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arXiv:1708.05339 · Physical Review A

Complexity bounds on quantum search algorithms in finite-dimensional networks

Stefan Boettcher, Shanshan Li, Tharso D. Fernandes, Renato Portugal

Published Jul 17, 2018• 7 citations
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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 pinpointing a marked element in the network. The RG further encompasses an optimization scheme devised by Tulsi that allows us 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.

DOI
10.1103/physreva.98.012320

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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, ds{d}_{s}ds​. 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 dwQ{d}_{w}^{Q}dwQ​ and the spatial (fractal) dimension df{d}_{f}df​ 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 pinpointing a marked element in the network. The RG further encompasses an optimization scheme devised by Tulsi that allows us 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.
Abstract We find a lower bound on the computational complexity of Grover’s quantum search algorithms in low-dimensional networks using the renormalization group (RG). It highlights the competition between Grover’s abstract algorithm, i.e., a rotation in Hilbert space, and quantum transport in an actual geometry. It can be characterized in terms of the quantum walk dimension <?CDATA dwQ{d}_{w}^{Q}dwQ​?> d w Q and the spatial (fractal) dimension df or, alternatively, the spectral dimension of the network, ds , even when translational invariance is broken.
We consider the quantum search problem with a continuous-time quantum walk for networks characterized by a finite spectral dimension ds{d}_{s}ds​ of the network Laplacian. For general networks of fractal (integer or noninteger) dimension df{d}_{f}df​, for which in general ${d}_{f}\ensuremath{\ne}{d}_{s}$, it suggests that it is ds{d}_{s}ds​ that determines the computational complexity of the quantum search. Our results continue those of A. M. Childs and J. Goldstone [Phys. Rev. A 70, 022314 (2004)] for lattices of integer dimension, where d=df=dsd={d}_{f}={d}_{s}d=df​=ds​. Thus, we find for general fractals that the Grover limit of quantum search can be obtained whenever ds>4{d}_{s}>4ds​>4. This complements the recent discussion of mean-field (i.e., ${d}_{s}\ensuremath{\rightarrow}\ensuremath{\infty}$) networks by S. Chakraborty et al. [Phys. Rev. Lett. 116, 100501 (2016)] showing that for all those networks, spatial search by quantum walk is optimal.
Abstract We consider discrete-time evolution equations in which the stochastic operator of a classical random walk is replaced by a unitary operator. Such a problem has gained much attention as a framework for coined quantum walks that are essential for attaining the Grover limit for quantum search algorithms in physically realizable, low-dimensional geometries. In particular, we analyze the exact real-space renormalization group (RG) procedure recently introduced to study the scaling of quantum walks on fractal networks. While this procedure, when implemented numerically, was able to provide some deep insights into the relation between classical and quantum walks, its analytic basis has remained obscure. Our discussion here is laying the groundwork for a rigorous implementation of the RG for this important class of transport and algorithmic problems, although some instances remain unresolved. Specifically, we find that the RG fixed-point analysis of the classical walk, which typically focuses on the dominant Jacobian eigenvalue λ 1 , with walk dimension d w R W = log 2 ⁡ λ 1 , needs to be extended to include the subdominant eigenvalue λ 2 , such that the dimension of the quantum walk obtains d w Q W = log 2 ⁡ λ 1 λ 2 . With that extension, we obtain analytically previously conjectured results for d w Q W of Grover walks on all but one of the fractal networks that have been considered.
We introduce a framework to analyze quantum algorithms with the renormalization group (RG). To this end, we present a detailed analysis of the real-space RG for discrete-time quantum walks on fractal networks and show how deep insights into the analytic structure as well as generic results about the long-time behavior can be extracted. The RG flow for such a walk on a dual Sierpinski gasket and a Migdal-Kadanoff hierarchical network is obtained explicitly from elementary algebraic manipulations, after transforming the unitary evolution equation into Laplace space. Unlike for classical random walks, we find that the long-time asymptotics for the quantum walk requires consideration of a diverging number of Laplace poles, which we demonstrate exactly for the closed-form solution available for the walk on a one-dimensional loop. In particular, we calculate the probability of the walk to overlap with its starting position, which oscillates with a period that scales as NdwQ/df{N}^{{d}_{w}^{Q}/{d}_{f}}NdwQ​/df​ with system size NNN. While the largest Jacobian eigenvalue ${\ensuremath{\lambda}}_{1}$ of the RG flow merely reproduces the fractal dimension, ${d}_{f}={log}_{2}{\ensuremath{\lambda}}_{1}$, the asymptotic analysis shows that the second Jacobian eigenvalue ${\ensuremath{\lambda}}_{2}$ becomes essential to determine the dimension of the quantum walk via ${d}_{w}^{Q}={log}_{2}\sqrt{{\ensuremath{\lambda}}_{1}{\ensuremath{\lambda}}_{2}}$. We trace this fact to delicate cancellations caused by unitarity. We obtain identical relations for other networks, although the details of the RG analysis may exhibit surprisingly distinct features. Thus, our conclusions---which trivially reproduce those for regular lattices with translational invariance with df=d{d}_{f}=ddf​=d and dwQ=1{d}_{w}^{Q}=1dwQ​=1---appear to be quite general and likely apply to networks beyond those studied here.
The task of finding an entry in an unsorted list of NNN elements famously takes O(N)O(N)O(N) queries to an oracle for a classical computer and O(N)O(\sqrt{N})O(N​) queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large NNN, the random walks converge to the same evolution, both taking Nln⁡(1/ϵ)N \ln(1/\epsilon)Nln(1/ϵ) time to reach a success probability of 1−ϵ1-\epsilon1−ϵ. In contrast, the discrete-time quantum walk asymptotically takes πN/22\pi\sqrt{N}/2\sqrt{2}πN​/22​ timesteps to reach a success probability of 1/21/21/2, while the continuous-time quantum walk takes πN/2\pi\sqrt{N}/2πN​/2 time to reach a success probability of 111.
In the first part of the thesis we construct a new, simple framework which amplifies to a constant the success probability of any abstract search algorithm. The total query complexity is given by the quantum hitting time of the resulting operator, which we show that it is of the same order as the quantum hitting time of the original algorithm. As a major application of our framework, we show that for any reversible walk PPP and a single marked state, the quantum walk corresponding to PPP can find the solution using a number of queries that is quadratically smaller than the classical hitting time of PPP. Our algorithm is more general and simpler to implement than the solution known previously in the literature (Krovi, Magniez, Ozols, and Roland, 2015), which was developed specifically for quantum walks; we also prove that, for the particular case of quantum walks, we can embed their algorithm into our framework, thus simulating it exactly. Finally, we show that we can implement amplitude amplification using our tool. In the second part of the thesis we give a new lower bound in the query model which proves that Grover's algorithm for unordered searching is exactly optimal. Similar to existing methods for proving lower bounds, we bound the amount of information we can gain from a single oracle query, but we bound this information in terms of angles. This allows our proof to be simple, self-contained, based on only elementary mathematics, capturing our intuition, while obtaining at the same an exact bound. We then turn our attention to non-adaptive algorithms for the same problem of searching an unordered set. In this model, we obtain a lower bound and we give an algorithm which matches the lower bound, thus showing that the lower bound is exactly optimal.