This chapter details the mathematical tools and techniques required by some of the advanced algorithms. Beginners may choose to skip this section and refer back to it as needed. The chapter discusses the spectral theorem, density matrices and the partial trace, Schmidt decomposition and state purification, as well as various operator decompositions.

This section details several optimization algorithms. The variational quantum eigensolver is presented, which allows finding a minimum eigenvalue for a given Hamiltonian. This section also includes extensive notes on performing measurements in arbitrary bases. After a brief introduction of the quantum approximate optimization algorithm, the chapter further discusses the quantum maximum cut algorithm and the quantum subset sum algorithm in great detail.

The algorithms presented in this chapter were the first to establish a query complexity advantage for quantum algorithms. The list of algorithms includes the Bernstein-Vazirani algorithm, Deutsch’s algorithm, and Deutsch-Jozsa algorithm. Quantum oracles and their construction are being introduced.

This chapter lays out a more complete software framework, including a high-performance simulator. It discusses transpilation, a powerful compiler-based technique that allows seamless porting of circuits to other frameworks. The methodology further enables the implementation of key features found in quantum programming languages, such as automatic uncomputation or conditional blocks. An elegant sparse representation is also being introduced.

A quantum walk algorithm is the quantum analog to a classical random walk with potential applications in search problems, graph problems, quantum simulation, and even machine learning. In this section, we describe the basic principles of this class of algorithms on a simple one-dimensional topology.

This brief chapter discusses the minimum mathematical background required to fully understand the derivations in this text. Basic familiarity with matrices and vectors is assumed. The chapter reviews key properties of complex numbers, the Dirac notation with inner and outer products, the Kronecker product, unitary and Hermitian matrices, eigenvalues and eigenvectors, the matrix trace, and how to construct the Hermitian adjoint of matrix–vector expressions.

This chapter discusses Grover’s fundamental algorithm, which enables searching over a domain of N elements with complexity of the square root of N. Several derivative algorithms and applications are being discussed, including amplitude amplification, amplitude estimation, quantum counting, Boolean satisfiability, graph coloring, and quantum mean, medium, and minimum finding.

Quantum algorithms operate on inputs encoded as quantum states. Preparing these input states can be quite complicated. The section discusses the trivial basis and amplitude encoding schemes, as well as Hamiltonian encoding. It also discusses smaller circuits for two- and three-qubit states. Then this chapter presents two of the most complex algorithms in this book, the general state preparation algorithms from Möttönen and the Solovay–Kitaev algorithm for gate approximation. Beginners may decide to skip these two algorithms on a first read.

This chapter discusses the terms overlap and similarity between quantum states and introduces the important swap test, as well as the Hadamard test and the inversion test. The mathematical derivations in this chapter are still very detailed.

This chapter discusses quantum noise and techniques for quantum error correction, a necessity for quantum computing. It discusses bit-flip errors, phase-flip errors, and their combination. The formalism of quantum operations is introduced, along with the operator-sum representation and the Kraus operators. With this in mind, the chapter discusses the depolarization channel and imprecise gates, as well as (briefly) amplitude and phase damping. For error correction, repetition codes are introduced to motivate Shor’s 9-qubit error correction technique.

We have introduced a compact infrastructure for exploration and experimentation, but all at the level of individual gates. Higher levels of abstraction are needed to scale to larger programs. The chapter discusses several quantum programming languages, including their specific tooling, such as hierarchical program representations or entanglement analysis. General challenges for compilation are discussed as well as compiler optimization techniques.

The appendix contains a detailed description of the sparse simulation infrastructure.