A relational time-symmetric framework for analyzing the quantum computational speedup

TL;DR

This paper introduces a time-symmetric, relational framework for quantum algorithms that explains the quantum speedup by considering boundary conditions and pre- and post-selection effects, providing new insights into quantum computation.

Contribution

It extends the standard quantum algorithm representation to include problem-setting and solution, highlighting the role of time symmetry and relational quantum mechanics in understanding quantum speedup.

Findings

Quantum algorithms can be viewed as influenced by both initial and final boundary conditions.

The framework shows that Alice effectively has partial advanced knowledge, explaining the speedup.

The approach applies to various quantum algorithms, elucidating their efficiency.

Abstract

The usual representation of quantum algorithms is limited to the process of solving the problem. We extend it to the process of setting the problem. Bob, the problem setter, selects a problem-setting by the initial measurement. Alice, the problem solver, unitarily computes the corresponding solution and reads it by the final measurement. This simple extension creates a new perspective from which to see the quantum algorithm. First, it highlights the relevance of time-symmetric quantum mechanics to quantum computation: the problem-setting and problem solution, in their quantum version, constitute pre- and post-selection, hence the process as a whole is bound to be affected by both boundary conditions. Second, it forces us to enter into relational quantum mechanics. There must be a representation of the quantum algorithm with respect to Bob, and another one with respect to Alice, from whom the outcome of the initial measurement, specifying the setting and thus the solution of the problem, must be concealed. Time-symmetrizing the quantum algorithm to take into account both boundary conditions leaves the representation to Bob unaltered. It shows that the representation to Alice is a sum over histories in each of which she remains shielded from the information coming to her from the initial measurement, not from that coming to her backwards in time from the final measurement. In retrospect, all is as if she knew in advance, before performing her problem-solving action, half of the information that specifies the solution of the problem she will read in the future and could use this information to reach the solution with fewer computation steps (oracle queries). This elucidates the quantum computational speedup in all the quantum algorithms examined.