Why adiabatic quantum annealing is unlikely to yield speed-up

Aarón Villanueva; Peyman Najafi; Hilbert J. Kappen

doi:10.1088/1751-8121/ad0439

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arXiv · 2212.13649

Why adiabatic quantum annealing is unlikely to yield speed-up

Aarón Villanueva·Peyman Najafi·Hilbert J. Kappen

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Abstract

We study quantum annealing for combinatorial optimization with Hamiltonian $H = z H_f + H_0$ where $H_f$ is diagonal, $H_0=-|\phi \rangle \langle \phi|$ is the equal superposition state projector and $z$ the annealing parameter. We analytically compute the minimal spectral gap as $\mathcal{O}(1/\sqrt{N})$ with $N$ the total number of states and its location $z_*$. We show that quantum speed-up requires an annealing schedule which demands a precise knowledge of $z_*$, which can be computed only if the density of states of the optimization problem is known. However, in general the density of states is intractable to compute, making quadratic speed-up unfeasible for any practical combinatoric optimization problems. We conjecture that it is likely that this negative result also applies for any other instance independent transverse Hamiltonians such as $H_0 = -\sum_{i=1}^n \sigma_i^x$.

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Aarón Villanueva, Peyman Najafi, Hilbert J. Kappen. 2022-12-28. Why adiabatic quantum annealing is unlikely to yield speed-up. https://doi.org/10.1088/1751-8121/ad0439

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