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Lower bounds on circuit depth of the quantum approximate optimization algorithm

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Abstract

The quantum approximate optimization algorithm (QAOA) is a method of approximately solving combinatorial optimization problems. While QAOA is developed to solve a broad class of combinatorial optimization problems, it is not clear which classes of problems are best suited for it. One factor in demonstrating quantum advantage is the relationship between a problem instance and the circuit depth required to implement the QAOA method. As errors in noisy intermediate-scale quantum (NISQ) devices increase exponentially with circuit depth, identifying lower bounds on circuit depth can provide insights into when quantum advantage could be feasible. Here, we identify how the structure of problem instances can be used to identify lower bounds for circuit depth for each iteration of QAOA and examine the relationship between problem structure and the circuit depth for a variety of combinatorial optimization problems including MaxCut and MaxIndSet. Specifically, we show how to derive a graph, G, that describes a general combinatorial optimization problem and show that the depth of circuit is at least the chromatic index of G. By looking at the scaling of circuit depth, we argue that MaxCut, MaxIndSet, and some instances of vertex covering and Boolean satisfiability problems are suitable for QAOA approaches while knapsack and traveling salesperson problems are not.

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Acknowledgements

This work was supported by DARPA ONISQ program under award W911NF-20-2-0051. J. Ostrowski acknowledges the Air Force Office of Scientific Research award, AF-FA9550-19-1-0147. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan. (http://energy.gov/downloads/doe-public-access-plan).

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Tennessee at Knoxville, Knoxville, Tennessee, 37996-2315, USA

    Rebekah Herrman & James Ostrowski

  2. Quantum Computing Institute, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 37830, USA

    Travis S. Humble

  3. Department of Physics and Astronomy, University of Tennessee at Knoxville, Knoxville, Tennessee, 37996-1200, USA

    George Siopsis

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  1. Rebekah Herrman
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  2. James Ostrowski
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  3. Travis S. Humble
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  4. George Siopsis
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Correspondence to James Ostrowski.

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Herrman, R., Ostrowski, J., Humble, T.S. et al. Lower bounds on circuit depth of the quantum approximate optimization algorithm. Quantum Inf Process 20, 59 (2021). https://doi.org/10.1007/s11128-021-03001-7

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