The rhythm of quantum algorithms

Abstract

Quantum algorithms can be generally represented as the dynamical evolution of an input quantum register, with the action of each logical gate, as well as of any transmission channel, defined by some quantum propagator. From a global viewpoint, this unitary dynamics is ruled by the flow of a continuous time, and the possible splitting into shorter logical sub-units is nothing but a harmless, though useful, zooming process. On the other hand, understanding how elementary units of the quantum register, namely single qubits, are actually hauled along the algorithm, is a more complex matter, as it involves the dynamical entanglement generation entailed in the action of two-qubit gates. In this work, we first review how the essential elements of quantum algorithms can be described in terms of dynamical processes, and then analyze the corresponding non-unitary dynamics of single qubits, by referring to the formalism adopted in the study of open quantum systems. We show that single qubits evolution cannot be split into intervals shorter than the typical time needed by two-qubit gates for accomplishing their task, which somehow gives a rhythmical structure to the algorithm itself. We further point out that the local evolution entails a memory, in that the way each qubit takes an infinitesimally small step forward in time, is set by its previous history, back to the instant when it entered the last two-qubit gate. This memory originates from quantum correlations, and it is suggested to play an essential role in quantum information processing. As a concluding remark, we just touch on the idea that a similar analysis could be put forward for getting a clue on how we extract meaningful contents out of complex informational input.