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.

Voltage and current sources, both independent and dependent, are introduced, along with resistors and their equivalent circuit laws. The Thevenin and Norton theorems are presented. Several examples of resistor applications are given. Various techniques for solving circuit problems are discussed, including Kirchhoff’s laws, the mesh loop method, superposition, and source transformation. Input resistance of measuring instruments is discussed and the various types of AC signals are presented.

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.

For different types of environmental conditions, the logarithmic changes in each concentration Xj, denoted by δXj(E), are proportional for almost all components, over a wide range of perturbations, where the proportionality coefficient is given by the ratio of change in cell growth rate δμ(E). Then consider the evolution after applied environmental changes. Let the change in log concentration be δXj(G) and the change in growth rate be δμ(G). The theory suggests that δX_j(G)/ δX_j(E)= δμ(G)/ δμ(E), as confirmed experimentally. With evolution, the right hand term gradually moves toward 0, accordingly the change in concentrations does. This is a process similar to the Le Chatelier principle of thermodynamics. The relationships described above arise because phenotypic changes due to environmental perturbations, noise, and genetic changes are constrained to a common low-dimensional manifold as a result of evolution. This is because the adapted state after evolution should be stable against a variety of perturbations, while phenotypes retain plasticity to change, in order to have evolvability. To achieve this dimensional reduction, there is a separation of a few slow modes in the dynamics for phenotypes. The variance of phenotypes due to noise and mutation is proportional over all phenotypes, leading to the possibility of predicting phenotypic evolution.

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.

The other facet of adaptation, immutability or homeostasis, is discussed. Dynamical system models that buffer external changes in a few variables to suppress changes in other variables are presented. In this case, some variable makes a transient change depending on the environmental change before returning to the original state. This transient response is shown to obey fold-change detection (or Weber–Fechner law), in which the response rate by environmental changes depends only on how many times the environmental change is to the original value. As for the multicomponent cell model, a critical state in which the abundances of each component are inversely proportional to its rank is maintained as a homeostatic state even when the environmental condition is changed. In biological circadian clocks, the period of oscillation remains almost unchanged against changes in temperature (temperature compensation) or other environmental conditions. When several reactions involved in the cyclic change use a common enzyme, enzyme-limited competition results. This competition among substrates explains the temperature compensation mentioned above. In this case, the reciprocity between the period and the plasticity of biological clocks results.

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.