The universal quantum driving force to speed up a quantum computation -- The unitary quantum dynamics

TL;DR

This paper argues that unitary quantum dynamics is the fundamental universal force enabling exponential speedup in quantum computation, emphasizing the role of Hilbert space symmetry and structure in this process.

Contribution

It introduces a new quantum speedup theory based on unitary dynamics and Hilbert space symmetry, explaining the mechanisms behind exponential quantum speedup.

Findings

Unitary quantum dynamics is the universal driving force for quantum speedup.

Hilbert space symmetry is a key resource in quantum computing.

Existing algorithms like hidden-subgroup problems are less affected by Hilbert space symmetry.

Abstract

It is shown in the paper that the unitary quantum dynamics in quantum mechanics is the universal quantum driving force to speed up a quantum computation. This assertion supports strongly in theory that the unitary quantum dynamics is the fundamental and universal principle in nature. On the other hand, the symmetric structure of Hilbert space of a composite quantum system is the quantum-computing resource that is not owned by classical computation. A new quantum-computing speedup theory is set up on the basis of the unitary quantum dynamics. Both the unitary quantum dynamics and the symmetric structure and property of the Hilbert space of the quantum system are mainly responsible for an exponential quantum-computing speedup for a general efficient quantum algorithm. The inherent importance for the unitary quantum dynamics to speed up a quantum computation lies in the unique ability of the unitary quantum dynamics to build the effective interaction between the symmetric structure of the Hilbert space of the quantum system and the mathematical symmetric structure of a problem to be solved on the quantum system. This unique ability could result in an essential difference of computational power between quantum and classical computations by combining the symmetric structure and property of the Hilbert space. The new quantum-computing speedup theory also provides reasonable mechanisms for exponential quantum-computing speedup for the existing efficient quantum algorithms based on the quantum parallel principle. These existing quantum algorithms including the hidden-subgroup-problem quantum algorithms and conventional quantum search algorithms have the common character that the symmetric structure of the Hilbert space does not have any effective effect on these quantum algorithms. This could be the main reason why these quantum algorithms are quite special and considered to be semiclassical.