Deriving Grover's lower bound from simple physical principles

Abstract

Groverʼs algorithm constitutes the optimal quantum solution to the search problem and provides a quadratic speed-up over all possible classical search algorithms. Quantum interference between computational paths has been posited as a key resource behind this computational speed-up. However there is a limit to this interference, at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power? Sorkin has defined a hierarchy of possible interference behaviours-currently under experimental investigation-where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. In this work, we consider how Groverʼs speed-up depends on the order of interference in a theory. Surprisingly, we show that the quadratic lower bound holds regardless of the order of interference. Thus, at least from the point of view of the search problem, post-quantum interference does not imply a computational speed-up over quantum theory. Groverʼs algorithm [12] provides the optimal quantum solution to the search problem and is one of the most versatile and influential quantum algorithms. The search problem-in its simplest form-asks one to find a single 'marked' item from an unstructured list of N elements by querying an oracle which can recognise the marked item. The importance of Groverʼs algorithm stems from the ubiquitous nature of the search problem and its relation to solving NP-complete problems [6]. Classical computers require ( ) O N queries to solve this problem, but quantum computers-using Groverʼs algorithm-only require ( ) O N queries. Quantum interference between computational paths has been posited [32] as a key resource behind this computational 'speed-up'. However, as first noted by Sorkin [29,30], there is a limit to this interference-at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power?Sorkin has defined a hierarchy of possible interference behaviours-currently under experimental investigation [24, 27, 28]-where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. To get a greater understanding of the role of interference in computation, we consider how Groverʼs speed-up depends on the order of interference in a theory.Restriction to the second level of this hierarchy implies many 'quantum-like' features, which, at first glance, appear to be unrelated to interference. For example, such interference behaviour restricts correlations [11] to the 'almost quantum correlations' discussed in [21], and bounds contextuality in a manner similar to quantum theory [14,23]. This, in conjunction with interference being a key resource in the quantum speed-up, suggests that...

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