataset, consisting of 28×28 pixel images of digits 0–9, is a natural playground for learning. Each image becomes a vector of 784 numbers, and the netwo...

From Neural Networks and Deep Learning

Once we can measure how wrong our network’s predictions are, we need a way to make it less wrong. That is the role of gradient descent, one of the most elegant tools in optimization. We define a cost function — a quantitative measure of how poorly the network performs over its training examples — an...

From Neural Networks and Deep Learning

Every journey into quantum computation begins with mathematics, the language that gives form to physical intuition. In the second chapter, we establish this language—linear algebra, vector spaces, and operators on Hilbert spaces. Quantum mechanics, at its heart, is a theory of vectors and transforma...

From Quantum Computation and Quantum Information

The concept of the qubit is our starting point for reimagining information. Whereas a classical bit exists as 0 or 1, a qubit inhabits both possibilities simultaneously, described by a vector a|0⟩ + b|1⟩ in a two-dimensional complex space. The coefficients a and b, constrained by normalization, hold...

From Quantum Computation and Quantum Information