The key issue of two-qubit gates is discussed in this chapter: there are two basic approaches: direct interaction (which is easy but short-ranged) and using a quantum data bus, which is the key ingredient of the Cirac-Zoller gate.

Using linear algebra, the mathematical techniques needed for describing and manipulating qubits are laid out in detail, including quantum circuits. Moreover, the chapter also explains the state evolution of an isolated quantum system, as is predicted by the Schrödinger equation, as well as non-unitary irreversible operations such as measurement. More details of classical and quantum randomness and their mathematical representation is discussed, leading to the density matrix. representation of a quantum state.

This is the chapter that gets down to applying concepts from the previous chapters about qubits to construct a quantum computer. It teaches how numbers can be stored in quantum computers and how their functions can be evaluated. It also demonstrates the computational speed-up that quantum computers offer over their classical counterparts through the study of Deutsch, Deutsch-Jozsa, and Bernstein-Vazirani algorithms. Finally, it gives a practical demonstration of speed-up in search algorithms provided by Grover’s search algorithm.

The chapter defines the notion of a generator and its hardness, and formulates the hardness conjecture. It also defines a stronger notion of pseudosurjectivity of a generator and states the key conjecture about it. It examines some consequences of the two conjectures for the dWPHP problem. It also relates the hardness conjecture to feasible interpolation, gives a model-theoretic view of the issues and discusses a relation to pseudorandomness.

The chapter gives several consistency results related to the dWPHP problem. It also considers the hardness conjecture for feasibly infinite NP sets. It relates witnessing of dWPHP to various computational complexity conjectures.

Quantum entanglement requires a minimum of two quantum systems to exist, and each quantum system has to have a minimum of two levels. This is exactly what a two-qubit system is, which in this chapter is explored on various levels: state description, entanglement measures, useful theorems, quantum gates, hidden variable theory, quantum teleportation.

This final chapter offers a number of topics for further research involving ordinary PHP, S-T computations, a new notion of PLS-infinite NP sets, proof search algorithms, an exponential time weakening of generators and the function inversion problem.

The chapter concentrates on the pivotal case of extended resolution. It recalls some characterizations of its lengths-of-proofs function and formulates a framework for lower bounds proofs using expansions of pseudofinite structures. It gives an example of a specific candidate construction.