Abstract
We study both classical and quantum algorithms to solve a hard optimization problem, namely 3-XORSAT on 3-regular random graphs. By introducing a new quasi-greedy algorithm that is not allowed to jump over large energy barriers, we show that the problem hardness is mainly due to entropic barriers. We study, both analytically and numerically, several optimization algorithms, finding that entropic barriers affect in a similar way classical local algorithms and quantum annealing. For the adiabatic algorithm, the difficulty we identify is distinct from that of tunneling under large barriers, but does, nonetheless, give rise to exponential running (annealing) times.
Highlights
Hard discrete optimization problems are ubiquitous in scientific disciplines and practical applications
This physically reasonable expectation implies that quantum algorithms may be faster in the search for optimal configurations than their classical counterparts, and has fueled interest in quantum algorithms that could benefit from this phenomenon: quantum annealing [17,18,19,20,21,22], its more recent variant, the quantum approximate optimization algorithm [21], and population transfer [23,24,25] are some well-known examples
For all the algorithms analyzed, we find that the time to reach a solution scales exponentially with the system size and quantum dynamics seem to suffer from the presence of entropic barriers as much as the classical algorithmic dynamics
Summary
Hard discrete optimization problems are ubiquitous in scientific disciplines and practical applications. Based on the above picture, it is often believed that a quantum evolution—that allows for tunneling events—may escape local minima more efficiently than a classical stochastic dynamics This physically reasonable expectation implies that quantum algorithms may be faster in the search for optimal configurations than their classical counterparts, and has fueled interest in quantum algorithms that could benefit from this phenomenon: quantum annealing [17,18,19,20,21,22], its more recent variant, the quantum approximate optimization algorithm [21], and population transfer [23,24,25] are some well-known examples. As the effort to build a quantum computer are giving up some results [30], we believe it is important to identify all possible stumbling blocks for quantum architectures
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