Necessity of Superposition of Macroscopically Distinct States for Quantum Computational Speedup

TL;DR

This paper investigates whether superpositions of macroscopically distinct states are necessary for quantum speedup, extending previous conjectures to broader models and confirming their validity for Grover's algorithm.

Contribution

The authors generalize the indices for superpositions of macroscopically distinct states and extend the conjecture to various quantum algorithms, including Grover's search.

Findings

The generalized conjecture applies to a wide class of quantum algorithms.

Grover's algorithm satisfies the extended conjecture despite its quadratic speedup.

The results support the necessity of superpositions of macroscopically distinct states for quantum speedup.

Abstract

For quantum computation, we investigate the conjecture that the superposition of macroscopically distinct states is necessary for a large quantum speedup. Although this conjecture was supported for a circuit-based quantum computer performing Shor's factoring algorithm [A. Ukena and A. Shimizu, Phys. Rev. A69 (2004) 022301], it needs to be generalized for it to be applicable to a large class of algorithms and/or other models such as measurement-based quantum computers. To treat such general cases, we first generalize the indices for the superposition of macroscopically distinct states. We then generalize the conjecture, using the generalized indices, in such a way that it is unambiguously applicable to general models if a quantum algorithm achieves exponential speedup. On the basis of this generalized conjecture, we further extend the conjecture to Grover's quantum search algorithm, whose speedup is large but quadratic. It is shown that this extended conjecture is also correct. Since Grover's algorithm is a representative algorithm for unstructured problems, the present result further supports the conjecture.