Abstract
Clifford's calculus was treated in the fourth lesson. Each quantum operation can be seen as a rotation in the Hilbert space of register states. If we restrict ourselves to π/2 rotations, we obtain the so-called Clifford group of the register, which is a subgroup of the complete group of the register denoted SU(2N), where N is the number of qubits. Each operation on the register is now a discrete element of a finite group, and quantum computation can be seen in this miniature quantum mechanics as a simple non-commutative generalization of Boolean computation. We began the lesson with a thorough review of the properties of the stabilizers on which Clifford's operations are performed. The logarithm of their number, which can be seen as the measure of quantum information, is super-extensive. It increases as the square of the number of qubits. This property clearly indicates the superiority of quantum information over classical information. However, it is also the basis of Gottesman and Knill's theorem, which shows that if we restrict ourselves to Clifford operations, quantum algorithms can only offer a polynomial advantage over the corresponding classical algorithms. In the last part of the lesson, we discussed a fundamental characteristic of quantum gates : that of being symmetrical with respect to the set of qubits involved. Unlike classical gates, where control bit and target bit play intrinsically irreconcilable roles, quantum bits in a gate play both roles simultaneously, which is in fact called for by the universal principle of action and reaction. If classical bits seem to escape this basic principle of physics, which transcends the classical-quantum dichotomy, it's because they are in fact implemented by a very large number of effective degrees of freedom, most of which are hidden.
