More Ingredients

Additionally to the fundamental description of superposition and entanglement, a few more tools are necessary for properly describing the quantum phenomena whose control is mandatory for running a quantum computer.

The so-called qubits are based on Two-Level Systems (TLS) that are quantum systems that inherently possess two energy levels. These systems may have additional levels, but these must not be attained by the system else it would spoil the function of the quantum computer. Hence the energy gap between these two levels has to be sensibly different from the other levels. An isolated single qubit will evolve in the two-dimensional Hilbert space of this TLS.

Qubits must be allowed to interact with each other and form a single quantum system grouping all of them. This interaction involves a medium between the qubits that passes quantas of energy: this medium is itself a part of the quantum system, and is basically a quantized field such as light (e.g. photons in cavity QED) or vibrational modes (e.g. phonons in ion traps). These fields can be nicely described with the Harmonic Oscillator model.

To fit the qubits and their interaction all together, the quantization of the TLS and the field can be united into own powerful framework, that describes the state and the evolution of the system in terms of its energy: the Hamiltonian. The Quantization of the electromagnetic field is a prerequisite to formulating the atom-light interaction and is a fascinating description of light in terms of the harmonic oscillator. The Second Quantization is a way to formulate the atomic Hamiltonian in analog terms to the quantized electromagnetic field. It involves ladder operators such as the creation and annihilation operators, and describes a matter particle in the same terms then a field.

In his pioneering experiments in cavity QED, Serge Haroche called the qubits and the mediating entities between them the “spins and springs”, as its TLS under consideration were described in terms of 1/2 spins, while springs are illustrative for the harmonic oscillator.

Finally, the concept of quantum computing is based on an ideally isolated quantum system, but paradoxically this quantum system must be feed and controlled from the outside world: the environment. Beside this desirable interaction, the quantum system can and will sporadically interact with the environment in an uncontrolled way: the noise. There is a sound framework for describing the action of the Environment on a quantum system. It is well modelled, and a mandatory tool to describe the Noise & Errors in a Quantum Computer.