Adding color: Visualization of energy landscapes in spin glasses
AAdding color: Visualization of energy landscapes in spin glasses
Katja Biswas and Helmut G. Katzgraber Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA Microsoft Quantum, Redmond, Washington 98052, USA
Disconnectivity graphs are used to visualize the minima and the lowest energy barriers between the minimaof complex systems. They give an easy and intuitive understanding of the underlying energy landscape and,as such, are excellent tools for understanding the complexity involved in finding low-lying or global minimaof such systems. We have developed a classification scheme that categorizes highly-degenerate minima of spinglasses based on similarity and accessibility of the individual states. This classification allows us to condensethe information pertained in different dales of the energy landscape to a single representation using color todistinguish its type and a bar chart to indicate the average size of the dales at their respective energy levels.We use this classification to visualize disconnectivity graphs of small representations of different tile-plantedmodels of spin glasses. An analysis of the results shows that different models have distinctly different featuresin the total number of minima, the distribution of the minima with respect to the ground state, the barrier heightand in the occurrence of the different types of minimum energy dales.
PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+q, 64.60.-i
I. INTRODUCTION
Originally introduced to describe energy landscapes oftetrapeptides [1], disconnectivity graphs have become a pow-erful tool in visualizing the complex relationships betweenthe different energy minima of bio-polymers and nano-cluster [2–4]. Disconnectivity graphs represent a complexmulti-dimensional energy landscape via a set of minima andthe connections between them. The connections illustrate thelowest energy barrier a system has to overcome to be able totransition between different minima. In this respect, the dis-connectivity graphs give a clear visual aid and may also serveas a tool in understanding some of the difficulties when deal-ing with the optimization of complex systems and processes.Moreover, with the insights gained, bespoke nontrivial ran-dom benchmark problems can be designed to benchmark op-timization tools both classical [5–10] and quantum [11–16].In spin glasses complexity originates from the interplay offrustration and disorder [17–19], and can have an effect on thedynamics and success rates of optimization routines. Tryingto gain an understanding of the underlying energy structuresof spin glasses has a recurring history. Garstecki et al. [20]used enumeration to draw disconnectivity graphs and dynam-ical connectivity graphs for small two-dimensional Ising spinsystems. Amoruso et al. [21] used a hierarchical approach tocalculate minimum energy barriers. Burda et al. used the lidalgorithm for the barriers [22] and a steepest descent methodto find internal structures [23]. The branch and bound algo-rithm was used to gain information about the energy land-scape for spin systems up to the third excitation [24]. Dalland Sibani [25] used aging dynamics to gain some insightinto the structure of valleys and barriers of the energy land-scape of Ising spin glasses. Barrier trees [26, 27] have beenused to describe p -spin models. Seyed-Allaei et al. [28] useddisconnectivity graphs to study the energy-landscape of up to spin Ising models. Finally, Zhou et al. [29, 30] utilizedrandom walks to obtain minima and barriers and thus producedisconnectivity graphs.Minimum-energy configurations that have regions of spins for which the local Hamiltonian is zero, broaden the minimaand form dales (wide valleys) of equal energy in the energylandscape. For larger systems, these dales [31] add difficul-ties to the visualization of disconnectivity graphs as they es-sentially consist of a vast number of minima connected viatheir own energy and thus do add very little to an intuitiveunderstanding of the underlying energy structure of the sys-tem. For this reason we propose to simplify the visualizationof these structures by merging them into representations thatwe refer to as “dale minima.” We further distinguish betweendifferent types of dale minima, which are then visualized viacolors in the disconnectivity graph, and indicate their respec-tive sizes by bar charts for the different energy levels. Thisgreatly enhances the intuitive understanding of the underlyingenergy landscapes and simplifies the representations, allowingfor the mapping of larger structures and higher energies whilepreserving the core information of the landscape. The ideaof grouping minima has been utilized in the ballistic searchmethod [32–35] to find ground-state configurations for vari-ous problems. In ballistic search, minima that are connectedby a single variable or single spin flip at no energy cost aregrouped into one general category which in Refs. [32–35] arecalled “clusters.” However, our approach is different in thatwe further distinguish between the different ways the minimacan be connected.A different approach to obtain information about groundstates was given by Landry and Coppersmith [36] who definea cluster as groups of minima connected in a way analogousto the ballistic search method, but further distinguish betweendifferent structures that can occur within a cluster which theycall a “bunch.” In their definition, a bunch is a local structureof spins that are either zero-energy spins or under certain ful-filled conditions can become zero-energy spins. In this way,a single configuration can have multiple bunches, which arethen used to obtain information about the structure of the in-dividual ground states. Our approach also differs from theaforementioned bunches in that, to ease the visualization, weuse the overall connectivity between the individual minima.In other words, although the connection between minima is a r X i v : . [ c ond - m a t . d i s - nn ] A p r influenced by local structures of spins within the particularminimum energy configurations, we are not interested in char-acterizing these local structures. We characterize the entireminimum to belong to a certain type which we call either aregular, or a dale minimum. This approach allows us to ef-fectively enhance and reduce the visualization of minima indisconnectivity graphs, which is the main goal of this paper.To demonstrate the visualization approach we propose inthis work, we use our enhanced disconnectivity graphs tostudy the underlying energy landscape of instances of pla-nar tile-planted spin glasses [37, 38]. These are special typesof tunable spin glasses used to benchmark optimization tech-niques that have been constructed with a known ground-state solution. Note that other types of Ising Hamiltoni-ans with planted solutions have been constructed on Chimeragraphs [13, 39–41], using a random adaptive optimizationmethod [42], as well as via patch planting [43].The article is structured as follows. In section II A we in-troduce and explain our classification scheme. Section II Bexplains the basics of disconnectivity graph. Section III givesa short overview of the tile-planted spin glasses constructedwith different elementary plaquettes and which we refer to as C -type model, C -type model, and C -type model. In sec-tion IV we discuss our results, followed by conclusions. II. METHODSA. Classification
Disconnectivity graphs are two-dimensional representa-tions of high-dimensional potential energy landscapes. Intheir simplest form, they depict the minima of the landscapeand the lowest high-energy barriers between any two minima.The lowest high-energy barrier is the minimum increase inenergy necessary to transition from one minimum to another.When dealing with disordered spin systems several factors areto be taken into account that complicate the generation of dis-connectivity graphs.For big enough systems of spin-glasses in which the spin-spin interactions are drawn from a discrete and finite distri-bution, there exist arrangements of the spin-spin interactionthat allow some of the spins to take any orientation withoutchanging the energy of the system. This is typically the casewhen the local interactions centered around a spin s i and theircorresponding connecting spins sum to zero, i.e. H E i = 0 .We call the spins s i which show this effect zero-energy spins,thus highlighting the fact that their contribution to the totalenergy of the system is zero. These effects can influence iso-lated spins or groups of connected spins. Furthermore, certainarrangements of spins and interactions can “transfer” the ef-fects of zero local energy to other spins, thus leading to path-ways in the energy landscape that can only be traversed inspecific directions without increasing the energy . A differentorientation of a single spin constitutes a different configura-tion of the whole system. Hence, configurations that only dif-fer by spin orientations of the described form constitute dalesin the energy landscape. A dale here is a region on the energy landscape that has the same energy for different configura-tions of the system [44], i.e., the configurations are linked toeach other by paths in the energy landscape that do not requirean intermediate increase in energy. From a practical point ofview when generating disconnectivity graphs, the minima be-longing to the same dale increase the storage of data substan-tially while adding very little information to the energy struc-ture of the system. In addition, dales clutter disconnectivitygraphs with unnecessary information. From an energy con-nectivity perspective, one can think of each such a dale as anindividual but degenerate state. This has led us to develop acolor scheme for the disconnectivity graphs to distinguish be-tween regular (nondegenerate) minima and various types ofdales.We distinguish between different types of minima. Thetypes are based on similarity arguments and accessibility ofthe minima within the dales. We distinguish the minima asfollows: Regular minimum : A regular minimum is a minimumfor which the flip of any spin of the system increases theenergy.
Type-1 dale minimum : A dale minimum of type-1 is adegenerate minimum in which any combination of flips of thezero-energy spins preserves the energy of the system.
Type-2 dale minimum : A dale minimum of type-2 isa degenerate minimum in which only some, but not all,combinations of flips of the zero-energy spins preserve theenergy of the system.
Type-3 dale minimum : A dale minimum of type-3represents a connection of dale minima accessible only viaspecific transitions between the minima.We further separate type-3 dale minima into two subcate-gories, which we call “type-3 dale minima with a simple path”and “type-3 dale minima with a split path.” These subcate-gories are based on the specific ways the energy landscapehas to be traversed in order to preserve the energy of the sys-tem. Furthermore, note that the classification into regular min-ima, type-1 dale minima and type-2 dale minima is based onthe overall connectivity of the individual configurations of thewhole configuration space, i.e., while the zero-energy spinscan form multiple small clusters within the systems, the classi-fication according to the types is determined by the occurrenceof the structures of connecting spins of the highest type. Thisis justified by the fact that these individual structures are theones that determine the connectivity of the individual configu-rations on the configuration space. We also emphasize that thepresent study is performed for sparse graphs with fixed con-nectivity. A different, more complex, picture might emergefor Ising Hamiltonians on random, as well as fully-connectedgraphs.In the disconnectivity graphs, we distinguish the differenttypes of minima using colors. The regular minima are de-picted in black, whereas type-1 and type-2 dale minima arecolored in blue and green, respectively. Because type-3 dale
FIG. 1: Visualization of type-1 dale minima. The interactions J ij = − , , and are represented by wiggly lines ( ), dashed lines ( ),and solid lines ( ), respectively. White triangles represent spinup and black triangles represent spin down. The red triangleslabeled and are zero-energy spins. These can be flipped in anycombination at no energy cost.FIG. 2: Visualization of type-2 dale minima. The interactions J ij = − , , and are represented by wiggly lines ( ), dashed lines ( ),and solid lines ( ), respectively. White triangles represent spinup and black triangles represent spin down. The red triangleslabeled and are zero-energy spins. They can be flipped onlyin certain combinations at no energy cost. minima are two dales joined together, we represent them byconnecting the corresponding dales by a horizontal line (col-ored in red) at their respective energies indicating the zero-energy transition between the two connecting dales.Figures 1 to 6 illustrate via examples the different dale min-ima. Upward triangles represent Ising spins with the value +1 (up) and spins with the value − (down) are represented bydownward-pointing triangles. Zero-energy spins are drawn inred. The interactions J ij between spins i and j are drawn assolid lines when J ij = +2 , dashed when J ij = +1 , and wig-gly lines when J ij = − . Note that the examples given inFigs. 1 to 6, only show the sections necessary to illustrate thedifferent types of dale minima and are part of larger systems.Figure 1 shows an example of a type-1 dale minimum. Inthis example, there are two zero-energy spins (red) As can beseen from the figure, any combination of the two zero-energyspins (spin 1 and spin 2) preserves the energy of the system.Without additional zero-energy spins such a minimum has adegeneracy of originating from the free choice of orienta- FIG. 3: Visualization of type-3 dale minima with a simple path. Theinteractions J ij = − , , and are represented by wiggly lines ( ),dashed lines ( ), and solid lines ( ), respectively. White trianglesrepresent spin up and black triangles represent spin down. Inthe presented configuration only the red triangle labeled is azero-energy spin and can be flipped at no energy cost. However, oncespin has been flipped, the blue spin labeled as becomes azero-energy spin. Lastly, only after flips the spin labeled canbe flipped at no energy cost. The direction of this simple path isindicated by the green arrows ( ).FIG. 4: Visualization of type-3 dale minima with a simple path inthe reverse direction of Fig. 3. The interactions J ij = − , , and are represented by wiggly lines ( ), dashed lines ( ), and solidlines ( ), respectively. White triangles represent spin up andblack triangles represent spin down. In the presented configura-tion only the red triangle labeled is a zero-energy spin and canbe flipped at no energy cost. However, once spin has been flippedthe blue spin labeled with becomes a zero-energy spin. Andlastly, only after flips the spin labeled can be flipped at no en-ergy cost. The direction of this simple path is indicated by the greenarrows ( ). Note that this path is in the reverse direction of Fig. 3. tion, i.e., both spins up ( ↑↑ ), only one of the spins up ( ↑↓ and ↓↑ ) or both spins down ( ↓↓ ). In this example the free choice offlipping any combination of the zero-energy spins originatesfrom their separation, i.e., that they are not nearest neighbors.Therefore, each choice of spin up or down does not affect theother, they are independent of each other.In Fig. 2 a type-2 dale minimum is illustrated. Here the twozero-energy spins are nearest neighbors, which together withthe arrangement of the spin-spin interactions leads to the fol-lowing scenario: A flip of one spin influences the flipping ofthe other spin, i.e., if spin 1 is flipped then spin 2 will havenegative local energy and thus can no longer be flipped with-out an increase in energy. Similarly, if spin 2 is flipped, then FIG. 5: Visualization of type-3 dale minima with a split path. Theinteractions J ij = − , , and are represented by wiggly lines ( ),dashed lines ( ), and solid lines ( ), respectively. White trianglesrepresent spin up and black triangles represent spin down. Inthe configuration shown, only the red triangle is a zero-energyspin and can be flipped at no energy cost. Once the zero-energy spinhas been flipped the spins drawn in blue which are labeledbecome zero-energy spins, and if have been flipped, the bluespins labeled can be flipped at no energy cost. This then allows toflip spin . Starting with flipping the center spin , this dale splitsinto two paths, indicated by the green arrows . spin 1 will in its up-configuration be in a local minimum (hasnegative local energy) and can no longer be flipped. In thisexample the degeneracy originating from the zero-energy spineffects is , corresponding to the orientations of both spins up( ↑↑ ) or only one of them up and the other down ( ↑↓ and ↓↑ ).Note that in general, a necessary condition for the occurrenceof this type of dale minimum is that at least two of the zero-energy spins have to be neighboring.Figures 3 and 4 illustrate a type-3 dale minimum with a sim-ple path. In Fig. 3 only spin 1 is a zero-energy spin. However,if spin 1 is flipped, then spin 2 becomes a zero-energy spinand can be flipped without an increase in energy. Further, ifspin 1 and 2 are flipped, then spin 3 is a zero-energy spin andcan be flipped at zero energy cost leading to the configurationdepicted in Fig. 4. Reversely, if the system starts in a config-uration of Fig. 4, then the order of the flipping of the spins is → → , i.e., representing the spins in the order (1 , , the accessibility of the minima of this dale can be depicted as ↑↑↑ ←→ ↓↑↑ ←→ ↓↓↑ ←→ ↓↓↓ . (1)Note that, while each of the configurations in the schematicrepresentation, Eq. (1), is a minimum on its own, each canonly access a limited number of neighboring minima belong-ing to the same dale in the energy landscape. This means thatthe direct accessibility of the minima is dependent on the cur-rent minimum. The overall dale is pathway dependent. Path-way dependent in this example means that there are only twopathways (one in each direction) that connect all the minimabelonging to the same dale minimum. In this example, thelocal degeneracy of the dale minimum is , representing thefour subminima. FIG. 6: Visualization of type-3 dale minima with a split path inthe reverse direction of Fig. 5. The interactions J ij = − , , and are represented by wiggly lines ( ), dashed lines ( ), and solidlines ( ), respectively. White triangles represent spin up andblack triangles represent spin down. In the configuration shownthe two red triangles and labeled are zero-energy spinsand can be flipped at no energy cost. Once they have been flippedthe spins drawn in blue which are labeled become zero-energy spins, which if flipped allow the spins labeled to be flippedat no energy cost. They form two paths, which are indicated by thegreen arrows . Only if both the paths have been completed, i.e.,both the spins labeled have been flipped, the center spin indicatedin green can be flipped at zero energy cost. This double conditionon the flippability of the center spin is indicated by the pink arrows. Another variation of the type-3 dale minima is illustratedin Figs. 5 and 6. We call this subclass type-3 dale minimumwith a split path, denoting that the transition between the twoextreme minima (i.e., minima in Fig. 5 and minima in Fig. 6in our example) do not follow a single path but rather a paththat can be thought of to be split into two (or more) segments.A path here denotes a sequence of flipping of spins which areneighboring to each other. In the minima depicted in Fig. 5only the center spin (drawn in red) is a zero-energy spin andcan be flipped at no cost. However, once the center spin hasbeen flipped it opens the two paths denoted by 1, 2, and 3,which can be either flipped in sequence or alternating betweenthe two paths while maintaining the directional order. A flipof the zero-energy spin makes spin labeled 1 a zero-energyspin, which when flipped allows to flip spin number 2 andlastly spin number 3, if 2 has been flipped. This directionalorder has to be maintained in both directions in order for thetransition to occur without an increase in energy. Flipping thespins along both the paths leads to the minimum depicted inFig. 6. The reverse transition is shown in Fig. 6. Here the twooutermost spins (depicted in red) are zero-energy spins andcan be flipped at no cost. This allows then the flipping of thespins marked with 2 and 3 in subsequent order. Only after bothof the paths have been completed, i.e., if both the spins markedwith 3 are flipped, the center spin (drawn in green) becomesa zero-energy spin and can be flipped without an increase inenergy. The resulting minima after all the spin flips have beencompleted is the minima depicted in Fig. 5. These transitionscan also be visualized as ↑↑↑↑↓↓↓(cid:108)↑↑↑↓↓↓↓(cid:37)(cid:46) (cid:38)(cid:45)↑↑↓↓↓↓↓ ↑↑↑↓↑↓↓(cid:108) (cid:108)↑↓↓↓↓↓↓ ↑↑↑↓↑↑↓(cid:108) (cid:108)↓↓↓↓↓↓↓ ↑↑↑↓↑↑↑(cid:38) (cid:46) (cid:46) ↓↓↓↓↑↑↑ , (2)representing schematically the transition corresponding toFig. 5, and ↓↓↓↓↑↑↑(cid:37)(cid:46) (cid:38)(cid:45)↑↓↓↓↑↑↑ ↓↓↓↓↑↑↓(cid:108) (cid:108)↑↑↓↓↑↑↑ ↓↓↓↓↑↓↓(cid:108) (cid:108)↑↑↑↓↑↑↑ ↓↓↓↓↓↓↓(cid:38) (cid:46) (cid:46) ↑↑↑↓↓↓↓(cid:108)↑↑↑↑↓↓↓ , (3)representing schematically the transition depicted in Fig. 6.The wedged arrows denote the requirement that both of thepreceding paths have to be completed before the next step.Note, that both the representations, Eq. (2) and Eq. (3), aresimple representation of an underlying highly-dimensionalnature, meaning that the transitions between the two outer-most minima (the minima depicted in Fig. 5 and Fig. 6) havemultiple possible realizations corresponding to a large varietyof pathways across the dale in the energy landscape. Eachstep or point in the configuration space is a subminimum sit-uated within the dale. The dale in our example is comprisedof subminima, where the two minima depicted in Fig. 5and Fig. 6 are its outermost points that can only be accessedfrom one minimum but have two possible transitional minimafor leaving in both the cases. The paths shown here have beenselected based on their significance in highlighting the impor-tant aspects of the transitions. B. Disconnectivity Graphs
In order to draw a disconnectivity graph, first, all the min-ima of the system have to be calculated. We do this via a com-plete enumeration of all possible spin configurations. Then,the height of the lowest energy barrier connecting the minimais calculated by considering the possible paths between theminima. For this purpose the procedure of Cieplack et al. [20] for the calculation of possible pathways was followed. Fur-ther, to deal with the large number of minima in a computa-tionally effective manner not all barriers [45] are calculatedbut an approximation is used. This approximation is based onthe structural difference between the minima: For each mini-mum only the barriers to the nearest n minima are calculated.With nearest, we mean the number of differences in the orien-tation of the spins between any two minima and n is a givennumber of calculated barriers. We reason that when searchingfor low-energy pathways a system is likely to transition overintermediate minima or funnels of intermediate minima. Thisapproximation greatly reduced the number of necessary bar-rier computations and thus allows for a complete mapping ofthe energy landscape to a disconnectivity graph. A similar ap-proach was used in the analysis of potential energy landscapesof hexapeptides [46]. In general, the validity and the extent towhich this approximation can be applied depends on the spe-cific problem studied. We discuss this in more detail below inSec. III.Figures 7, 8, and 9 show examples of disconnectivitygraphs obtained for the planted spin-glass models discussedbelow. The minima are represented by vertical bars, whoselowest points denote their energy. The branching points rep-resent the lowest high-energy barrier needed to overcome totransition between the adjacent minima. The black bars repre-sent regular minima, i.e., minima for which a flip of any spinwould lead to an increase in energy. The blue bars representtype-1 dale minima, the green bars represent type-2 dale min-ima, and red horizontal lines joining two or more minima atthe bottom indicate type-3 dale minima.The minima are joined and sorted into cluster structures ac-cording to their energy barriers. Within each individual clus-ter, the minima are arranged based on the number of spinsup. Minima with the highest number of spins up are drawntowards the left and sequentially minima with an increasingnumber of spins down are arranged toward the right [47].To indicate the size of the different types of minima, i.e.,the number of subminima belonging to the individual dales,we add a bar chart to the right-hand side of the disconnectivitygraph. This chart is arranged such that the vertical axis repre-sents the energies of the minima and dales, and the horizontalaxis represents the average number of subminima belongingto the types, which we call the size of the types of minima. III. MODEL
For the numerical experiments we focus on two-dimensional spin glasses on a square lattice with tile-plantedsolutions consisting of combinations of either C , C or C unit cells, as shown in Fig. 10 Unit cell C consists of one J ij = − and three J ij = +2 interactions between spins i and j that together form a square and on whose vertices thespins s i are located. If all spins are either oriented in the upor all in the down configuration the local Hamiltonian of thisconfiguration has ground-state energy H C = − . Unit cell C consists of one J ij = − , one J ij = +1 and two J ij = +2 spin-spin interactions, which can be arranged in three distinct regulartype-1 daletype-2 daletype-3 dale e n e r g y FIG. 7: Example of a disconnectivity graph for a planted spin-glass model ( C , see below) on a two-dimensional square lattice with N = 36 spins. The black vertical bars represent regular minima, the blue bars represent type-1 dale minima, the green bars represent type-2 daleminima and the red vertical bars indicate type-3 dale minima. The bar chart on the right shows the average size (i.e., the number of subminimacomprising the dales) of the types of minima at their respective energy levels. The axis denoting the average size of types of minima onlyapplies to the bar chart on the right-hand side of the disconnectivity graph. This axis does not represent the total number of minima in thesystem. ways shown in Fig. 10. This unit cell has four ground stateconfigurations with an energy H C = − . Unit cell C hasone J ij = − , two J ij = +1 and one J ij = +2 spin-spin in-teractions. Their possible arrangements are shown in Fig. 10.Unit cell C has 6 ground states, corresponding to an energyof H C = − . The total system is constructed by placing theunit cells in a checker-board fashion such that only the verticesare joined, see Fig. 11. Within a subgroup C , C , or C theunit cells are chosen and rotated randomly to form the largersystem. Note that arranged in this way, the larger systems willhave two known ground-state solutions, where all spins areeither pointing up or all spins are pointing down. The totalHamiltonian is then given by the sum over all interactions J ij centered around the individual spins H = − (cid:88) (cid:104) ij (cid:105)∈ E J ij s i s j = 12 (cid:88) i H E i s i . (4)In Eq. (4) the sum is taken over all vertices, s i = ± denotethe values of the individual spins in their up or down configu-rations, respectively, and H E i is the local Hamiltonian of theedges centered around spin i .While some of the solutions of the ground states are con-structed, each of these types represents a model of differentcomplexity. The hardness in terms of time to solution (TTS)of pure and of mixed types of these models has been studiednumerically in Refs. [37, 38]. In this work we map out the underlying energy landscapes of these tile-planted systems. IV. RESULTS
Results have been obtained for systems with N = 6 × spins, i.e., unit cells. To obtain sufficient statistics for eachof the C , C and C types of planted spin glasses a sample of systems is randomly generated and evaluated. The min-ima of the systems are obtained by complete enumeration andfor each of the minima, barriers are calculated to the closest minima. Although we only study one system size in thiswork, the obtained results should be qualitatively representa-tive for the underlying behavior of the model system.Figure 12 shows the distribution of the minimum hammingdistance for the three models of planted spin glasses. Thehamming distance here is understood as the minimum numberof differences in the orientation of spins between two minima.For type-1 and type-2 dale minima, only the hamming dis-tances between the two closest subminima are represented,i.e., between the subminima of each dale with the shortesthamming distance. To take into account the complexity intransitioning between the subminima belonging to type-3 daleminima, for this class only the hamming distances betweenthe minima farthest apart on the energy landscape have beentaken into account. Since for each of the models studied mul- regulartype-1 daletype-2 daletype-3 dale e n e r g y FIG. 8: Example of a disconnectivity graph for a planted spin-glass model ( C , see below) on a two-dimensional square lattice with N = 36 spins. The black vertical bars represent regular minima, the blue bars represent type-1 dale minima, the green bars represent type-2 daleminima and the red vertical bars indicate type-3 dale minima. The bar chart on the right shows the average size of the types of minima at theirrespective energy levels. Note, that the axis denoting the average size of types of minima only applies to the bar chart on the right side of thedisconnectivity graph. tiple samples were analyzed, each of which varies in the totalnumber of minima, the normalization is in two steps. First, foreach sample separately the distribution of the minimum ham-ming distance is determined and normalized to unity. Subse-quently, an average over samples is performed and the distri-butions normalized. The standard deviation is computed forthe average over samples as are statistically more signifcant.As can be seen from Fig. 12, for the three models studied,the shortest hamming distance that occurred most often was with a relative occurrence of . for C -based models, . for C -based models, and . for C -based mod-els . Overall, the C model has a larger number of minimathan C and C , see Tab. I. This leads to an overall smallerdifference in the orientation of the spins between neighbor-ing minima and therefore shorter hamming distances. Note,that the configuration space consists of all possible combina-tions of the orientations of the spins of the systems and there-fore has the same size for all the three models. This requiresthe minima of the C model — which on average have fewerminima than C and C — to be more sparsely distributedbetween the two outermost configurations consisting of eitherall spins up or all spins down. The remaining minima are dis-tributed between the extreme cases of all spins up and all spinsdown in model C , thus explaining why there are less statesonly two spin flips apart than for the other two models, andthe rest requiring three ( . ), four ( . ), up to a maxi-mum of nine required spin flips between neighboring minima ( . ). While these numbers will change with differentsystem sizes, we estimate that the generic trend of the datawhere two spin flips are dominant remains. The maximumnumber of required spin flips between neighboring minimafor the C model was found to be seven in our sample of 100systems with an occurrence of . , and for C it is six at . .To verify the validity of our approximation of includingonly up to barriers for each minimum, we introduce a bar-rier field. The barrier field is the storage allocation of the bar-riers which are sorted according to their hamming distances,i.e., the hamming distances are sorted according to their sizeand allocated to the hamming field such that small hammingfield values store the information about the barriers for thesmallest hamming distances. Figures 13, 14, and 15 showthe distribution of the hamming distances for the barrier fieldfor the C , C and the C model systems, respectively. Thefigures show the results normalized for each value of the bar-rier field separately and give an indication of the sizes of thehamming distances that are covered by calculating barriers in-cluding only neighboring states.As can be seen from Figs. 13, 14, and 15 , in all of themodel systems, only a few allocations (i.e., < ) are nec-essary to include all the minima within the shortest hammingdistance. By obtaining just about barriers for each of theminima hamming distances up to spin flips are covered forthe C model system with the highest occurrence of or regulartype-1 daletype-2 daletype-3 dale e n e r g y FIG. 9: Example of a disconnectivity graph for a planted spin-glass model ( C , see below) on a two-dimensional square lattice with N = 36 spins. The black vertical bars represent regular minima, the blue bars represent type-1 dale minima, the green bars represent type-2 daleminima and the red vertical bars indicate type-3 dale minima. The bar chart on the right shows the average size of the types of minima at theirrespective energy levels. Note, that the axis denoting the average size of types of minima only applies to the bar chart on the right side of thedisconnectivity graph.FIG. 10: Schematic representation of the planar unit cells C , C ,and C in their invariant form. The solid lines ( ) represent J ij =+2 , the dashed lines ( ) represent J ij = +1 , and the wiggly lines( ) J ij = − . required flips within the th’s barrier calculation. For the C model system hamming distances up to occur, with or spin flips having the highest occurrence. For the C modelsystem hamming distances of up to occur, with the highest FIG. 11: Schematic representation of the construction pattern of a × planar system with periodic boundaries. The filled squaresrepresent the tiles in which the unit cells are planted. occurrence of and necessary flips in the barrier calcu-lation. The difference in the covered hamming distances ofthe models is due to the difference in the number of minimabetween the different models, see Tab. I.Figures 16, 17, and 18 show the distribution of the aver- hamming distance ρ ( h m i n ) C C C FIG. 12: Distribution of the minimum hamming distance for the C , C , and C models. The vertical bars denote the standard deviation.
10 20 30 40 50 60 70 80 barrier field h a mm i n g d i s t a n c e FIG. 13: Distribution of hamming distances vs the barrier field perminima of the C model. age barrier heights versus the hamming distance of the threemodel systems C , C , and C . They visualize the calculatedrange of the barrier field. Note that a dale minimum consistsof multiple minima that are joined by zero-energy spin flips,i.e., they form dales on the energy landscape on which the sys-tem can transit without an increase in energy. In order to avoidunnecessary double counting, only the barriers correspondingto the shortest hamming distance of minima, belonging to thesame dales for type-1 dales and type-2 dales, have been takeninto account. Due to the dependence on the pathway, for type-3 dale minima the hamming distance is determined betweenthe endpoints of the individual dales. The figures are normal-ized to unity for the full distribution.As can be seen, in all model systems, the barrier heights for
10 20 30 40 50 60 70 80 barrier field h a mm i n g d i s t a n c e FIG. 14: Distribution of hamming distances vs the barrier field perminima of the C model.
10 20 30 40 50 60 70 80 barrier field h a mm i n g d i s t a n c e FIG. 15: Distribution of hamming distances vs the barrier field perminima of the C model. small hamming distances are low. This is due to the fact thatwhen only a few spin flips are necessary to change betweenminimum energy configurations then only a few spin-spin in-teractions contribute to the increase in energy of the system.Minima separated by larger hamming distances show a largerdistribution of the energy barriers but are also still dominatedby small barrier heights. Furthermore, comparing Figs. 16,17, and 18, a significant contribution of high-energy barriersoccurs only at intermediate hamming distances for the modelsystems C and C . The energy of the highest barrier is largerfor C than for C and C , with C having the lowest highbarrier. The results from C originate from two factors. First,0 hamming distance › E b fi FIG. 16: Distribution of the barrier heights versus the hammingdistance for the C model. hamming distance › E b fi FIG. 17: Distribution of the barrier heights versus the hammingdistance for the C model. due to the large number and huge degeneracy of dale minimain the C model system a larger section of the configurationspace is covered than for the other models. Thus, on average asmaller number of spin flips is necessary to traverse betweenthe minima considered in the barrier calculation, also explain-ing the shorter hamming distances. Second, the C modelsystem has more spin-spin interactions of smaller value thanthe two other models, which leads to that even if a barrier hasto be crossed, the necessary increase in energy between twominima is, in general, smaller than for the other two models.This explains the difference in the observed barrier heights.These together with the distribution of the hamming dis- hamming distance › E b fi FIG. 18: Distribution of the barrier heights versus the hammingdistance for the C model. tance for the barrier field (see Figs. 13, 14 and 15) the dis-tribution of the energy barriers confirm with our approxima-tion of the calculation of the barriers. This approximation isbased on the assumption that minimum energy transitions be-tween minima, which are far apart on the energy landscape,will most likely occur through pathways that are leading alongor through dales of intermediate minima. During such a tran-sition the necessary increase in energy for the total transitionwill be already captured by intermediate barriers. While low-energy barriers had the highest occurrence indicating the higharrangement of minima into first-order cluster structures, ourreduced computation of barriers (i.e., up to a maximum of barriers for each minimum) also covered higher-energy transi-tions. This allows for transitions out of the local cluster struc-tures. It indicates that for each local cluster there are minimafor which the calculated barriers include high-energy transi-tions, and hence enables the overall connectivity of the min-ima.Table I shows the number of minima for the different typesof minima of the C , C and the C model system. As can beseen from the table, the actual number of minima N ∗ is muchhigher than the number of minima in the reduced description N . This is especially striking for the C model, whose en-ergy landscape exhibits a huge number of type-2 dale minimawith large sizes of energy dales. The size S gives the av-erage number of minima belonging to a dale. The averagetotal number of minima, counting each state on a minimumenergy dale as individual minimum, is . for C , . for C and . for C (post sample average). Especiallyfor C , drawing these huge numbers of minima into discon-nectivity graphs would lead to problems. However, using ourclassification scheme, the average total number of minima inthe reduced description is . for C , . for C and . for C , thus making it much simpler to display the en-ergy structure of the minima in the disconnectivity graphs. In1 TABLE I: Number of the minima types N in the reduced descrip-tion, percentage of the minima % N in the reduced description, av-erage size of minima types S , actual number of minima types N ∗ , percentage of the actual number of minima % N ∗ . The values areaverages obtained from our samples of 100 systems for each of the C , C and C models of tile-planted spin glasses. R denotes regu-lar minima, D1, D2, and D3 denote type-1, type-2, and type-3 daleminima, respectively [48].regular type-1 type-2 type-3 C N . . . . N S N ∗ . . . . N ∗ C N . . . . N S N ∗ . . . . N ∗ C N . . . . N S N ∗ . . . . N ∗ the reduced description of the minima, in the C model regu-lar minima and type-1 and type-2 dale minima seem to haveequally likely occurred. However, this observation is to betaken with precaution since the reduced description does nottake the size of the dales into consideration. Taking the size S of the minimum energy dales into account, one can see fromTab. I, that most of the individual minimum energy configura-tions belong to the type-2 dale minima. Similarly, for the C model, while in the reduced description the number of type-1dale minima dominates, taking into account the different sizes S of the dales shows that the majority of the individual mini-mum energy configurations belong to the type-2 dale category.Lastly, compared to the other two model systems, the C model has the smallest percentage of regular minima and thehighest percentage of type-2 dale minima. Together with thehigh occurrence of low-energy states, this indicates that theenergy landscape of this model is easier to traverse using stan-dard optimization routines than the other two models.Figure 19 shows the average distribution of the energy ofthe minima E min in the reduced description and their stan-dard deviation relative to the ground state energy E ground forthe three models of tile-planted spin glasses. The large stan-dard deviation in the data is due to the large variation in thenumber of minima in the generated systems of the three mod-els. As can be seen from Fig. 19 and comparing with the dis-connectivity graphs in Figs. 7, 8, and 9, the C model systemis distinguished by a large gap between the ground state andhigher-order minima. This is not the case for the two othermodel systems. The majority of the minima of the C modelsystem are found at medium energies, whereas the C model E min − E ground d i s t r i b u t i o n o f m i n i m a C C C FIG. 19: Distribution of the energy of the minima relative to the en-ergy of the ground state for the different planted spin-glass modelsystems C (black circles), C (red triangles) and C (blue dia-monds). The lines are Gaussian fits and are guides to the eyeFIG. 20: (a) Possible contributions of spin-spin interactions of anindividual C unit cells to a vertex. The solid lines ( ) represent J ij = +2 spin-spin interactions and the wiggly lines ( ) J ij = − spin-spin interactions. (b) Possible combinations of the spin-spininteractions to a vertex of model system C . No zero-energy spinsare possible if all neighboring spins point in the same direction. is distinguished by a large number of minima energeticallyclose to the ground state.Figures 7, 8 and 9 show examples of the disconnectivitygraphs of the C , C , and the C model systems, respectively.As can be seen from Fig. 7 (see also Fig. 21), the ground stateof the C model system has an energy of E ground = − .This is due to the fact that each unit cell has an energy of H = − corresponding to a configuration in which either allspins are up or all spins are down and there are unit cellsin the lattice studied. Furthermore, the ground state of the C model system is always a regular minimum. Figure 20(a)shows the possible combinations of spin-spin interactions toa vertex belonging to a single unit cell. Each unit cell cancontribute to a vertex either two J ij = +2 or one J ij = +2 and one J ij = − spin-spin interaction. Since two unit cellsjoin at each vertex, this gives the possible combinations ofspin-spin interactions shown in Fig. 20(b). As can be seen,none of the possible combinations of spin-spin interactionsleads to a zero-energy vertex if all neighboring spins have thesame orientation. Hence, the ground states of the C modelare always regular minima.2
95 90 85 80 75 70 65 60 55 50 E min d i s t r i b u t i o n o f t y p e s o f m i n i m a regulartype-1 daletype-2 daletype-3 dale FIG. 21: Distribution of the different dale minima types for the C model system of planted spin glasses. The lines are Gaussian fits andguides to the eye [49].
90 85 80 75 70 65 60 55 50 E min a v e r a g e s i z e o f t y p e s o f m i n i m a regulartype-1 daletype-2 daletype-3 dale FIG. 22: Average size of the different dale minima types for the C model of planted spin glasses. Figure 21 shows the distribution of the minima types of the C model system for all the minima. One can see in Fig. 21that the ground states of this model are always regular min-ima and are separated by a large energy gap to higher energystates. It appears that the C model system has on average thehighest number of states belonging to type-2 dale minima (seeTab. I). This suggests that with a simple optimization routineone is most likely to encounter type-2 dale minima lying in themedian regions of the energetically possible configurations.Figure 22 shows the average size of the minima and typesof dale minima at their respective energies. With size we un-derstand here the number of subminima that make up the dif-ferent types of dales. Regular minima consist of only onesubminimum, hence their size is always one at any energylevel on which they occur. The size of the different types ofdales is always larger than that of the regular minimum, with type-3 dales having the highest number of subminima becausethese are composed of two dales of type-1 or type-2 joined to-gether. The occurrence that type-1 dale minima are smallerthan type-2 dale minima at all energy levels (Fig. 22) is nota general property of the types of dales. It is a specific fea-ture of the model studied. Furthermore, as can be seen fromFig. 22, the number of subminima belonging to the individ-ual dales is especially large around medium energies (at about E min = − ) with a second peak at E min = − . Taking intoaccount the findings of the distribution of the types of min-ima (Fig. 21) only the first peak considerably contributes tothe features of the energy landscape. The second peak, whilecomparably large in the size of type-2 and type-3 dale minima,has a low occurrence (less than . , see Fig. 21) and thus haslittle significance for general optimization procedures. Thesize of type-1 dale minima does not vary much and is alwaysless than four for the system sizes studied.The ground states of the C model are either regular min-ima or type-1 dale minima and have a ground-state energy of E ground = − , because each of the cells has an energyof H = − . For the planted ground-state solutions the occur-rence of type-1 dale minima can be understood by consider-ing the possible combinations of spin-spin interactions to thespins. Figure 23 illustrates this situation. The possible contri-butions of spin-spin interactions of a single unit cell to a ver-tex are shown in Fig. 23(a). In Fig. 23(b) the possible distinctcombinations of spin-spin interactions to a vertex are shown.Only one of these combinations allows for a zero-energy spinin the planted solutions, i.e., two unit cells need to be orientedin such a way, that from each of the cells one J ij = +1 andone J ij = − spin-spin interaction join at the vertex. Onlythen the energy of the vertex will be balanced if all neighbor-ing spins point in the same direction. However, because eachof the two unit cells already contributes with their J ij = +1 and J ij = − spin-spin interactions, the remaining spin-spininteractions of each of the two unit cells are J ij = +2 as il-lustrated in Fig. 23. This makes it impossible for any of theneighboring vertices to the zero-energy vertex to be energeti-cally balanced in the planted solution. This prohibits the for-mation of connected zero-energy spins, and hence the plantedsolutions of this model will always be regular minima or type-1 dale minima.Figure 24 shows the distribution of minima types in the re-duced description of all the minima in the C model. Differentto the C model system, the C model does not have a largegap to the ground state. Similarly to C , the majority of theminima populate the energetically medium regions. Togetherwith the higher number of minima, this suggests that this typewill be harder to solve in optimization routines than C and C . This was also observed in the study of Perera et al. [38],which compares the time to solution of different models oftile-planted spin glasses.Figure 25 shows the average size of the types of minima attheir respective energies. The size of types of minima followsthe same order as in the C model, with regular minima al-ways having size one, followed by type-1 dale minima, type-2dale minima, and type-3 dale minima having the largest size.In contrast to C in the C model, no distinct peak in the size3 FIG. 23: (a) Possible contributions of spin-spin interactions of asingle C unit cell to a vertex. The bonds J ij = − , , and areindicated by wiggly lines ( ), dashed lines ( ), and solid lines( ) respectively. (b) Possible combinations of the spin-spin interac-tions to a vertex of model system C . Only one of the combinations(framed) allows for zero-energy spins for the planted solutions. (c)The zero-energy vertex of case (b) only leaves J ij = +2 spin-spininteractions for each joining unit cell, hence this system cannot haveneighboring zero-energy spins in the planted solution.
75 70 65 60 55 50 45 40 E min d i s t r i b u t i o n o f t y p e s o f m i n i m a regulartype-1 daletype-2 daletype-3 dale FIG. 24: Distribution of the different dale type minima for the C model of planted spin glasses. The lines are Gaussian fits and guidesto the eye [49]. is visible. The largest sizes of the dales seem to span a rangefrom E min = − to E min = − for the system size studied.However, taking into account the high occurrence of dales atthe energy of E min = − (see Fig. 24) indicates that espe-cially energies in this range have on average the highest num-ber of states pertaining to minimum energy configurations.Figure 9 shows an example of the disconnectivity graph forthe C model. Due to the small values of the spin-spin inter-actions, the ground state of the C model is highly degenerate.It exhibits a large number of minima of all types and has theenergy E ground = − as each of the cells has an energy of
75 70 65 60 55 50 45 40 E min a v e r a g e s i z e o f t y p e s o f m i n i m a regulartype-1 daletype-2 daletype-3 dale FIG. 25: Average size of the different dale minima types for the C model of planted spin glasses. H = − . Figure 26 (a) shows the possible contributions of thespin-spin interactions of a single unit cell to a vertex. If twounit cells are joined at a vertex, the possible distinct combina-tions of spin-spin interactions to a vertex are shown in Fig. 26.Note that from these combinations, similar to the C modelsystem, only one configuration of spin-spin interactions leadsto a zero-energy spin. This combination is framed in green inthe figure. However, different from the C case, it allows forlarger structures of connected zero-energy spins (see Fig. 26).These connected structures allow for type-2 dale minima andsince the spin-spin interactions to spins that are connected tozero-energy spins can be balanced once the zero-energy spinshave been flipped, this allows also for type-3 dale minima tooccur. Hence, the planted solutions of model system C canbe regular minima or dale-minima of any type.Figure 27 shows the distribution of the minima types inthe reduced description of the C model system. Differentto the C and C model systems, a large number of min-ima is energetically close to the ground state. This model isdominated by type-2 dale minima suggesting the formationof large minimum-energy dales on the energy landscape, i.e.,on average each dale consists of individual configurationsconnected by zero-energy spins (see Tab. I). Furthermore, thismodel has the least number of regular minima at any energy.These results suggest that, using standard optimization proce-dures, a configuration belonging to the ground state will beeasier to find for the C model than for the other models. Thisis in agreement with the numerical results of Perera et al. [38].Figure 28 shows the average size of the types of minimaat their respective energies of the C model. As can be seen,similar to the C and C model, regular minima always havesize one, i.e., they consist of only one subminimum, and type-3 dale minima have the largest number of subminima. How-ever, different from both the C and C models, type-2 daleminima are not always larger than type-1, i.e., only for en-ergies up to E min = − the type-2 dale minima consist ofmore subminima than the type-1 dale minima, but at the high-4 FIG. 26: (a) Possible contributions of spin-spin interactions of a sin-gle C unit cell to a vertex. The bonds J ij = − , , and are indi-cated by wiggly lines ( ), dashed lines ( ) and solid lines ( ), re-spectively. (b) Possible combinations of the spin-spin interactions toa vertex of model system C . Only one of the combinations (framed)allows for zero-energy spins of the planted solutions. (c) Possiblecombinations of the spin-spin interactions for the unit cells joined ata zero-energy vertex. One of these combinations (framed) allows fora zero-energy spin at a second vertex and hence for the formation ofconnected structures of zero-energy spins.
55 50 45 40 35 E min d i s t r i b u t i o n o f t y p e s o f m i n i m a regulartype-1 daletype-2 daletype-3 dale FIG. 27: Distribution of the different dale types for the C modelsystem of planted spin glasses. The lines are Gaussian fits and guidesto the eye [49]. est occupied energy level at E min = − type-1 dale minimaare on average larger than type-2 dales. The size of type-1dale minima is relatively constant ( ≤ ) for all energy lev-els. Type-2 and type-3 dales are largest at the lowest energies,which together with their very high occurrence (compare toFig. 27) is a further indication of the relative ease at whichlowest-energy states can be found.
55 50 45 40 35 E min a v e r a g e s i z e o f t y p e s o f m i n i m a regulartype-1 daletype-2 daletype-3 dale FIG. 28: Average size of the different dale types for the C modelsystem of planted spin glasses. V. SUMMARY
We introduce a classification scheme for the different typesof connected degenerate minima, which we call dale min-ima. We distinguish between regular minima, i.e., minima forwhich a flip of any spin would increase the energy, and daleminima. The dale minima form broad valleys in the energylandscape and are composed of multiple minima connectedvia zero-energy spin flips. We distinguish between three dif-ferent types of dale minima based on similarity and accessibil-ity of the states to each other. This procedure effectively helpsto reduce the number of stored minima during the computationand eases the visualization of disconnectivity graphs. In thedisconnectivity graphs, the dale minima are distinguished bycolors. Furthermore, we add a bar chart depicting the averagenumber of subminima pertaining to the dales at the respectiveenergy levels. The added classification into different typesof minima allows for an enhanced version of disconnectivitygraphs that give an intuitive understanding of the importantaspects of the potential energy landscape of spin systems.We apply our classification scheme to a spin-glass modelwith planted solutions and differentiate between the three el-ementary problem classes, namely C , C , and C . The re-sulting disconnectivity graphs and subsequent analysis showdistinctly different features for the different classes. C sys-tems only have two ground states, corresponding to the ferro-magnetic solution of all spins up or all spins down. Its higher-energy states are separated by a large gap to the ground stateand are highly degenerate. All of the states, including theground states, of the C and C model systems are degen-erate, but do not have a large gap to the ground-state. Com-pared to C and C , C instances have the highest numberof ground-states relative to higher energy states, leading us toconclude that C instances are the easiest to solve when us-ing conventional optimization routines. Plots of the differentminima types versus their energy show distinct Gaussian fea-tures, with the mean of the C and C models being centered5at energies above the ground state. Only for C instances arethe majority of the minima located close to the ground state.While our analysis of these types of planted spin glasses islimited to a system size of spins, we estimate that the gen-eral features of the energy landscape, such as the occurrenceof ground-state solutions and the relative distribution of theminima across different energy levels, is preserved for largersystem sizes. Similar arguments hold for the occurrence ofthe different types of dales and their distribution over the dif-ferent energy levels. Note, however, that our conclusions arebased on relatively small system sizes because the phase spacegrows exponentially with the number of spins. This meansthat further types of dales might occur when the system sizesare increased. However, the developed methods can be gener-ally applied, thus presenting a simple visual approach to char-acterize the phase space of statistical mechanical systems. Acknowledgments
We would like to thank Dilina Perera for feedback on themanuscript. H.G.K. would like to thank Izumi Kirkland for in- spiration. This research is based upon work supported in partby the Office of the Director of National Intelligence (ODNI),Intelligence Advanced Research Projects Activity (IARPA),via MIT Lincoln Laboratory Air Force Contract No. FA8721-05-C-0002. The views and conclusions contained herein arethose of the authors and should not be interpreted as necessar-ily representing the official policies or endorsements, eitherexpressed or implied, of ODNI, IARPA, or the U.S. Govern-ment. The U.S. Government is authorized to reproduce anddistribute reprints for Governmental purpose notwithstand-ing any copyright annotation thereon. We thank the TexasA&M University for providing high performance computingresources. [1] O. M. Becker and M. Karplus,
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