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.

The bipolar junction transistor is introduced and its operation is explained. DC and switching applications are given. The need for DC biasing for AC amplification is illustrated and then satisfied by the Universal DC bias circuit. The thermal stability of this circuit is discussed and resulting constraints on resistor selection are developed. Amplifier gain, input impedance, and output impedance are defined and their usefulness is explained. The AC equivalents for the bipolar transistor are developed and then used to derive the properties of the common-emitter, common-collector, and common-base amplifiers. The concepts of distortion and feedback are introduced.

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.

After a brief review of dynamical systems theory, which is a key to understanding the dynamic process of biological states, we present the methodology adopted in this volume. It consists of (A) macroscopic phenomenological theory based on biological robustness, (B) universal statistical laws at the microscopic level, (C) general laws derived as a consequence of macro-micro consistency, (D) hierarchies with different time scales, and (E) experimental approaches to uncover universal properties and laws, as well as (F) consequences of a possible breakdown of consistency. To illustrate the consistency between cellular growth and molecular replication, we present examples of general statistical laws in gene expressions and the correlated change of expression levels across genes in response to environmental changes, together with their experimental confirmation. Later chapters explain the application of the methodology (A–F) to reveal fundamental properties in life.

The operational amplifier is introduced and the basic rules for its operation are given. Nonlinear operation is explained and the golden rules for linear operation are derived. Several examples of linear operation are given, including amplifiers, buffer, adder, differential amplifier, integrator, and differentiator. Practical considerations for using op-amps are discussed, including bias currents, offset voltages, slew rate limits, and frequency response. As a final non-linear example, an oscillator circuit, the astable multivibrator, is presented and analyzed.

Junction- and metal oxide-field effect transistors are introduced and their operation is explained. Governing equations are presented. DC and switching applications are given. The Universal DC bias circuit is used to provide DC biasing for AC amplification circuits. The AC equivalents for the field-effect transistor are developed and then used to derive the properties of the common-source, common-drain, and common-gate amplifiers.

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.