Quantifying Grover Speed-ups Beyond Asymptotic Analysis

Chris Cade^1,2, Marten Folkertsma³, Ido Niesen^1,2, and Jordi Weggemans³

¹QuSoft & University of Amsterdam (UvA), Amsterdam, the Netherlands
²Fermioniq, Amsterdam, the Netherlands
³QuSoft & CWI, Amsterdam, the Netherlands

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Abstract

Run-times of quantum algorithms are often studied via an asymptotic, worst-case 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 detailed complexity bounds that include all constants. We simulate 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 and prove that it obtains upper bounds on the true expected complexity of the quantum algorithms.
We apply our method to some simple quantum speedups of classical heuristic algorithms for solving the well-studied MAX-$k$-SAT optimization problem. This requires rigorous bounds (including all constants) on the expected- and worst-case complexities of two important quantum sub-routines: Grover search with an unknown number of marked items, and quantum maximum-finding. These improve upon existing results and might be of broader interest.
Amongst other results, we found that the classical heuristic algorithms we studied did not offer significant quantum speedups despite the existence of a theoretical per-step speedup. This suggests that an empirical analysis such as the one we implement in this paper already yields insights beyond those that can be seen by an asymptotic analysis alone.

Featured image: Numerical results for the query counts on randomly generated max-k-sat instances, with n variables and m = rn weighted (uniformly random between 0 and 1) clauses, of the proposed classical and quantum algorithms that implement a hill climber search routine.

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Popular summary

Run-times of quantum algorithms are often studied via rigorous worst-case analyses. Whilst useful, such comparisons 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. For sufficiently small input sizes, direct quantum (circuit) simulations can be performed on a quantum device or a simulation thereof. However, in order to investigate how such algorithms perform for larger inputs of practical interest, alternative approaches are required.

In this paper we consider an approach that combines classical emulation with detailed complexity bounds. We ‘simulate’ quantum algorithms indirectly: 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. This approach has the advantage that (1) it is not limited to small problem sizes, and (2) it provides a means to estimate input-dependent (upper bounds) to quantum algorithm runtimes, thereby allowing for a comparison of the performance of quantum and classical algorithms on particular inputs of interest.

We apply our method to some simple quantum speedups of classical heuristic algorithms for solving the well-studied MAX-k-SAT optimization problem. Amongst other results, we find that the heuristic algorithms we studied did not offer significant quantum speedups despite the existence of a theoretical speedup. This suggests that such an empirical analysis already yields insights beyond those that can be seen by a worst-case analysis alone.

► BibTeX data

@article{Cade2023quantifyinggrover, doi = {10.22331/q-2023-10-10-1133}, url = {https://doi.org/10.22331/q-2023-10-10-1133}, title = {Quantifying {G}rover speed-ups beyond asymptotic analysis}, author = {Cade, Chris and Folkertsma, Marten and Niesen, Ido and Weggemans, Jordi}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1133}, month = oct, year = {2023}<br /> }

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Cited by

[1] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, “Quantum algorithms: A survey of applications and end-to-end complexities”, arXiv:2310.03011, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-10-10 16:08:31). The list may be incomplete as not all publishers provide suitable and complete citation data.

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This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Source: https://quantum-journal.org/papers/q-2023-10-10-1133/

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Quantifying Grover speed-ups beyond asymptotic analysis

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