Quantum Severalty: Speed-up and Suppress Effect in Searching Problem
QQuantum Severalty: Speed-up and Suppress Effect in Searching Problem
Jin-Hui Zhu, Li-Hua Lu, and You-Quan Li
1, 2, ∗ Zhejiang Province Key Laboratory of Quantum Technology & Device andDepartment of Physics, Zhejiang University, Hangzhou 310027, P.R. China. Collaborative Innovation Center of Advanced Microstructure, Nanjing University, Nanjing 210008, R.R. China. (Received September 17, 2019)The idea that a search efficiency can be increased with the help of a number of autonomous agents is oftenrelevant in many situations, which is known among biologists and roboticists as a stigmergy. This is due to thefact that, in any probability-based search problem, adding information provides values for conditional probabil-ities. We report new findings of a speed-up and suppression effects occurring in the quantum search problemthrough the study of quantum walk on a graph with floating vertices. This effect is a completely counterintuitivephenomenon in comparison to the classical counterpart and may facilitate new insight in the future informationsearch mechanisms that were never been perceived in classical picture. In order to understand the first passageprobability, we also propose a method via ancillary model to bridge the measurement of the time dependenceof the total probability of the complimentary part and the first passage probability of the original model. This isexpected to provide new ideas for quantum simulation by means of qubit chips.
Searching and foraging has long been crucial topics in theecology and microbiology [1, 2], it becomes enormously im-portant in physics as well as in information science nowa-days [3]. The idea that a search efficiency can be increasedwith the help of a number of autonomous agents was oftenrelevant in many situations. Swarm communication is widelyadopted among animals [4] and insects [5], and it is knownamong biologists [6] and roboticists [7] as a stigmergy. Forexample, in the search strategies of some predatory animals,which mix essentials of random-walk model with concen-tration on seasonal hot spots to find preys [8]. Similarly,in biological cells, peptide binding to transmembrane recep-tors relies on hydrophobic attraction superimposed on randomBrownian motion [9]. L´evy walks are characterized by trajec-tories that have straight stretches for extended lengths whosevariance is infinite [10]. Comparing to classical randomwalks, the L´evy walks were shown to being advantageousin optimality of search [11, 12]. Group L´evy foraging withan artificial pheromone communication between robots wasstudied [7] where the experimental results showed that if aninteraction or exchange of information between the searchersis allowed the averaged search time can be decreased substan-tially. All those situations will continue to provide variousquestions for theorists [13].Recently, modelling protein folding as a quantum walk ondefinite graphs reveals a fast protein-folding time [14]. Thismotivate us raise a general question whether quantum walkstrategy can provide us any new insight on the searching effi-ciency [15]. Here we show our new findings that there arespeed-up and suppression effects occurring in the quantumsearch problem that does not occur in its classical counterpart.We introduce a bilayer graph in which a layer of switchablevertices is floating over a main chain of vertices. We studythe cases of the main chain only and compare successivelywith the case of one- and two-vertex on side chain connectingto the main chain. The mean first passage time starting fromone terminal to the other terminal are calculated and the effectcaused by connecting one or two vertices from the side-chain are investigated. As the search efficiency can be quantitativelymeasured by the mean first passage time [16, 17], we investi-gate various situations of our model in the framework of theclassical random walk and quantum walk, respectively. Wealso introduce reduced density matrix and calculate the vonNeumann entropy to get certain knowledge about the counter-intuitive effect. In order to understand the first passage proba-bility, we also propose a method via ancillary model to bridgethe measurement of the time dependence of the total probabil-ity of the complimentary part and the first passage probabilityof the original model. Quantum search’s feature is due to itsparallel calculation strategy and quantum coherence nature. Itis expected to provide new ideas for quantum simulation bymeans of qubit chips.
Model system and the mean first passage time:
We con-sider a two layer system that consists of a back layer (baselayer) of a chain of N vertices and a front layer (float layer) ofseveral initially isolated vertices which are switchable to con-nect with the contact-vertex in base layer. In the float layer,another neighbor vertex can also be switched on or off fromthe aforementioned connected vertex in the float layer (Fig. 1).Thus, we investigate a model of a main chain of N vertices FIG. 1.
Graphical illustration of the model Hamiltonian.
Here thedash-line circle denotes isolated vertex in the float layer that can bepushed to connect with a vertex in the base layer. If another neigh-bor in the float layer is sheared to connect with, a side chain of twovertices is tuned on then. along with a side chain of S vertices mounted at the center.In the model, the number of the side chain can be changed bytuning up one more vertex, that means the S can be changedfrom odd number to even number, or vise versa. We explorehow does the side chain affect the mean first passage time that a r X i v : . [ qu a n t - ph ] S e p costs from initial site, saying vertex- , to target site, sayingvertex- N . The mean first passage time is defined as an inte-grand, namely, τ = ˆ τ tF ,N ( t )d t (cid:46) ˆ τ F ,N ( t )d t, (1)where τ = ∞ in the classical random walk while it oughtto be determined by F ,N ( τ ) = 0 in quantum-walk prob-lem [14]. Here F ,N ( t ) is determined by the convolution re-lation [18–22], P ,N ( t ) = ˆ t F ,N ( t (cid:48) ) P N,N ( t − t (cid:48) )d t (cid:48) , (2)where P a,b ( t ) is the time-dependent probability distribution atvertex- b evolving from a initial distribution located merely atvertex- a , i.e., P a,b (0) = δ ab . The classical random walk:
Random walks on a graph(Fig. 1) is described by the probability distribution over thevertices, namely, p a ( t ) with a = 1 , , · · · , N, N + 1 , N +2 , · · · N + S obeying the following master equation, dd t p a ( t ) = (cid:88) b K ab p b ( t ) . (3)Here K ab = J ab / deg( b ) − δ ab with J ab being the adja-cency matrix of the graph and deg( b ) = (cid:80) c J cb the degreeof vertex-b. Let us first observe the classical random walkon the graph (Fig. 1) for S = 0 , and one by one. Foreach case, we solve the master equation (3), respectively, un-der the initial condition p (0) = 1 and another initial condi-tion p N (0) = 1 . Those solutions provide us P ,a ( t ) and P N,a that determine the so-called first passage probabilities [18–22] F ,N ( t ) through the convolution relation (2). Strictlyspeaking, the F a,b ( t ) measures a probability per unit time. Af-ter solving the time dependence F ,N ( t ) , we can evaluate themean first passage time as an integrand defined by Eq. (1).We numerically solve the random walk on systems with dif-ferent number of vertices on the main chain, saying N = 3 , N = 5 , · · · , till N = 43 . The solution of N = 9 is plot-ted in Fig. S1 and the relevant quantities for solving the firstpassage probabilities are plotted in Fig. 2. We calculate themean first passage time from vertex- to vertex- N , and de-note the obtained value for S = 0 by τ c , that for S = 1 by τ (cid:48) c and that for S = 2 by τ (cid:48)(cid:48) c . Here the subscript “c” refersthe magnitude from classical random walk. Our results ex-hibit that τ (cid:48) c is always larger that τ c and τ (cid:48)(cid:48) c is always largerthat τ (cid:48) c (Fig. 2C and Table S1). This manifests that an addi-tional vertex connected from the float layer always retards themean first passage time [19] of classical random walk in somesense, which just fits with the daily conventional intuition. Quantum severalty:
The quantum walk [23–25] on agraph is described by the time-dependent wavefunction | Ψ( t ) (cid:105) = (cid:80) N + Sa =1 ψ a ( t ) | a (cid:105) that obeys the Schr¨odinger equ-ation, i (cid:126) dd t | Ψ( t ) (cid:105) = ˆ H | Ψ( t ) (cid:105) . (4) Here the Hamiltonian ˆ H = (cid:80) ab J ab | a (cid:105)(cid:104) b | is defined by theadjacency matrix J ab of the graph. After solving Eq. (4) underthe initial condition | Ψ(0) (cid:105) = | (cid:105) and another initial condition | Ψ(0) (cid:105) = | N (cid:105) , respectively, we get P ,N ( t ) = | ψ (1) N ( t ) | and P N,N ( t ) = | ψ ( N ) N ( t ) | (the superscript is introduced todistinguish solutions from different initial conditions), thenobtain F ,N ( t ) from the convolution relation (2). We solvethe quantum mechanical problem for the cases S = 0 , and one by one and furthermore evaluate the mean first passagetime in terms of the solved F ,N ( t ) . The time dependence of P ,N ( t ) , P N,N ( t ) and F ,N ( t ) in quantum case for N = 9 areplotted in Fig. 2D-2F.The calculated mean first passage time is given in the in-serted panel of Fig. 2F. We can see ∆ = τ (cid:48) − τ < , whichimplies that the mean first passage time for S = 1 is shorterthan that for S = 0 . Unlike the classical case where an ad-ditional vertex at side chain will increase the mean first pas-sage time (retard the searching rapidity), quantum mechan-ically, it will speed up the searching rapidity (decrease themean first passage time). Our result from the quantum walkreveals a novel effect that is a completely counterintuitive phe-nomenon. In comparison to its classical counterpart, we sug-gest to call it “quantum severalty”. This is true not only for N = 9 , but also for other number of vertices. We also calcu-late N = 3 , , · · · , till (Table S1) and plot the speed-upratio ∆ /τ as a function of N , the number of the vertices onthe main chain (Fig. 2G). The fitted curve fulfills a power lowbehavior (see the inserted panel of Fig. 2G), namely, ∆ τ ≈ − . N − . . If connecting one more vertex to the already connected oneon the side chain, we find that ∆ = τ (cid:48)(cid:48) − τ turns to be a tinymagnitude. This means tuning on a second vertex on the sidechain, the original speed up effect will be suppressed down atonce. The suppression magnitude ∆ (cid:48) = τ (cid:48)(cid:48) − τ (cid:48) = ∆ − ∆ > . The relative suppression ratio ∆ /τ versus the site number N is shown in Fig. 2H. Thus, if one float vertex is pushed toconnect with the base layer, we attain a significant speed upeffect; furthermore, if a second nearby float vertex is sheeredto connect with the connected float vertex, it causes a suppres-sion effect. It is also worthwhile to know what happens if theside chain is not mounted at the center of the main chain. De-noting the central position as c , we discuss the case when theside chain is on the position c ± , c ± etc.. Our calculationresults exhibit that the speed-up and suppress effect still ex-ists if the the side chain is mounted nearby the center, and themagnitude changes are just affected slightly (Table S2).To help an understanding about the aforementioned quan-tum severalty, we consider a reduced 2 by 2 density ma-trix ˜ ρ in the spirit of Ref. [26] that may maintain certain in-formation of quantum coherence. Because the graph char-acterising our model system is actually a two-color graph(i.e., minimally, two colors are needed to dye every verticeswithout the occurrence of neighbor vertices sharing the samecolor), we are able to define the 2 by 2 density matrix ˜ ρ in Entropy t ( 1 / J )
S = 0
S = 1
S = 2 S I Integrand
F1,9 ( t ) t ( 1 / J ) S = 0
S = 1
S = 2 F (cid:1) s P9,9 ( t ) t ( 1 / J ) S = 0
S = 1
S = 2 E P1,9 ( t ) t ( 1 / J ) S = 0
S = 1
S = 2 D F1,9 ( t ) (cid:5)(cid:1)(cid:3)(cid:2)(cid:4)(cid:6) S = 0
S = 1
S = 2 C (cid:1) s P9,9 ( t ) t ( 1 / J ) S = 0
S = 1
S = 2 B P1,9 ( t ) (cid:5)(cid:1)(cid:3)(cid:2)(cid:4)(cid:6) S = 0
S = 1
S = 2 A ln(| D (cid:1) ) l n ( N ) D t N G (cid:1) E' ' (cid:2) - (cid:1) E (cid:2) D t N H QuantumClassical
FIG. 2.
Classical and quantum solutions, their features of the mean first passage time:
The time dependence of ( A ) P , ( t ) , ( B ) P , ( t ) and ( C ) F , ( t ) solved from classical random walk; the time dependence of ( D ) P , ( t ) , ( E ) P , ( t ) and ( F ) F , ( t ) solved from quantumwalk for N = 9 . ( G ) The quantum speed-up ratio ∆ /τ versus the number of sites N , the inserted panel is the logarithm scaled plot indicatinga power low behavior. ( H ) The relative suppression ratio ∆ /τ versus the number of sites N , presented by small triangular, together with themagnitude difference between the averaged von Neumann entropies for S = 2 and S = 0 presented by histogram which is scaled by the rightvertical axis. ( I ) The time dependence of von Neumann entropies for S = 0 , , when N = 9 . Their integrands are shown in the insertedpanel. terms of ˜ ρ = (cid:80) a ψ a ψ ∗ a , ˜ ρ = (cid:80) a ψ a ψ ∗ a , ˜ ρ = (cid:80) a a ψ a ψ ∗ a / √ n n and ˜ ρ = ˜ ρ ∗ where n and n standfor, respectively, the total numbers of the vertices in the samecolor and the square-root-denominator factor in the summa-tion guarantees the reduced density matrix such defined is ofnon-negatively definite. Then the von Neumann entropy [27]is given by E (˜ ρ ) = − tr(˜ ρ log ˜ ρ ) whose time dependencefor N = 9 is plotted in Fig. 2I. We denote the von Neumannentropies for S = 0 , , respectively by E , E (cid:48) and E (cid:48)(cid:48) , ofwhich the integrand over the whole period τ is indicated byhistograms in the inserted panel. One can also make an av-erage of the von Neumann entropy, (cid:104) E (cid:105) = ´ τ E ( t )d t/τ ,clearly, the difference (cid:104) E (cid:48)(cid:48) (cid:105) − (cid:104) E (cid:105) have something to do withthe ∆ /τ - N dependence (Fig. 2H). All these features are truealso for N = 3 , , · · · (Fig. S2). Implications of the first passage probability:
Classically,the first passage probability F ( t ) can be obtained directly viaensemble simulation. A numerical simulation of one millionensembles gives rise to a first-passage probability that fit wellwith the convolution results (Fig. S3). We know that the onedimensional random walk can be demonstrated by billiardsscattered by periodically placed nails (Fig. 3A). In order to un-derstand the physics implication of the first passage probabil-ity F ,N ( t ) , we make some changes on the set-up frequentlyadopted in science and technology museum. Pouring in bil-liards from the hole close to the left edge, keeping the right-side boundary open and placing an array of collectors to re-ceive, successively, the billiard scattered toward outside, wewill have a F ,N ( t ) after sufficient billiards were poured in.The magnitudes of F ,N ( t ) is valued by accounting the bil- … incident billiard t t collectors incident light observe side tune up/to A B
N N A collectors FIG. 3.
Proposals for experiment set-up : ( A ) Demonstration exper-iment set-up for random walk. This set-up is mounted vertically sothat the gravitational force naturally provide us the time order direc-tion. The collectors on the right-side are labelled by this time order.( B ) Experiment scheme for quantum walk via optical fibre. Incidentbeam of light is applied from one side and the observation is made onthe other side of which the lateral fibre is cut to form a unreflectingsurface. The speed up effect, and furthermore a suppress effect willbe observed by tuning up or tuning to the additional fibre. F1,N (t) t ( 1 / J )
C o n v o l u t i o n S i m u l a t i o n A S i m u l a t i o n B t ( 1 / J )
C o n v o l u t i o n S i m u l a t i o n A S i m u l a t i o n B
C B NN A D
FIG. 4.
Quantum simulation scheme via qubit chip : ( A ) Ancillarymodel with a sticky tail, which is obtained by connecting to the N -thvertex with an additional sticky vertex; ( B ) Ancillary model dressedwith a ring, which is a prolongation of the N -th vertex to be a ringof several more vertices; ( C ) The first passage probability obtained,respectively, by means of the above proposals for N = 43 , and ( D )that for N = 9 . The datas obtained via convolution relation are alsoplotted to compare. liards received by those collectors labelled by the t . Thus, our simulation (Fig. S3) on classical random walk can be realizedby such a demonstration experiment (Fig. 3A).The aforementioned classical set-up motivated us, at once,to propose a quantum simulation experimental scheme. Onecan make a set-up (Fig. 3B) in terms of optical fibre [28] bycutting a lateral fibre to form a reflectionless surface, then theobserved intensity along the fibre will be the F ,N ( t ) . One canalso observe the new phenomena we found previously [29],i.e., the speed up and suppress effects caused by tuning to ortuning up side-chain optical fibre. Quantum simulation by qubit chips:
We propose possi-ble experiment design (Fig. 4) to carry out quantum simula-tions [30] via qubit chips. To measure the first passage proba-bility F ,N ( t ) at vertex- N , we can either connect an additionalsticky vertex to that vertex (Fig. 4A) or make a prolongationof that vertex to be a ring of several more vertices (Fig. 4B).For the former proposal (Fig. 4A), as we know, in the pres-ence of sticky, we need to solve the density matrix from theLindblad equation [31] dd t ˜ ρ = 1 i (cid:126) [ ˜ H, ˜ ρ ] + λ (cid:0) L ˜ ρ L † − ˜ ρ L † L − L † L ˜ ρ (cid:1) , (5)where ˜ H = ˆ H + V | N + 1 (cid:105)(cid:104) N + 1 | with V being thenegative potential on the sticky vertex, and L = | N (cid:105)(cid:104) N + 1 | the Lindblad operator. Once the density matrix ˜ ρ ab ( t ) ( a, b =1 , , · · · , N, N + 1 ) is solved for the ancillary model with asticky tail (Fig. 4A), we are able to obtain the first passageprobability: F ,N ( t ) = − A dd t σ ( t ) , (6)where σ ( t ) = (cid:80) N − a =1 ˜ ρ aa ( t ) and A is a normalization con-stant so that ´ τ F ,N ( t )d t = 1 , i.e., A = σ ( τ ) . If the re-alization of the sticky vertex in some experimental system isuneasy, an alternative proposal is to prolong the vertex N tobe a ring of several qubits (Fig. 4B), we call it the ancillarymodel dressed with a vertices ring. With this ancillary model,one is still able to simulate the F ,N ( t ) by the same formula(6), but, with the σ ( t ) given by σ ( t ) = (cid:80) N − a =1 | ˜ ψ (1) a ( t ) | ( t ) .Our quantum simulation results for N = 43 together with theconvolution result are plotted in Fig. 4C, where the parameterchoices are λ = 4 . , V = − . J and the ring of vertices.Fig. 4D shows that for N = 9 with parameter choices λ = 5 , V = − . J and the ring of vertices. Therefore, there is abridge between the probability measurement of the ancillarymodel and the first passage probability of the original model,which will shed new light on the research area of quantumsimulation.This work is supported by National Key R & D Programof China, Grant No. 2017YFA0304304 and NSFC, Grant No.11935012. ∗ email: [email protected] [1] M. F. Shlesinger, Mathematical physics: Search research. Nature , 281-282 (2006).[2] O. B´enichou, C. Loverdo, M. Moreau, & R. Voituriez, Two-dimensional intermittent search processes: An alternative toL´evy flight strategies.
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