Quantifying Grover speed-ups beyond asymptotic analysis

Abstract

The usual method for studying run-times of quantum algorithms is via an asymptotic, worstcase analysis. Whilst useful, such a comparison can often fall short: it is not uncommon for algorithms with a large worst-case run-time to end up performing well on instances of practical interest. To remedy this it is necessary to resort to run-time analyses of a more empirical nature, which for sufficiently small input sizes can be performed on a quantum device or a simulation thereof. For larger input sizes, alternative approaches are required.In this paper we consider an approach that combines classical emulation with rigorous complexity bounds: simulating quantum algorithms by running classical versions of the sub-routines, whilst simultaneously collecting information about what the run-time of the quantum routine would have been if it were run instead. To do this accurately and efficiently for very large input sizes, we describe an estimation procedure that provides provable guarantees on the estimates that it obtains. A nice feature of this approach is that it allows one to compare the performance of quantum and classical algorithms on particular inputs of interest, rather than only on those that allow for an easier mathematical analysis.We apply our method to some simple quantum speedups of classical heuristic algorithms for solving the well-studied MAX-k-SAT optimization problem. To do this we first obtain some rigorous bounds (including all constants) on the expected-and worst-case complexities of two important quantum sub-routines, which improve upon existing results and might be of broader interest: Grover search with an unknown number of marked items, and quantum maximum-finding. Our results suggest that such an approach can provide insightful and meaningful information, in particular when the speedup is of a small polynomial nature.

Quantum Algorithms for Community Detection and their Empirical Run-times

“…We design several quantum algorithms for community detection on graphs by building quantum versions of the Louvain algorithm, and analyse their complexities in the usual way, finding, as is not uncommon for heuristic quantum algorithms, a per-step speedup. We then apply the bounds and methodology from [9] to estimate the complexities of the algorithms on practical inputs, and investigate whether the speedups promised by the asymptotic analyses manifest in practice.…”

“…In [9], we introduced the necessary tools and methodology for obtaining accurate numerical estimates of the complexities of quantum algorithms with a reasonably generic form that is common to many quantum speedups of classical heuristic algorithms. In particular, we considered algorithms with the form shown in Algorithm 1.…”

“…As such, it is likely that it will be necessary to study the performance of quantum algorithms from a more empirical standpoint in order to assess their usefulness in the time before large, fault tolerant quantum computers become widespread. In [9], we suggested a general framework for doing so that was based on the combination of carefully derived upper bounds on the complexities of quantum sub-routines with classical simulation of the entire algorithm, in a way that allowed for the estimation of the quantum run-time. We gave evidence for the utility of this approach by studying the potential quantum speed-ups that might arise from quantum versions of classical heuristic algorithms for solving maxsat, an optimisation problem that generalises the sat problem.…”

“…In this paper, we apply the methodology developed in [9] to a problem of more practical interest. We consider quantum speedups of a popular heuristic algorithm that goes by the name of the Louvain algorithm 2 which forms one of the main tools for tackling a problem ubiquitous in the study of complex networks: that of community detection.…”

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We apply our recent work [9] on empirical estimates of quantum speedups to the practical task of community detection in complex networks. We design several quantum variants of a popular classical algorithm -the Louvain algorithm for community detection -and first study their complexities in the usual way, before analysing their complexities empirically across a variety of artificial and real inputs.

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