The quantum speed up as advanced knowledge of the solution

TL;DR

This paper explains quantum speed ups as resulting from the algorithm's effective gain of 50% of the solution information in advance, linking quantum advantage to time symmetry and problem-solution interdependence.

Contribution

It provides a simple, general explanation for quantum speed ups based on advanced knowledge and time symmetry, applicable to both quadratic and exponential cases.

Findings

Quantum algorithms are equivalent to classical algorithms with 50% of solution information known in advance.

Speed up is due to information gain from time symmetry and problem-solution interdependence.

The approach unifies understanding of quadratic and exponential quantum speed ups.

Abstract

With reference to a search in a database of size N, Grover states: "What is the reason that one would expect that a quantum mechanical scheme could accomplish the search in O(square root of N) steps? It would be insightful to have a simple two line argument for this without having to describe the details of the search algorithm". The answer provided in this work is: "because any quantum algorithm takes the time taken by a classical algorithm that knows in advance 50% of the information that specifies the solution of the problem". This empirical fact, unnoticed so far, holds for both quadratic and exponential speed ups and is theoretically justified in three steps: (i) once the physical representation is extended to the production of the problem on the part of the oracle and to the final measurement of the computer register, quantum computation is reduction on the solution of the problem under a relation representing problem-solution interdependence, (ii) 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 knowledge of the solution reduces the size of the solution space to be explored by the algorithm, (iii) if I is the information acquired by measuring the content of the computer register at the end of the algorithm, the quantum algorithm takes the time taken by a classical algorithm that knows in advance 50% of I, which brings us to the initial statement.