Complexity Bounds on Quantum Search Algorithms in finite-dimensional Networks

Abstract

We establish a lower bound concerning the computational complexity of Grover's algorithms on fractal networks. This bound provides general predictions for the quantum advantage gained for searching unstructured lists. It yields a fundamental criterion, derived from quantum transport properties, for the improvement a quantum search algorithm achieves over the corresponding classical search in a network based solely on its spectral dimension, $d_{s}$. Our analysis employs recent advances in the interpretation of the venerable real-space renormalization group (RG) as applied to quantum walks. It clarifies the competition between Grover's abstract algorithm, i.e., a rotation in Hilbert space, and quantum transport in an actual geometry. The latter is characterized in terms of the quantum walk dimension $d_{w}^{Q}$ and the spatial (fractal) dimension $d_{f}$ that is summarized simply by the spectral dimension of the network. The analysis simultaneously determines the optimal time for a quantum measurement and the probability for successfully pin-pointing a marked element in the network. The RG further encompasses an optimization scheme devised by Tulsi that allows to tune this probability to certainty, leaving quantum transport as the only limiting process. It considers entire families of problems to be studied, thereby establishing large universality classes for quantum search, which we verify with extensive simulations. The methods we develop could point the way towards systematic studies of universality classes in computational complexity to enable modification and control of search behavior.