Nanoscale Photonic Network for Solution Searching and Decision Making Problems

2724IEICE TRANS. COMMUN., VOL.E96–B, NO.11 NOVEMBER 2013

INVITED PAPER Special Section on Progress in Information Network Science

Nanoscale Photonic Network for Solution Searching and DecisionMaking Problems

Makoto NARUSE†a), Masashi AONO††, and Song-Ju KIM†††, Members

SUMMARY Nature-inspired devices and architectures are attractingconsiderable attention for various purposes, including the development ofnovel computing techniques based on spatiotemporal dynamics, exploitingstochastic processes for computing, and reducing energy dissipation. Thispaper demonstrates that networks of optical energy transfers between quan-tum nanostructures mediated by optical near-field interactions occurring atscales far below the wavelength of light could be utilized for solving a con-straint satisfaction problem (CSP), the satisfiability problem (SAT), and adecision making problem. The optical energy transfer from smaller quan-tum dots to larger ones, which is a quantum stochastic process, dependson the existence of resonant energy levels between the quantum dots or astate-filling effect occurring at the larger quantum dots. Such a spatiotem-poral mechanism yields different evolutions of energy transfer patterns inmulti-quantum-dot systems. We numerically demonstrate that networksof optical energy transfers can be used for solution searching and deci-sion making. We consider that such an approach paves the way to a novelphysical informatics in which both coherent and dissipative processes areexploited, with low energy consumption.key words: nanophotonics, optical energy transfer, nature-inspired archi-tecture, solution searching, decision making

1. Introduction

There is great demand for novel computing devices andarchitectures that can overcome the limitations of conven-tional technologies based solely on electron transfer, interms of reducing power dissipation, solving computation-ally intractable problems, and so on [1]. Also, nature-inspired architectures are attracting significant attentionfrom various research areas, such as brain-like computingand computational neurosciences [2], stochastic-based com-puting and noise-based logic [3], and spatiotemporal com-putation dynamics [4].

Among these research topics, Aono et al. demonstrated“amoeba-based computing” by utilizing the spatiotemporaloscillatory dynamics of the photoresponsive amoeboid or-ganism Physarum combined with external optical controlto solve a constraint satisfaction problem (CSP) [4] and thetraveling salesman problem (TSP) [5]. Besides such experi-

Manuscript received April 9, 2013.Manuscript revised June 25, 2013.†The author is with Photonic Network Research Institute, Na-

tional Institute of Information and Communications Technology,Koganei-shi, 184-8795 Japan.††The author is with Earth-Life Science Institute, Tokyo Insti-

tute of Technology, Tokyo, 152-8550 Japan.†††The author is with International Center for Materials Nanoar-

chitectonics, National Institute for Materials Science, Tsukuba-shi,305-0044 Japan.

a) E-mail: [email protected]: 10.1587/transcom.E96.B.2724

mental demonstrations, Leibtnitz et al. showed an algorithmfor selecting the most suitable and robust network by utiliz-ing fluctuations, inspired by biological experiments wherethe speed of fluorescence evolution of proteins in bacteria isobserved to have a positive correlation with the phenotypicfluctuation of fluorescence over clone bacteria [6].

These demonstrations indicate that we can utilize theinherent spatial and temporal dynamics appearing in phys-ical processes in nature for novel computing architecturesand applications. Such arguments should also be applicableto nanometer-scale light–matter interactions. In fact, Naruseet al. demonstrated nanophotonic computing based on opti-cal near-field processes at scales below the wavelength oflight [7]. In particular, energy transfer between quantumnanostructures mediated by optical near-field interactions,detailed in Sect. 2 below, plays a crucial role. Optical near-field interactions, which are described by a Yukawa-type po-tential, have been used to realize energy transfer that in-volves conventionally dipole-forbidden energy levels. Itstheoretical foundation has been explained by the dressedphoton model [8], and the process has been experimentallydemonstrated in various quantum nanostructures, such as In-GaAs [9], ZnO [10], and CdSe [11]. In particular, Kawazoeet al. recently demonstrated room-temperature optical en-ergy transfer using two-layer InGaAs quantum dots (QDs)[12]. In addition, the optical energy transfer has been shownto be 104-times more energy efficient than that of the bit-flipenergy required in conventional electrically wired devices[13].

This article demonstrates that a network of optical en-ergy transfers between quantum dots mediated by opticalnear-field interactions can be utilized for solving the CSP,the satisfiability problem (SAT), and the multi-armed ban-dit problem (BP), which is a decision making problem.The optical energy transfer from smaller quantum dots tolarger ones depends on the existence of resonant energy lev-els between the quantum dots or a state-filling effect oc-curring at the larger destination quantum dots. Also, asindicated by quantum master equations, the energy trans-fer process is fundamentally probabilistic. Such a spa-tiotemporal mechanism yields different evolutions of energytransfer patterns combined with certain control mechanisms,which we call bounceback control, similarly to the evolu-tion of the shape of Physarum demonstrated by Aono et al.in [4]. At the same time, in contrast to biological organ-isms, optical energy transfer is implemented by highly con-trolled engineering means for designated structures, such

Copyright c© 2013 The Institute of Electronics, Information and Communication Engineers

NARUSE et al.: NANOSCALE PHOTONIC NETWORK FOR SOLUTION SEARCHING AND DECISION MAKING PROBLEMS2725

as semiconductor quantum nanostructures fabricated by, forinstance, molecular beam epitaxy [14] or DNA-based self-assembly [15]. The operating speed of such optical-near-field-mediated quantum dot systems, which is on the orderof nanoseconds when radiative relaxation processes are in-volved, is significantly faster than those based on biologicalorganisms, which is on the order of seconds or minutes [4],[5]. The energy efficiency [13], as indicated already above,and the possibility of room-temperature operation [12] arealso strong motivations behind the investigations describedin this paper. In addition, we should emphasize that the con-cept and the principles discussed in this paper are fundamen-tally different from those of conventional optical computingor optical signal processing, which are limited by the proper-ties of propagating light [16]. The concept and principles arealso different from the quantum computing paradigm wherea superposition of all possible states is exploited to lead toa correct solution [17]. The optical-near-field–mediated en-ergy transfer is a coherent process, suggesting that an opti-cal excitation could be transferred to all possible destinationQDs via a resonant energy level, but such a coherent interac-tion between QDs results in a unidirectional energy transferby an energy dissipation process occurring in the larger dot,as described in Sect. 2 below. Thus, our approach opens upthe possibility of another computing paradigm where bothcoherent and dissipative processes are exploited.

This paper is organized as follows. Section 2 charac-terizes a nanoscale network of optical energy transfers viaoptical near-field interactions. Sections 3, 4, and 5 respec-tively demonstrate solving CSP, SAT, and decision makingproblems. Section 6 concludes the paper.

2. Nanoscale Network of Optical Energy Transfer

Here we assume two cubic quantum dots whose side-lengthsare a and

√2a, which we call QDS and QDL1, respectively,

as shown in Fig. 1(a). There exists a resonance between thelevel of quantum number (1,1,1) in QDS, denoted by S inFig. 1(a), and that of quantum number (2,1,1) in QDL1, de-noted by L(U)

1 . Note that the (2,1,1)-level in QDL1 is a dipole-forbidden energy level, meaning that propagating light can-not populate this level via optical excitations. However, op-tical near-fields allow this level to be populated thanks tothe localized inhomogeneous fields in the vicinity of QDS.Therefore, an exciton in the (1,1,1)-level in QDS could betransferred to the (2,1,1)-level in QDL1. In QDL1, due tothe sublevel energy relaxation with a relaxation constant Γ,which is faster than the near-field interaction, the excitonrelaxes to the (1,1,1)-level, denoted by L(L)

1 , from where itradiatively decays. As a result, we find unidirectional opti-cal excitation transfer from QDS to QDL1.

When the lower energy level of the destination quan-tum dot is filled via another excitation (called “state filling”),an optical excitation occurring in a smaller QD (QDS) can-not move to a larger one (QDL1). This suggests two differentpatterns of optical energy transfer appear depending on theoccupation of the destination quantum dot (Fig. 1(b)).

Fig. 1 (a,b) Optical energy transfer between QDs mediated by near-fieldinteractions. (c,d) Network of optical near-field interactions for solutionsearching.

The key to achieving solution searching and decisionmaking is to formulate a network of optical energy trans-fers. For instance, in the case of solving a CSP, shown inSect. 3, we design an architecture where a smaller QD, la-beled QDS, is surrounded by four larger QDs, labeled QDL1,QDL2, QDL3, and QDL4, as indicated in Fig. 1(c). Fig-ure 1(d) shows representative parameterizations associatedwith the system. The (1,1,1)-level in QDS is denoted by S,and the (2,1,1)-level in QDLi is denoted by L(U)

i . These levelsare resonant with each other and are connected by inter-dotinteractions denoted by USLi (i = 1, . . . , 4). The lower levelin QDLi, namely the (1,1,1)-level, is denoted by L(L)

i , whichcould be filled via the sublevel relaxation from L(U)

i , denotedby ΓLi. The radiations from the S and Li levels are respec-tively represented by the relaxation constants γS and γLi. Inthe following description, we call the inverse of the relax-ation constant the radiation lifetime. We also assume thatthe photon radiated from the lower level in QDLi can be sep-arately captured by photodetectors. In addition, we assumecontrol light beams, denoted by CLi in Fig. 1(c), that can in-duce a state filling effect at L(L)

i . Summing up, Figs. 1(c)and (d) schematically represent the basic architecture of thesystem to be studied for solving a CSP, described in Sect. 3,and an SAT problem, described in Sect. 4.

First, we assume that the system initially has one exci-ton in S. From the initial state, through the inter-dot inter-actions USLi, the exciton in S can be transferred to L(U)

i (i =1, . . . , 4). Correspondingly, we can derive quantum masterequations in the density matrix formalism [8]. The Liouvilleequation for the system is then given by

dρ(t)dt= − i�

[Hint, ρ(t)] − NΓρ(t) − ρ(t)NΓ, (1)

where ρ(t) is the density matrix with respect to the five en-ergy levels, Hint is the interaction Hamiltonian, and NΓ indi-cates relaxations. In the numerical calculation, we assume


Fig. 2 (a) Calculated energy transfer probabilities depending on the con-trol light beams. (b) Schematic representation of possible states of the sys-tem. States (7) and (10) correspond to the correct solutions.

U−1SLi = 100 ps, Γ−1

i =10 ps, γ−1Li =1 ns, and γ−1

S ≈ 2.92 ns as atypical parameter set [18].

Based on the above modeling and parameterizations,we can calculate the populations involving L(L)

1 , L(L)2 , L(L)

3 ,

and L(L)4 , which are relevant to the radiation from the larger

QDs. Also, when QDLi is subjected to state filling by controllight CLi, the energy transfer from QDS to QDLi behavesdifferently.

We assume that the probability of energy transfer toQDLi is correlated with the integral of the population ofL(L)

i , as summarized in Fig. 2(a). We should note that suchtime integrals of the populations are indeed a figure-of-merit(FoM) indicating the trend of optical energy transfer fromthe smaller dot to the four larger ones. Therefore, the law ofconservation of probability does not hold; namely, the sum-mation of the transition probabilities to QDLi is not unity.Instead, we see that the energy transfer to QDLi occurs if arandom number generated uniformly between 0 and 1 is lessthan the transition probability to QDLi shown in Fig. 2(a);for example in the case of Fig. 2(a,(3)), the energy trans-fer to QDL4 is highly likely, whereas the transfers to QDL1,QDL2, and QDL3 are less likely. One minor remark regard-ing Fig. 2(a,(1)) is that the energy transfer probability toQDL3 is higher than those to QDL2 and QDL4. This arisesfrom the network structure shown in Fig. 1(d) where thephysical distance from QDL1 to QDL3 is larger than to QDL2

and QDL4; and thus QDL3 is less likely affected by the statefilling induced in QDL1 compared with QDL3 and QDL4.

The idea for problem solving is to control the opticalenergy transfer by controlling the destination QD using con-trol light with a suitable mechanism, what we call bounce-back control. The notion of bounceback, rather than feed-back, implies that the system does not know the preferredstatus beforehand, in contrast to feedback control, whichutilizes the difference between the present and the intendedstates.

3. Solving the Constraint Satisfaction Problem

We consider the following constraint satisfaction problem asan example regarding an array of N binary-valued variablesxi (i = 1, . . . ,N) [19]. The constraint is that xi = NOR(xi−1,

xi+1) should be satisfied for all i. That is, variable xi shouldbe consistent with a logical NOR operation of its two neigh-bors. For i=1 and N, the constraints are respectively givenby x1 = NOR(xN , x2) and xN = NOR(xN−1, x1). We callthis problem the “NOR problem” in this paper. Taking ac-count of the nature of an individual NOR logic operation,one important inherent property is that, if xi = 1, then its twoneighbors should both be zero, i.e., xi−1 = xi+1 = 0. Now,we suppose that a photon radiated, or observed, from the en-ergy level L(L)

i corresponds to a binary value xi=1, whereasthe absence of an observed photon means xi=0. Therefore,xi=1 should mean that optical energy transfer to both L(L)

i−1and L(L)

i+1 is prohibited, so that xi−1 = xi+1 = 0 is satisfied.Therefore, the bounceback mechanism is:[Bounceback rule for the NOR problem] If xi = 1 at timet, then the control light beams Ci−1 and Ci+1 are turned on attime t = t + 1.

In the case of N=4, there are in total 24 optical energytransfer patterns from the smaller dot to the larger ones. Inthis case, variables satisfying the constraints exist, and theyare given by {x1, x2, x3, x4} = {0, 1, 0, 1} and {1, 0, 1, 0},which we call “correct solutions”. Figure 2(b) schematicallyrepresents some of the possible states, where States (7) and(10) respectively correspond to the correct solutions.

There are a few remarks that should be made regardingthe NOR problem. One is about potential deadlock, analo-gous to Dijkstra’s “dining philosophers problem”, as alreadyargued by Aono et al. in [4]. Starting with an initial statexi=0 for all i, and assuming a situation where optical energyis transferred to all larger QDs, we observe photon radiationfrom all energy levels L(L)

i , namely, xi=1 for all i. Then,based on the bounceback mechanism shown above, all con-trol light beams are turned on. If such a bounceback mech-anism perfectly inhibits the optical energy transfer from thesmaller QD to the large ones at the next step t+1, the vari-ables then go to xi=0 for all i. This leads to all control lightbeams being turned off at t+2. In this manner, all variablesconstantly repeat periodic switching between xi=0 and xi=1in a synchronized manner. Consequently, the system cannever reach the correct solutions. However, as indicatedin Fig. 2(a), the probability of optical energy transfer to thelarger dots is in fact not zero even when all larger QDs areilluminated by control light beams, as shown in Fig. 2(a,(4)).Also, even for a non-illuminated destination QD, the energytransfer probability may not be exactly unity. Such stochas-tic behavior of the optical energy transfer plays a key rolein solving the NOR problem. This nature is similar to whatwas demonstrated in the amoeba-based computer [4], wherefluctuations of chaotic oscillatory behavior involving spon-taneous symmetry breaking in the amoeboid organism guar-antees such a critical property.

The operating dynamics cause one pattern to change toanother one every iteration cycle. Thanks to the stochasticnature, each trial could exhibit a different evolution of theenergy transfer patterns. In particular, the transition proba-bility, shown in Fig. 2(a), affects the behavior of the tran-sitions. Therefore, we introduce a gain factor (G) to be


Fig. 3 (a) The evolution of the ratio of the output appearance from QDLi,and (b) the ratio of the states corresponding to correct solutions. (c,d) Time-averaged traces of (b) and (c), respectively. (e) The evolution of the ratioof the output appearance from QDLi, and (f) the ratio of the states corre-sponding to correct solutions, with the initial state (7).

multiplied by the energy transfer probability summarized inFig. 2(a).

The curves in Fig. 3(a) represent the evolution of theoutput appearance from QDLi, namely, the incidence ratiowhen xi=1 among 1,000 trials evaluated at each cycle. Thecurves in Fig. 3(b) characterize the ratio of appearance ofthe states that correspond to the correct solutions: {0,1,0,1}(State (7)) and {1,0,1,0} (State (10)), respectively. When weclosely examine the evolutions of xi in Fig. 3(a), we can seethat the pair x1 and x3 exhibit similar behavior, as do the pairx2 and x4. Also, as the former pair exhibit larger values,the latter pair exhibit smaller values, and vice versa. Thiscorresponds to the fact that correct solutions are likely to beinduced as the number of iteration cycles increases.

Such a tendency is more clearly represented when weevaluate the time-averages of the characteristics in Figs. 3(a)and (b). Figure 3(c) shows the evolutions of the ratio ofthe incidences when xi=1, and Fig. 3(d) shows the ratios ofState (7) and State (10) averaged over every 5 cycles. Wecan clearly observe a similar tendency to the one describedabove. Also, we should emphasize that, thanks to the prob-abilistic nature of the system, the states of correct solutionsappear in an interchangeable manner. This is a clear indi-cation of the fact that the probabilistic nature of the systemautonomously seeks the solutions that satisfy the constraintsof the NOR problem; the state-dependent probability of en-ergy transfer plays a critical role in this. In other words, itshould be emphasized that a non-local correlation is mani-fested in the evolution of xi; for instance, when the systemis in State (7), {0,1,0,1}, the probabilities of energy trans-fer to QDL1 and QDL3 are equally comparably low (due tostate filling), whereas those to QDL2 and QDL4 are equallycomparably high, indicating that the probability of energy

transfer to an individual QDLi has inherent spatial patterns ornon-local correlations. At the same time, the energy transferto each QDLi is indeed probabilistic; therefore, the energytransfer probability to, for instance, QDL1 is not zero evenin State (7), and thus, the state could transition from State(7) to State (10), and vice versa. In fact, starting with theinitial condition of State (7), the ratio of output appearancefrom QDL1 and the ratio of the correct solutions evolve asshown in Figs. 3(e) and (f), where States (7) and (10) occurequally in the steady state at around 20 time cycles.

4. Solving the Satisfiability Problem (SAT)

SAT is the first problem proven to be nondeterministic poly-nomial time (NP)-complete, i.e., the most difficult problemamong those that belong to the complexity class NP [20].Given a logical formula φ, which consists of N Boolean vari-ables xi ∈ {0 {false), 1(true)} (i ∈ I = {1, 2, . . . ,N}), SAT isthe problem of determining whether there exists at least one“satisfying” assignment of the truth values (0 or 1) to thevariables represented by xi such that it makes the formulaevaluate to true (φ = 1). Roughly speaking, φ represents alogical proposition, and the existence of a satisfying assign-ment verifies that the proposition is self-consistent. For ex-ample, the formula φex = (x1∨¬x2)∧(¬x2∨ x3∨¬x4)∧(x1∨x3)∧(x2∨¬x3)∧(x3∨¬x4)∧(¬x1∨x4) has a unique solution(x1, x2, x3,x4) = (1,1,1,1) that makes φex = 1. This sectiondescribes a SAT problem solver inspired by the spatiotem-poral dynamics of the network of optical energy transfers,what we call “NanoPS” [21].

SAT is called 3-SAT when φ consists of M clauses thatare connected by ∧ (logical AND), and each clause connectsat most three literals by ∨ (logical OR). Any SAT instancecan be transformed into a 3-SAT instance, and 3-SAT is alsoNP-complete. A powerful SAT solver has great potentialfor a wide range of applications, such as artificial intelli-gence, information security, and bioinformatics, because theNP-completeness implies that all NP problems, includingmany practical real-world problems, can be transformed tothe SAT problem [20].

In solving SAT by networks of optical excitation trans-fers, we assign two larger-sized quantum dots to a singlevariable xi; namely, a QDL for representing xi=0, and an-other QDL for xi=1. Therefore, to solve N-variable 3-SAT,we use 2N QDLs. QDi,v denotes the variable correspondingto xi = v, where v is either 0 or 1 and i ∈ I = {1, 2, . . . ,N}.When optical energy is transferred from QDS to QDi,v andradiation is subsequently observed at a time step t, we writethis as Ri,v(t)=1, whereas Ri,v(t)=0 indicates that no radiationoccurs. When state-filling stimulation is applied to QDi,v,we denote this as Fi,v(t)=1, whereas Fi,v(t)=0 denotes nostate-filling. As discussed in Sect. 2 and 3, radiation fromQDi,v depends stochastically on the energy transfer proba-bility as follows:

Ri,v(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1 with probability p+i,v if Fi,v(t) = 11 with probability p−i,v if Fi,v(t) = 00 otherwise.

(2)


Here p+i,v and p−i,v respectively indicate the radiation proba-bility from QDi,v when it is state-filled and non-state-filled,where the former is much smaller than the latter in gen-eral. Note that each of p+i,v and p−i,v is determined not onlyby the presence or absence of the local state-filing stimu-lus but also by the nonlocal condition of the system as wesaw in Fig. 2(a); the probability varies as a function of thenumber of state-filled QDLs in the whole system [21]. Eachradiation event Ri,v(t) is accumulated by a newly introducedvariable Xi,v(t) ∈ {−1, 0, 1} as follows:

Xi,v(t + 1) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Xi,v(t) + 1 if Ri,v(t) = 1 and Xi,v(t) < 1Xi,v(t) − 1 if Ri,v(t) = 0 and Xi,v(t) > −1Xi,v(t) otherwise.

(3)

The above dynamics can be implemented either in the formof combinations of QDs or external electrical circuits. Ateach step t, the variable Xi,v(t) yields the estimated variablesxi as follows:

xi(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 if Xi,0(t) = 1 and Xi,1(t) ≤ 01 if Xi,0(t) ≤ 0 and Xi,1(t) = 1

xi(t − 1) otherwise.(4)

The state-filling stimulations Fi,v are updated syn-chronously according to the following dynamics:

Fi,v(t + 1) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

1if ∃(P,Q) ∈ B s.t.∀( j, u) ∈ P, Xj,u(t) = 1 and (i, v) ∈ Q

0 otherwise(5)

where B is a set of bounceback rules to be defined shortly.Each element (P,Q) in B implies the following statement: ifall the Xj,us specified by P are positive at step t, then stimu-late all QDi,vs specified by Q to inhibit their radiation at stept + 1.

To understand the meaning of the bounceback rules, letus consider the example formula φex. To satisfy the formulaφex=1, every clause in φex should be true. For example, sup-pose the system tries to assign x1=0, i.e., X1,0(t)=1, as indi-cated by the black broken circle in Fig. 4. Now let us focuson the first clause (x1∨¬x2) of φex. To make this clause true,if x1=0 then x2 should not be 1. Therefore, we apply state-filling stimulation F2,1(t+1)=1 to inhibit radiation R2,1(t+1)from QD2,1, as indicated by the gray broken circle in Fig. 4.At the same time, x3 in the third clause (x1 ∨ x3) should notbe 0, and so we apply F3,0(t+1) = 1 (the gray dotted circle).In addition, we must apply F1,1(t + 1) = 1 (the solid circle)because x1=0 necessitates that x1 should not be 1. Likewise,the set of all bounceback rules B is determined by scanningall clauses in φex.

The elements of the bounceback rule B formally con-sist of three parts: B = INTRA ∪ INTER ∪ CONTRA.

INTRA forbids each variable i from taking two values0 and 1 simultaneously:

INTRA = {{(i, v)}, {(i, 1 − v)})| i ∈ I ∧ v ∈ {0, 1}}. (6)

Each clause c = (x∗j ∨ x∗k ∨ x∗l ) has its literals x∗imapped to i∗ = i if x∗i = xi and to −i otherwise,

Fig. 4 Schematic representation of the QD formation and the bounce-back control for the solution search of a four-variable SAT problem. Thebounceback control applies state-filling stimulations F1,1(t+1) = F2,1(t+1)= F3,0(t + 1) =1 if X1,0(t) = 1.

and the formula φ is expressed equivalently by a set Φ,which includes all the clauses as its elements. For exam-ple, the example formula φex is transformed into Φex =

{{1,−2}, {−2, 3,−4}, {1, 3}, {2,−3}, {3,−4}, {−1, 4}}. For eachC in Φ and each variable i in C, INTER blocks the radiationthat makes C false [either Ri,0(t + 1) or Ri,1(t + 1)]:

INTER = {(P, {(i, 0)})|i ∈ C} ∪ {(P, {(i, 1)})| − i ∈ C} (7)

where P = {( j, 0)| j ∈ C ∧ j � i} ∪ {( j, 1)| − j ∈ C ∧ j � i}.Some rules in INTER may imply that neither 0 nor 1 can beassigned to a variable. To avoid this contradiction, for eachvariable i, we build CONTRA by checking all the relevantrules in INTER:

CONTRA = {(P ∪ P′, P ∪ P′)|i ∈ I ∧ (P, {(i, 0)}) ∈ INTER ∧ (P′, {(i, 1)}) ∈ INTER}. (8)

Before we start solving the given problem, B is obtained inpolynomial time O(N •M) by generating all the bouncebackrules in INTRA, INTER, and CONTRA based on the aboveprocedures.

The calculation starts from the condition thatXi,v(0)=Ri,v(0)=Fi,v(0)=0 for all (i, v), and the time evolutionof the system is simulated by calculating the above equa-tions iteratively. Figure 5 shows that the system successfullyfound the solution of the example formula φex at step t=13.

The performance of the proposed NanoPS was com-pared with that of the best-known search algorithm, Walk-SAT [22]. In WalkSAT, an assignment x(0) = (x1(0),x2(0), . . . , xN(0)) is initially randomly chosen. At each timestep t, by checking whether each clause is satisfied by thecurrent assignment x(t), WalkSAT randomly chooses oneof the unsatisfied clauses and satisfies it by flipping one ofits variables chosen at random. This routine is iterated un-til a satisfying assignment is obtained. Schoning estimatedthe average number of iterations that WalkSAT required forfinding a solution to a 3-SAT problem as the exponentialfunction (4/3)Npoly(N) [23].

We used benchmark problem instances provided by


Fig. 5 Simulated time evolution of Xi,v(t) in the SAT solver using a net-work of optical near-fields. The system found the solution (x1, x2, x3, x4)= (1, 1, 1, 1) at t = 13.

Fig. 6 Performance comparison of NanoPS and WalkSAT. For each al-gorithm and each N, 100 instances were evaluated. The instances weresorted from easiest to the most difficult in ascending order of the averagenumber of iterations that WalkSAT required to find a solution.

SATLIB online [24], which were the most difficult 3-SATinstances obtained by randomly generating three-literal con-junctive normal form formulae, where the difficulty can bemaximized by setting the ratio between the number of vari-ables N and the number of clauses M at the phase transitionregion around M/N=4.26 [25], [26]. We chose 100 instancesfrom each of the test sets uf75-325 and uf100-430, whichtook [N=75, M=325] and [N=100, M=430] formulae fromthe most difficult region where M/N is about 4.333 and 4.3,respectively.

For each instance, we conducted 500 trials consistingof Monte Carlo simulations to obtain the average number ofiterations (time steps t) required to find a solution. As shownin Fig. 6, NanoPS, which is a nanophotonic network, founda solution after a much smaller number of iterations thanWalkSAT. Also, the advantage of NanoPS over WalkSATincreased as the number of variables N increased.

5. Solving the Decision Making Problem

Consider a number of slot machines, each of which re-wards the player with a coin at a certain probability Pk

(k ∈ {1, 2, . . . ,N}) when played. To maximize the totalamount of reward, it is necessary to make a quick and ac-curate judgment of which machine has the highest proba-bility of giving a reward. To accomplish this, the playershould gather information about many machines; however,in this process, the player should not fail to exploit the re-ward from the known best machine. These requirementsare not easily met simultaneously because there is a trade-off between “exploration” and “exploitation”, referred to asthe “exploration-exploitation dilemma”. Such a problem iscalled the multi-armed bandit problem (BP).

BP was originally described by Robbins [27], althoughthe same problem in essence was also studied by Thompson[28]. However, the optimal strategy is known only for alimited class of problems in which the reward distributionsare assumed to be known to the players [29], [30]. There area number of important practical applications of BP, such asMonte Carlo tree searches [31].

Biological organisms commonly encounter the “exp-loration-exploitation dilemma” in surviving uncertain envi-ronments. Inspired by the amoeba’s shape-changing processunder dynamic light stimuli, Kim et al. proposed an algo-rithm for BP called the “tug-of-war model” (TOW) [32].

TOW is a dynamical system model of an amoeba-likebody, which maintains a constant intracellular resource vol-ume while collecting environmental information by con-currently expanding and shrinking its branches. The con-servation law entails a “nonlocal correlation” among thebranches; that is, the volume increment in one branch is im-mediately compensated by volume decrement(s) in the otherbranch(es). This nonlocal correlation was shown to enhancethe performance in solving BP [32].

Here we show that a network of optical energy trans-fers among quantum dots can implement a variant of TOW,which we call a “nanophotonic decision maker” (NanoDM)[33]. Although we demonstrate only the two-armed case,NanoDM can be easily extended to N-armed (N > 2) cases.

We use three types of cubic QDs with side lengthsa,√

2a, and 2a, which are represented by QDS, QDM,and QDL, respectively. We assume that five QDs are one-dimensionally arranged in the order QDL-QDM-QDS-QDM-QDL as shown in Fig. 7. The QDL on the left and right handside are respectively denoted by QDLL and QDLR. Also, TheQDM on the left and right hand side are respectively denotedby QDML and QDMR. When an optical excitation is gener-ated in QDS, it is transferred to the lowest energy levels inboth QDLs through the inter-dot optical near-field interac-tion network; thus we observe negligible radiation from theQDMs. However, when the lowest energy levels of the QDLsare populated by control light, which induce state-filling ef-fects, an exciton at QDS is more likely to be radiated fromQDM.


Fig. 7 Nanophotonic decision maker (NanoDM) composed of five QDsmediated by inter-dot optical near-field interactions.

Fig. 8 Intensity adjuster (IA) and the difference between radiationprobabilities from QDMR and QDML.

We consider the radiation from the QDMs, namely, theleft one (QDML) or right one (QDMR), as the decision of se-lecting slot machine A or B, respectively. The intensity ofthe control light to induce state-filling at the left and rightQDLs is respectively modulated on the basis of the resul-tant rewards obtained from the chosen slot machine. Simi-lar to the demonstrations shown in Sect. 3 and 4, the funda-mentally probabilistic attributes of optical energy transfer ina multi-quantum-dot system are exploited for the decisionmaking.

In NanoDM, there are in total 11 energy levels, asschematically shown in Fig. 7. The energy levels are net-worked either by optical near-field interactions or sublevelenergy dissipation. The populations concerning the radia-tion from the QDMs are calculated based on a density ma-trix formalism taking account of the external control lightradiating the QDLs.

We adopt an intensity adjuster (IA) to modulate the in-tensity of incident light to the QDLs, as shown at the bottomof Fig. 8. Also, we consider that the effects of state fillingcan be equivalently represented by the value of sublevel re-laxation parameters, which has been validated in [19].

The initial position of the IA is zero. In this case, thesame intensity of light is applied to both the energy levelsLL1 and LR1 shown in Fig. 7. If we move the IA to theright, the intensity at the right increases and that at the leftdecreases. In contrast, if we move the IA to the left, the in-

Fig. 9 Performance comparison between NanoDM and the Softmaxalgorithm. (a) PA = 0.2, PB = 0.8; (b) PA = 0.4, PB = 0.6.

tensity at the left increases and that at the right decreases.This situation can be described by the following relaxationrate parameters as functions of the IA position j: ΓLR2 =

1/100 − j/10000 + 1/100000 and ΓLL2 =1/100 + j/10000 +1/100000. The radiation probabilities from ML1 and MR1,which are respectively denoted as S A( j) and S B( j), are de-rived by solving the master equation. The difference be-tween radiation probabilities, S B( j) − S A( j), is shown bythe solid line in Fig. 8.

The dynamics of the IA are defined as follows:

1. Set the IA position j to 0.2. Select machine A or B based on S A( j) and S B( j).3. Play the selected machine.4. If a coin is dispensed, then move the IA in the direction

of the selected machine, that is, j = j − D for A andj = j+D for B, where D is the amount of the increment.

5. If no coin is dispensed, then move the IA in the oppositedirection from the selected machine, that is, j = j+D forA and j = j − D for B.

6. Go back to step 2.

In this way, NanoDM selects A or B, and the IA movesto the right or left according to the reward.

We compared the performance of NanoDM with thatof the Softmax algorithm, which is known to be the best-fitting algorithm for human decision-making behavior in theBP [34]. Figures 9(a) and (b) demonstrate the efficiency (cu-mulative rate of correct selections) for NanoDM (solid line)and Softmax with the optimized parameter (broken line) inthe case where the reward probabilities of the slot machinesare (a) PA = 0.2 and PB = 0.8 and (b) PA = 0.4 and PB

= 0.6. In these cases, the correct selection is “B” becausePB is greater than PA. These cumulative rates of correct se-lections are average values for each 1,000 samples. Hence,each value corresponds to the average number of coins ac-quired from the slot machines. Even with a nonoptimizedparameter D, the performance of NanoDM was higher thanthat of Softmax with its optimized parameter in a wide pa-rameter range of D = 10 to 100, although we show only theD = 50 case in Figs. 9(a) and (b).

One remark is that in this study we dealt with restrictedproblems, namely, PA + PB = 1. General problems can,however, also be solved by an extended NanoDM, althoughthe IA dynamics become slightly complicated.


6. Conclusion

In summary, we have demonstrated that a nanoscale net-work of optical energy transfers between quantum nanos-tructures mediated by optical near-field interactions occur-ring at scales far below the wavelength of light has thepotential to solve solution searching and decision makingproblems. More specifically, we demonstrated solving aconstraint satisfaction problem, a satisfiability problem, anda multi-armed bandit problem. The key is that nanostruc-tured matter in the form of quantum dots are networkedvia optical near-fields; optical energy transfer from smallerquantum dots to larger ones, which is a quantum stochasticprocess, depends on the existence of resonant energy lev-els between the quantum dots or a state-filling effect oc-curring at the destination quantum dots. We exploit theseunique spatiotemporal mechanisms in optical energy trans-fer to solve solution searching and decision making prob-lems.

As indicated in the introduction, the concept and theprinciples demonstrated in this paper are based on both co-herent and dissipative processes on the nanoscale, which isnot the case with conventional optical, electrical, and quan-tum computing paradigms. The inherently non-local natureis also a unique attribute provided by the optical-near-field–mediated optical energy transfer network. This work shownin this paper paves the way to applying nanometer-scalephotonic networks to solving computationally demandingapplications and suggests a new computing paradigm.

Acknowledgments

The authors would like to thank many collaborators involvedin this work, in particular Drs. M. Ohtsu, T. Kawazoe,W. Nomura, H. Hori, and M. Hara. This work was sup-ported in part by the Strategic Information and Communica-tions R&D Promotion Programme (SCOPE) of the Ministryof Internal Affairs and Communications, and Grants-in-Aidfor Scientific Research from the Japan Society for the Pro-motion of Science.

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Makoto Naruse received the B.E., M.E.,and Ph.D. degrees from the University of To-kyo in 1994, 1996, and 1999, respectively. From1999 to 2000, he was a postdoctoral researcherof the JSPS. From 2000 to 2002, he was an as-sistant professor at the University of Tokyo. In2002, he joined the National Institute of Infor-mation and Communications Technology. From2003, he has been a senior researcher. From2001 to 2005, he concurrently served as a re-searcher of PRESTO funded by JST. From 2006

to 2011, he also served as a Visiting Associate Professor at the Universityof Tokyo. He has received a number of awards, such as the Optics Prize forExcellent Papers from the Japan Society of Applied Physics in 2008.

Masashi Aono received the B.E. degreefrom Keio University and the M.E. and Ph.D.degrees from Kobe University in 1999, 2001,and 2004, respectively. From April 2004 toMarch 2013, he served as a research scientistat RIKEN. From April 2013, he has been amember of Earth-Life Science Institute, TokyoInstitute of Technology. His research interestsinclude nonlinear dynamical systems, complexsystems, biological systems, and their applica-tion to computing and communication systems.

Song-Ju Kim received his BSc degree inphysics from Korea University, Japan, in 1994.He received his M.Sc. and Ph.D. degrees inphysics and applied physics from Waseda Uni-versity, Japan, in 1997 and 2001, respectively.From 2000 to 2002, he was a research fellow ofthe Advanced Research Institute for Science andEngineering, Waseda University, Japan. From2002 to 2008, he was a research fellow of theNational Institute of Information and Communi-cations Technology (NICT), Japan. From 2008

to 2013, he was a research scientist of RIKEN Advanced Science Institute,Japan. Currently, he is a special researcher of International Center for Ma-terials Nanoarchitectonics, National Institute for Materials Science, Japan.His research interests include nonlinear dynamical systems, complex sys-tems, and their application to computing and communication systems.

Nanoscale Photonic Network for Solution Searching and Decision Making Problems

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