[Submitted on 22 Oct 2022]

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    Abstract: Despite its popularity, several empirical and theoretical studies suggest
    that the quantum approximate optimization algorithm (QAOA) has issues in
    providing a substantial practical advantage. So far, those findings mostly
    account for a regime of few qubits and shallow circuits. In this work we extend
    on this by investigating the perspectives of QAOA in a regime of deep quantum

    Due to a rapidly growing range of classical control parameters, we consider
    local search routines as the characteristic class of variation methods for this
    regime. The behaviour of QAOA with local search routines can be best analyzed
    by employing Lie theory. This gives a geometrically nice picture of
    optimization landscapes in terms of vector and scalar fields on a
    differentiable manifold. Our methods are clearly borrowed from the field of
    optimal control theory.

    In the limit of asymptotic circuits we find that a generic QAOA instance has
    many favourable properties, like a unique local minimum. For deep but not close
    to asymptotically deep circuits many of those nice properties vanish. Saddle
    points turn into effective local minima, and we get a landscape with a
    continuum of local attractors and potentially exponentially many local traps.
    Our analysis reveals that statistical distribution properties of traps, like
    amount, sizes, and depths, can be easily accessed by solely evaluating
    properties of the classical objective function. As a result we introduce
    performance indicators that allow us to asses if a particular combinatorial
    optimization problem admits a landscape that is favourable for deep circuit

    Even though we see that there is no free lunch on general instances, certain
    problem classes like random QUBO, MAXCUT on 3-regular graphs, or a very
    unbalanced MAX-$k$-SAT have a chance to perform not too bad in the deep circuit

    Submission history

    From: Lennart Binkowski [view email]

    Sat, 22 Oct 2022 10:17:28 UTC (7,359 KB)

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