The Quantum Speed up as Advanced Cognition of the Solution

Abstract

Solving a problem requires a problem solving step (deriving, from the formulation of the problem, the solution algorithm) and a computation step (running the algorithm). The latter step is generally oblivious of the former. We unify the two steps into a single physical interaction: a many body interaction in an idealized classical framework, a measurement interaction in the quantum framework. The many body interaction is a useful conceptual reference. The coordinates of the moving parts of a perfect machine are submitted to a relation representing problem-solution interdependence. Moving an “input” part nondeterministically produces a solution through a many body interaction. The kinematics and the statistics of this problem solving mechanism apply to quantum computation—once the physical representation is extended to the oracle that produces the problem. Configuration space is replaced by phase space. The relation between the coordinates of the machine parts now applies to a set of variables representing the populations of the qubits of a quantum register during reduction. The many body interaction is replaced by the measurement interaction, which changes the population variables from the values before to the values after measurement (and the forward evolution into the backward evolution, the same unitary transformation but ending with the state after measurement). Quantum computation is reduction on the solution of the problem under the problem-solution interdependence relation.

The speed up is explained by a simple consideration of time-symmetry, it is the gain of information about the solution due to backdating, to before running the algorithm, a time-symmetric part of the reduction on the solution. This advanced cognition of the solution reduces the solution space to be explored by the algorithm. The quantum algorithm takes the time taken by a classical algorithm that knows in advance 50% of the information acquired by reading the solution (i.e. by measuring the content of the computer register at the end of the quantum algorithm).

From another standpoint, the notion that a computation process is condensed into a single physical interaction explains the fact that we perceive many things at the same time in the introspective “present” (the instant of the interaction in the classical case, the time interval spanned by backdated reduction in the quantum case).