Vertices cannot be hidden from quantum spatial search for almost all random graphs

Glos, Adam; Krawiec, Aleksandra; Kukulski, Ryszard; Puchała, Zbigniew

doi:10.1007/s11128-018-1844-7

Cited by 18 publications

(25 citation statements)

References 24 publications

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“…II, we prove that all Kronecker graphs generated by such regular, dominant-eigenvalue initiators asymptotically support optimal quantum search, reaching a success probability of 1 at time π √ N /2, and we give the critical jumping rate γ c that enables this. We prove this using properties of Kronecker products, a Lemma by Chakraborty et al [25], and an extension by Glos et al [26]. This general result proves Wong et al's conjecture with the complete initiator [9] as a special case.…”

Section: Introductionsupporting

confidence: 63%

“…Recall from the Lemma that c is an upper bound on the magnitudes of the non-principal eigenvalues of H 1 . Glos et al [26] noted that the supplemental material of [25] gives an explicit bound on r between −c/(1 + c) and c/(1 − c), inclusive.…”

Section: Dominant Eigenvalue Initiatorsmentioning

confidence: 99%

“…Glos et al [26] also noted that if c = o(1), then the success probability is not just any constant, but is asymptotically 1. Similarly, the runtime is not just Θ( √ N ), but is asymptotically π √ N /2.…”

Section: Dominant Eigenvalue Initiatorsmentioning

confidence: 99%

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Optimal quantum-walk search on Kronecker graphs with dominant or fixed regular initiators

Glos

Wong

2018

Phys. Rev. A

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In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such Kronecker graphs can be searched by a continuoustime quantum walk in optimal Θ( √ N ) time. First, if the initiator is regular and its adjacency matrix has a dominant principal eigenvalue, meaning its unique largest eigenvalue asymptotically dominates the other eigenvalues in magnitude, then the Kronecker graphs generated by this initiator can be quantum searched with probability 1 in π √ N /2 time, asymptotically, and we give the critical jumping rate of the walk that enables this. Second, for any fixed initiator that is regular, non-bipartite, and connected, the Kronecker graphs generated by it are quantum searched in Θ( √ N ) time. This greatly extends the number of Kronecker graphs on which quantum walks are known to optimally search. If the fixed, regular, connected initiator is bipartite, however, then search on its Kronecker powers is not optimal, but is still better than classical computer's O(N ) runtime if the initiator has more than two vertices.

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Section: Introductionsupporting

confidence: 63%

Section: Dominant Eigenvalue Initiatorsmentioning

confidence: 99%

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Optimal quantum-walk search on Kronecker graphs with dominant or fixed regular initiators

Glos

Wong

2018

Phys. Rev. A

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“…In recent years, many other graph topologies have been considered. For instance, the algorithm has been investigated on complete bipartite graphs [5], on balanced trees [6], on Erdös-Rényi graphs [7,8], on the simplex of the complete graph [9] and on graphs with fractal dimensions [10,11]. Moreover, it has been shown that high connectivity and global symmetry of the graph are not necessary for fast quantum search [12,13].…”

Section: Introductionmentioning

confidence: 99%

Quantum spatial search on graphs subject to dynamical noise

et al. 2018

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We address quantum spatial search on graphs and its implementation by continuous-time quantum walks in the presence of dynamical noise. In particular, we focus on search on the complete graph and on the star graph of order N, also proving that noiseless spatial search shows optimal quantum speedup in the latter, in the computational limit N 1. The noise is modeled by independent sources of random telegraph noise (RTN), dynamically perturbing the links of the graph. We observe two different behaviors depending on the switching rate of RTN: fast noise only slightly degrades performance, whereas slow noise is more detrimental and, in general, lowers the success probability. In particular, we still find a quadratic speed-up for the average running time of the algorithm, while for the star graph with external target node we observe a transition to classical scaling. We also address how the effects of noise depend on the order of the graphs, and discuss the role of the graph topology. Overall, our results suggest that realizations of quantum spatial search are possible with current technology, and also indicate the star graph as the perfect candidate for the implementation by noisy quantum walks, owing to its simple topology and nearly optimal performance also for just few nodes.

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“…Such problems involve running quantum algorithms for finding a marked node, known as quantum spatial search, in reverse. In fact, it has already been established that the problem of spatial search by quantum walk is optimal for G(n, p) [27][28][29]. Asymptotic dynamics of coined quantum walks on percolation graphs has been studied [30] while quantum dynamics on complex networks has also been numerically investigated [31][32][33].…”

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confidence: 99%

How Fast Do Quantum Walks Mix?

2020

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The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as mixing, finds widespread applications in several areas of quantum information and computation. Of particular interest in most of these applications, is the minimum time beyond which the instantaneous probability distribution of the quantum walk remains close to this limiting distribution, known as the quantum mixing time. However this quantity is only known for a handful of specific networks. In this letter, we prove an upper bound on the quantum mixing time for almost all networks, i.e. the fraction of networks for which our bound holds, goes to one in the asymptotic limit. To this end, using several results in random matrix theory, we find the quantum mixing time of Erdös-Renyi random networks: networks of n nodes where each edge exists with probability p independently. For example for dense random networks, where p is a constant, we show that the quantum mixing time is O(n 3/2+o(1) ). Besides opening avenues for the analytical study of quantum dynamics on random networks, our work could find applications beyond quantum information processing. Owing to the universality of Wigner random matrices, our results on the spectral properties of random graphs hold for general classes of random matrices that are ubiquitous in several areas of physics. In particular, our results could lead to novel insights into the equilibration times of isolated quantum systems defined by random Hamiltonians, a foundational problem in quantum statistical mechanics.The quantum dynamics of any discrete system can be captured by a quantum walk on a network, which is a universal model for quantum computation [1]. Besides being a useful primitive to design quantum algorithms [2][3][4][5][6], quantum walks are a powerful tool to model transport in quantum systems such as the transfer of excitations in light-harvesting systems [7][8][9]. Studying the long-time dynamics of quantum walks on networks is crucial to the understanding of these diverse problems. As quantum evolutions are unitary and hence distancepreserving, quantum walks never converge to a limiting distribution, unlike their classical counterpart. However, given a network of n nodes, one can define the limiting distribution of quantum walk on the network as the long-time average probability distribution of finding the walker in each node [10]. Of particular interest is the quantum mixing time: starting from some initial state, the minimum time after which the underlying quantum walk remains close to its limiting distribution.The importance of the problem of mixing for quantum walks cannot be overstated: this is at the heart of quantum speedups for a number of quantum algorithms [2,11] and is also key to demonstrating the equivalence between the standard (circuit) and Hamiltonian-based models of quantum computation [12,13]. Unfortunately, no general result exists for quantum mixing time on networks: it has been estimated for a handful of specific graphs (graphs and networks are ...

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Vertices cannot be hidden from quantum spatial search for almost all random graphs

Cited by 18 publications

References 24 publications

Optimal quantum-walk search on Kronecker graphs with dominant or fixed regular initiators

Optimal quantum-walk search on Kronecker graphs with dominant or fixed regular initiators

Quantum spatial search on graphs subject to dynamical noise

How Fast Do Quantum Walks Mix?

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