# Fileset

[acs.jpclett.6c02157.pdf](https://mdr.nims.go.jp/filesets/3b4593f0-5651-455c-b299-7450a4b70db7/download)

## Creator

[Ai Koizumi](https://orcid.org/0000-0003-4609-4586), [Tomofumi Tada](https://orcid.org/0000-0003-3093-3779), [Ryo Tamura](https://orcid.org/0000-0002-0349-358X)

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Kinetic MonteCarlo–Ising Machine Optimizationfor Atomistic Inverse Design of Solid Electrolytes](https://mdr.nims.go.jp/datasets/844dc94a-ca6d-44ef-8440-7c27ff5ce54d)

## Fulltext

Kinetic Monte Carlo−Ising Machine Optimization for AtomisticInverse Design of Solid ElectrolytesAi Koizumi,* Tomofumi Tada,* and Ryo Tamura*Cite This: J. Phys. Chem. Lett. 2026, 17, 10758−10762 Read OnlineACCESS Metrics & More Article Recommendations *sı Supporting InformationABSTRACT: Maximizing ionic conductivity remains a fundamental challenge in the atomisticdesign of solid electrolytes. To this end, we present a framework that combines kinetic MonteCarlo (KMC) and factorization machine with quadratic-optimization annealing (FMQA), anIsing-machine-based black-box optimization algorithm for large-scale combinatorial problems.KMC evaluates the ionic conductivity for a given dopant configuration, whereas FMQAiteratively learns a surrogate model from a small configuration-conductivity data set andproposes configurations expected to maximize conductivity. To address the severecombinatorial explosion in large KMC simulation cells, we partition the configuration spacefor parallel optimization. As a proof of concept, we apply this KMC−FMQA framework to bulk 8 mol % yttria-stabilized zirconia,identifying a dopant configuration with an order-of-magnitude higher conductivity than that of random configurations. Combinedwith experiments, this framework will enable the determination of microscopic structures from measured conductivity, providinginsight into the underlying transport mechanisms.Solid oxide fuel cells (SOFCs)1−6 are next-generation energyconversion devices, and ionic conductivity of solid electro-lytes, which are a key component of SOFCs, has been extensivelyinvestigated to improve their performance. In the design of solidelectrolytes, the microscopic structure plays a crucial role indetermining ionic conductivity, especially when SOFC devicesare miniaturized. For example, yttria-stabilized zirconia (YSZ) isa representative solid electrolyte used in SOFCs, and its ionicconductivity has been well investigated as a function of dopantconcentration, and 8 mol % YSZ (8YSZ) exhibits the highestionic conductivity.7 In bulk 8YSZ, the dopants are randomlydistributed, and the effect of the configuration on ionicconductivity is generally negligible. In contrast, thin film 8YSZexhibits a wide range of ionic conductivity,8 suggesting a strongdependence on various microscopic structural features,including dopant configuration. Thus, if microscopic structurescan be well controlled, the ionic conductivity of 8YSZ could befurther enhanced. However, elucidating this microscopicstructure−conductivity relationship experimentally remainschallenging, motivating computational approaches to investigateit. Recently, we used kinetic Monte Carlo (KMC) simula-tions9−11 to investigate drift currents arising from anisotropicoxygen-ion diffusion under an applied bias voltage while varyingthe dopant configuration in bulk 8YSZ, thereby demonstratingthat dopant configurations have a profound effect on ionicconductivity. Furthermore, we established a link betweenmicroscopic dopant configurations and ionic conductivitiesusing a structural descriptor related to ion-migration paths.11 Inother words, ionic conductivity was explicitly shown to dependsystematically on the microscopic dopant configuration.The ultimate goal is not solely to establish a link betweendopant configurations and ionic conductivity, but to identify theconfiguration that maximizes ionic conductivity based on thelink. This is the essence of atomistic inverse design.12−14However, since KMC only solves the forward problem ofpredicting ionic conductivity from a given configuration,achieving inverse design requires efficient exploration of thevast configuration space. To this end, black-box optimization(BBO)15 is particularly well studied. BBO uses a surrogatemodel constructed from a small configuration-conductivity dataset to iteratively propose and evaluate configurations, therebyidentifying configurations with the desired ionic conductivitywithout requiring knowledge of the full configuration−conductivity relationship. That said, identifying optimal atomicconfigurations in large multicomponent crystalline systems ischallenging due to the combinatorial explosion in the number ofpossible configurations, which renders conventional BBOmethods such as Bayesian optimization16,17 intractable. Insuch situations, Factorization Machine with Quadratic-opti-mization Annealing (FMQA)18,19 provides a powerful alter-native, combining the efficiency of BBO with the combinatorialsearch capability of Ising machines.20 Recently, we appliedFMQA with Vienna Ab initio Simulation Package (VASP) todetermine stable atomic configurations in a small supercell21 andin a relatively large grain boundary system,22 thereby elucidatingtheir atomic configurations. FMQA has thus emerged as a keyReceived: June 29, 2026Revised: August 23, 2026Accepted: August 24, 2026Published: September 3, 2026Letterpubs.acs.org/JPCL© 2026 The Authors. Published byAmerican Chemical Society10758https://doi.org/10.1021/acs.jpclett.6c02157J. Phys. Chem. Lett. 2026, 17, 10758−10762This article is licensed under CC-BY 4.0Downloaded from pubs.​acs.​org/​jpclcd/​article-pdf/​17/​37/​10758/​68334494/​acs.​jpclett.​6c02157.​pdf by NATL INST FOR MATLS SCIENCE (NIMS) user on 01 November 2026https://pubs.acs.org/action/doSearch?field1=Contrib&text1="Ai+Koizumi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Tomofumi+Tada"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Ryo+Tamura"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.jpclett.6c02157&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?goto=articleMetrics&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?goto=recommendations&?ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?goto=supporting-info&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=tgr1&ref=pdfhttps://pubs.acs.org/toc/jpclcd/17/37?ref=pdfhttps://pubs.acs.org/toc/jpclcd/17/37?ref=pdfhttps://pubs.acs.org/toc/jpclcd/17/37?ref=pdfhttps://pubs.acs.org/toc/jpclcd/17/37?ref=pdfpubs.acs.org/JPCL?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://pubs.acs.org?ref=pdfhttps://doi.org/10.1021/acs.jpclett.6c02157?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://pubs.acs.org/JPCL?ref=pdfhttps://pubs.acs.org/JPCL?ref=pdfhttps://creativecommons.org/licenses/by/4.0/technology in BBO, with successful applications ranging fromcrystal structure prediction to broader combinatorial config-uration problems.23−26Here, we demonstrate that the atomistic inverse designproblem for solid electrolytes can be solved by combiningKMC9−11 with FMQA,18,19 establishing an efficient structuraldesign framework (KMC−FMQA framework), as illustrated inFigure 1. KMC evaluates the ionic current for a given dopantconfiguration, and FMQA iteratively learns a surrogate modelfrom a small configuration-current data set and proposesconfigurations predicted to exhibit the desired ionic current.We apply this framework to optimize the dopant configurationin a 16-layer region of a bulk 8YSZ supercell (see Figure 1(c)).For efficient exploration of the vast configuration space, weintroduce a block model in which four lattice sites are groupedinto a single unit (block), and three types of blocks, Zr4, Y2Zr2,and Y4, are used to construct the 8YSZ supercell (see Figure1(a)). To maintain the 8YSZ dopant concentration, weenumerated all 81 possible block compositions (b = 0−80)that 320 Y atoms were included in 200 blocks within the 16layers, as shown in Figure 2(a) (see Supporting Note B fordetails). We then partitioned the search space into 81compositional subspaces and explored each subspace independ-ently. In this letter, we set the inverse-design target to identifythe block configuration exhibiting the highest ionic conductivity.Further details are described in Methods.Figure 2(b) shows the evolution of the highest ionicconductivity obtained in 81 parallel KMC−FMQA optimizationruns over 30 steps with 10 initial training data. FMQAconsistently identified dopant configurations with higher ionicconductivity than those obtained by the initial random sampling,demonstrating the effectiveness of the KMC-FMQA framework.Note that histograms of the ionic conductivities are shown inFigure S1. To further explore promising configurations, anadditional 70 optimization steps were performed for the threeblock compositions with b = 20, 14, and 8, that exhibited thehighest ionic conductivities during the first 30 steps (see FigureS2). As shown in Figure S2, the optimization trajectories appearto vary significantly depending on the initial data distribution,suggesting that there is room for improving the BBO algorithmto ensure more robust optimization (see Supporting Note E andF).The optimization identified a dopant configuration with anionic conductivity of 0.423 S cm−1, which is an order-of-magnitude higher than the average conductivity for randomdopant distributions (0.06 S cm−1).11 Note that the calculatedconductivity for random dopant distributions is close to theexperimentally reported conductivity of bulk 8YSZ (0.0505 Scm−1).27 Figures 3(a)-(c) show the three highest-conductivityconfigurations for the three block compositions with b = 20, 14,and 8, whereas Figures 3(d)-(f) show the configurations withionic conductivity close to the experimentally reported value,selected from the initial random sampling data sets for the sameblock compositions. Note that we searched for high-conductivity dopant configurations in the bulk 8YSZ system inwhich the dopants are confined to the 16 layers, that is, theinvestigated system corresponds to the ZrO2/YSZ/ZrO2 stripemodel which has two interfaces in the simulation cell. In thehigh-conductivity configurations, dopants are highly concen-trated near the edges of the 16 layers (i.e., at the YSZ/ZrO2interfaces); the importance of the interfacial structure was alsoconfirmed by the Random Forest regression analysis (seeSupporting Note G). Such layered dopant configurations areexpected to be fabricated using epitaxial growth techniques suchas pulsed laser deposition (PLD) and atomic layer deposition(ALD). The ease of ion migration is determined by the dopantconfiguration in the six-site ion-migration path shown in Figure1(d) and by the electric field (see our previous study11 fordetails). When dopants are locally concentrated in the path, theconfiguration-dependent migration barrier tends to be higher(see Table S2). We thus investigated the electrostaticcontribution to the activation barrier around the layered dopantconfigurations (see Supporting Note H), but a definitiveexplanation for the observed enhancement in conductivityremains elusive. To fully elucidate the configuration−conductivity relationship, further investigations are required,including exploring a broader range of dopant configurationsand identifying key structural features from high-dimensionalstructural descriptors governing ionic conductivity.11To benchmark the framework, we compared three approacheswith progressively reduced search spaces: the all-atom model,the block model without the composition-based partitioningbased on the block composition index b, and the block modelwith the partitioning (see Supporting Note A). The block modelwith composition-based partitioning achieved the best opti-mization performance among the three approaches, demon-strating that a block-based representation combined withsearch-space partitioning is more effective than simplyincreasing the number of trials in a vast search space.In summary, we developed KMC−FMQA framework andapplied it to search for high-conductivity dopant configurationsin the bulk 8YSZ system. As a proof of concept, wedemonstrated that the framework enables the atomistic inversedesign of solid electrolytes, identifying a dopant configurationFigure 1. KMC−FMQA framework. In (a), the factorization machine(FM) is trained using the relationship between the dopantconfigurations encoded as bit strings and their ionic conductivities,and the trained FM is used by an Ising machine (b) to explorepromising dopant configurations. In (c), KMC simulations areperformed on the systems corresponding to the proposed bitconfigurations, and the resulting ionic conductivities are added to thetraining data set for the next cycle. Green and gray spheres representzirconium (Zr) and yttrium (Y) lattice ions to be optimized,respectively, whereas red spheres represent oxygen ions migrating viaoxygen vacancies along paths consisting of six lattice sites (d).The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letterhttps://doi.org/10.1021/acs.jpclett.6c02157J. Phys. Chem. Lett. 2026, 17, 10758−1076210759Downloaded from pubs.​acs.​org/​jpclcd/​article-pdf/​17/​37/​10758/​68334494/​acs.​jpclett.​6c02157.​pdf by NATL INST FOR MATLS SCIENCE (NIMS) user on 01 November 2026https://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig1&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig1&ref=pdfpubs.acs.org/JPCL?ref=pdfhttps://doi.org/10.1021/acs.jpclett.6c02157?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-aswith an order-of-magnitude higher ionic conductivity than thatof random configurations. The framework efficiently exploresthe vast configuration space through a block-based representa-tion combined with composition-based partitioning, enablingrapid identification of high-conductivity configurations. Toensure more practical exploration, we plan to take energeticstability into account in the framework. In the future, byminimizing the discrepancy between experimental and calcu-lated conductivities, the framework will enable inverseestimation of microscopic structures, potentially providingdeeper microscopic insight into ionic transport mechanisms inSOFC electrolytes. The approach is readily extensible toelectrolyte-electrode interfaces, opening a pathway towardbroader atomistic materials design for SOFC performanceoptimization.■ METHODSAn overview of the KMC−FMQA framework is summarized inFigure 1. KMC evaluates the ionic conductivity for a givendopant configuration, whereas FMQA learns a surrogate modelfrom a small configuration-conductivity data set and optimizesthe dopant configuration to maximize ionic conductivity. Here,we first briefly explain the KMC evaluation and then describe theFMQA optimization.In KMC part, for given dopant configurations, KMC evaluatesdrift currents arising from the migration of mobile light oxygenions (Figure 1(d)), where the migration barriers mainly dependon the dopant configuration (see Table S2 for the barrierparameters). In contrast, FMQA updates dopant configurationsto achieve the desired current. That is, KMC is employed tosimulate the time evolution of light mobile-ions, whereas FMQAFigure 2. (a) Eighty-one block compositions satisfying the 8YSZ dopant concentration, where Nsite and Ndope denote the numbers of block placementsites and yttrium (Y) dopants in the system, respectively (Nsite = 200, Ndope = 320). b denotes the block composition index. (b) Optimizationtrajectories for each of the 81 block compositions. Each trajectory shows the highest ionic conductivity achieved at each iteration step. The ionicconductivity up to step 0 corresponds to the initial training data obtained from 10 evaluations through random sampling, and the subsequent values areobtained through FMQA optimization. The white solid and white dashed horizontal lines represent the average conductivity for random dopantdistributions11 and the experimentally reported conductivity of bulk 8YSZ,27 respectively.Figure 3. Cross-sectional images of high-conductivity configurations sliced into 16 layers. (a)-(c) The three highest-conductivity configurationsidentified by the KMC-FMQA framework, with ionic conductivities of 0.423, 0.400, and 0.371 S cm−1, respectively. (d)-(f) Configurations with ionicconductivity close to the experimentally reported value, selected from the initial random data sets corresponding to (a)-(c). Green and gray spheresrepresent zirconium (Zr) and yttrium (Y) atoms, respectively.The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letterhttps://doi.org/10.1021/acs.jpclett.6c02157J. Phys. Chem. Lett. 2026, 17, 10758−1076210760Downloaded from pubs.​acs.​org/​jpclcd/​article-pdf/​17/​37/​10758/​68334494/​acs.​jpclett.​6c02157.​pdf by NATL INST FOR MATLS SCIENCE (NIMS) user on 01 November 2026https://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig2&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig3&ref=pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?fig=fig3&ref=pdfpubs.acs.org/JPCL?ref=pdfhttps://doi.org/10.1021/acs.jpclett.6c02157?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asupdates the configuration of heavy lattice-atoms. When theoptimization targets energetic stability rather than ionicconductivity, the roles of KMC and FMQA become analogousto those of MD andMC in MC−MD algorithms,28 respectively.The KMC computational conditions are summarized in TableS1 and are described in detail in our previous study.11 In theprevious study, we observed diverse ionic conductivities whendopants were highly concentrated in 16 layers along an appliedbias voltage direction shown in Figure 1(c). Therefore, in thisstudy, the KMC−FMQA framework was applied to theoptimization of dopant configurations in the 16 layers betweenthe dashed lines in Figure 1(c).To efficiently handle the large configuration space, weintroduced a block model based on the six-site ion-migrationpaths that govern ionic conductivity11 (see Supporting Note Bfor details), even though this reduction of the search space maynot be appropriate for the inverse estimation of microscopicstructures when combined with experiments. This modelsignificantly reduced the search space, yielding a QuadraticUnconstrained Binary Optimization (QUBO) model that canbe efficiently solved using an Ising machine. The 16 layers wereconstructed using three types of blocks (see Figure 1(a)), and 81block compositions satisfying the 8YSZ dopant concentrationwere enumerated in advance (see Figure 2(a)). The search spacereduction is provided in Supporting Note C.FMQA was employed to optimize the block configuration tomaximize ionic conductivity. Since only binary variables (0/1)can be used as inputs for FMQA, the block configurations wererepresented as bit structures; the three blocks were representedas 100, 010, and 001, as illustrated in Figure 1(a). The bit qitsatisfies the following two constraints==q i1,tit13(1)where i and t denote the site and block-type indices, respectively,and when constructing the system by a block composition withindex b,==q N b( )iNit t1site(2)where Nt(b) denotes the number of blocks of type t, and Nsitedenotes the number of block placement sites. The firstconstraint ensures that each site is assigned exactly one blocktype (one-hot constraint), and the second fixes the blockcomposition to satisfy the 8YSZ dopant concentration. Theoptimization was performed as following four steps.1. Initial Training Data Generation: 10 block configurationswere generated by random sampling under the twoconstraints of eqs 1 and 2, and KMC simulations wereperformed to obtain the corresponding ionic conductivi-ties.2. Model Training: The factorization machine (FM) wastrained using the initial data set. The ionic conductivityprediction model is defined as follows.= * + * *= = = = = = =w q v v q qq( )iNtit itiNt jNu kKitk juk it ju1 131 131 131site site site(3)where q denotes a block configuration represented as a bitstructure, wit* and vitk* denote the trained parameters, and Kdenotes the dimension of the latent space (hyperparameter)which was set to 8 (default value).293. Ising Machine Optimization: Ising machine searches forthe global maximum for a given function. In this study,Fixstars Amplify AE,30 a classical annealer, was employedas the Ising machine and performed Simulated Annealing.A promising configuration q that maximizes the trainedFM is determined by an Ising machine under the twoconstraints of eqs 1 and 2. The most promising solution,q*, was selected in each cycle. If q* violated theconstraints or was already included in the training data,q* was regenerated by random sampling.4. Ionic Conductivity Calculation: The KMC simulation isperformed on the system identified by q*. The obtainedconductivity is added to the training data, and theprocedure then returns to Step 2.The crystalline system exhibits high symmetry; however, thephysically equivalent configurations are treated as distinct pointsin the bit structure space. Therefore, to cover all symmetry-equivalent configurations with a single evaluation, 100 symmetryoperations were applied (see Supporting Note D), therebyexpanding the training data set by a factor of 100 from a singleKMC run. In this study, for each of the 81 block compositions,we performed three independent optimization trials withdifferent initial training data sets. We then determined thehighest conductivity in three independent trials and plotted it asa function of optimization step in Figure 2(b).■ ASSOCIATED CONTENT*sı Supporting InformationThe Supporting Information is available free of charge athttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157.KMC condition; Migration barrier table; Comparison ofionic conductivity histograms from random sampling andFMQA;Optimization trajectories with three highest ionicconductivities; Optimization performance under differentconstraints; Block model; Search space reduction;Symmetry operations; Analysis of BBO performance;Prediction accuracy of FM; Analysis of structure-conductivity relationship; Electrostatic contribution(PDF)■ AUTHOR INFORMATIONCorresponding AuthorsAi Koizumi − Center for Basic Research on Materials, NationalInstitute for Materials Science, Tsukuba, Ibaraki 305-0044,Japan; Center for Computational Sciences, University ofTsukuba, Tsukuba, Ibaraki 305-8577, Japan; orcid.org/0000-0003-4609-4586; Email: koizumi.ai@ccs.tsukuba.ac.jpTomofumi Tada − Kyushu University Platform of Inter/Transdisciplinary Energy Research, Kyushu University,Fukuoka 819-0395, Japan; orcid.org/0000-0003-3093-3779; Email: tada.tomofumi.054@m.kyushu-u.ac.jpRyoTamura− Center for Basic Research on Materials, NationalInstitute for Materials Science, Tsukuba, Ibaraki 305-0044,Japan; Graduate School of Frontier Sciences, The University ofTokyo, Kashiwa, Chiba 277-8561, Japan; orcid.org/0000-0002-0349-358X; Email: TAMURA.Ryo@nims.go.jpComplete contact information is available at:https://pubs.acs.org/10.1021/acs.jpclett.6c02157The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letterhttps://doi.org/10.1021/acs.jpclett.6c02157J. Phys. Chem. Lett. 2026, 17, 10758−1076210761Downloaded from pubs.​acs.​org/​jpclcd/​article-pdf/​17/​37/​10758/​68334494/​acs.​jpclett.​6c02157.​pdf by NATL INST FOR MATLS SCIENCE (NIMS) user on 01 November 2026https://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?goto=supporting-infohttps://pubs.acs.org/doi/suppl/10.1021/acs.jpclett.6c02157/suppl_file/jz6c02157_si_001.pdfhttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Ai+Koizumi"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0003-4609-4586https://orcid.org/0000-0003-4609-4586mailto:koizumi.ai@ccs.tsukuba.ac.jpmailto:koizumi.ai@ccs.tsukuba.ac.jphttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Tomofumi+Tada"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0003-3093-3779https://orcid.org/0000-0003-3093-3779mailto:tada.tomofumi.054@m.kyushu-u.ac.jphttps://pubs.acs.org/action/doSearch?field1=Contrib&text1="Ryo+Tamura"&field2=AllField&text2=&publication=&accessType=allContent&Earliest=&ref=pdfhttps://orcid.org/0000-0002-0349-358Xhttps://orcid.org/0000-0002-0349-358Xmailto:TAMURA.Ryo@nims.go.jphttps://pubs.acs.org/doi/10.1021/acs.jpclett.6c02157?ref=pdfpubs.acs.org/JPCL?ref=pdfhttps://doi.org/10.1021/acs.jpclett.6c02157?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-asNotesThe authors declare no competing financial interest.■ ACKNOWLEDGMENTSThis study was supported by JST CREST Grant NumberJPMJCR2234, Japan. T.T. acknowledges the support from JSPSKAKENHI Grant Number JP24H02203. The authors wouldlike to thank Prof. Naoki Ohashi, Prof. Takashi Washio, andProf. Tomoyuki Higuchi for valuable discussions and insightfulcomments.■ ABBREVIATIONSFactorization machine with quadratic-optimization annealingFMQAQuadratic unconstrained binary optimizationQUBOKinetic Monte CarloKMCEight mol % yttria-stabilized zirconia8YSZSolid oxide fuel cellsSOFCs■ REFERENCES(1) Li, J.; Cai, Q.; Horri, B. A. Highly Conductive and StableElectrolytes for Solid Oxide Electrolysis and Fuel Cells: Fabrication,Characterisation, Recent Progress and Challenges. Mater. Adv. 2025, 6,39−83.(2) Minh, N. Q. Ceramic Fuel Cells. J. Am. Ceram. Soc. 1993, 76 (3),563−588.(3) Haile, S. M. Fuel Cell Materials and Components. Acta Mater.2003, 51 (19), 5981−6000.(4) Malavasi, L.; Fisher, C. A.; Islam, M. S. Oxide-Ion and ProtonConducting Electrolyte Materials for Clean Energy Applications:Structural and Mechanistic Features. Chem. Soc. Rev. 2010, 39 (11),4370−4387.(5) Yamamoto, O. Solid Oxide Fuel Cells: Fundamental Aspects andProspects. Electrochim. Acta 2000, 45 (15−16), 2423−2435.(6) Stambouli, A. B.; Traversa, E. Solid Oxide Fuel Cells (SOFCs): AReview of an Environmentally Clean and Efficient Source of Energy.Renew. Sustain. Energy Rev. 2002, 6 (5), 433−455.(7) Strickler, D. W.; Carlson, W. G. Ionic Conductivity of Cubic SolidSolutions in the System CaO�Y2 O3 �ZrO2. J. Am. Ceram. Soc. 1964,47 (3), 122−127.(8) Jiang, J.; Hertz, J. L. On the Variability of Reported IonicConductivity in Nanoscale YSZ Thin Films. J. Electroceramics 2014, 32(1), 37−46.(9) Tada, T.; Watanabe, N. Parallelized Meso-Scale Kinetic MonteCarlo Simulations for SOFC Characterization. ECS Trans. 2013, 57(1), 2437.(10) Tada, T. Full Atomistic Kinetic Monte Carlo and First PrinciplesStudy on Electromotive Force of SOFC with Direct CountingApproach. ECS Trans. 2017, 78 (1), 2815.(11) Koizumi, A.; Ivonina, M.; Tamura, R.; Tada, T. LinkingMicroscopic Structures to Ionic Currents in Solid Electrolytes viaKinetic Monte Carlo and Regression Analysis. J. Phys. Chem. C 2026,130 (10), 3897−3908.(12) Zunger, A. Inverse Design in Search of Materials with TargetFunctionalities. Nat. Rev. Chem. 2018, 2 (4), 0121.(13) Fung, V.; Zhang, J.; Hu, G.; Ganesh, P.; Sumpter, B. G. InverseDesign of Two-Dimensional Materials with Invertible Neural Net-works. Npj Comput. Mater. 2021, 7 (1), 200.(14) Wines, D.; Xie, T.; Choudhary, K. Inverse Design of Next-Generation Superconductors Using Data-Driven Deep GenerativeModels. J. Phys. Chem. Lett. 2023, 14 (29), 6630−6638.(15) Terayama, K.; Sumita, M.; Tamura, R.; Tsuda, K. Black-BoxOptimization for Automated Discovery. Acc. Chem. Res. 2021, 54 (6),1334−1346.(16) Yamashita, T.; Sato, N.; Kino, H.; Miyake, T.; Tsuda, K.; Oguchi,T. Crystal Structure Prediction Accelerated by Bayesian Optimization.Phys. Rev. Mater. 2018, 2 (1), 013803.(17) Boyar, O.; Gu, Y.; Tanaka, Y.; Tonogai, S.; Itakura, T.; Takeuchi,I. Crystal-Lsbo: Automated Design of de Novo Crystals with LatentSpace Bayesian Optimization. Neural Comput 2025, 37 (8), 1505−1527.(18) Kitai, K.; Guo, J.; Ju, S.; Tanaka, S.; Tsuda, K.; Shiomi, J.;Tamura, R. Designing Metamaterials with Quantum Annealing andFactorization Machines. Phys. Rev. Res. 2020, 2 (1), 013319.(19) Tamura, R.; Seki, Y.; Minamoto, Y.; Kitai, K.; Matsuda, Y.;Tanaka, S.; Tsuda, K. Black-Box Optimization Using Factorization andIsing Machines. Appl. Phys. Rev. 2026, 13, 021307.(20) Mohseni, N.; McMahon, P. L.; Byrnes, T. Ising Machines asHardware Solvers of Combinatorial Optimization Problems. Nat. Rev.Phys. 2022, 4, 363−379.(21) Lin, J.; Tada, T.; Koizumi, A.; Sumita, M.; Tsuda, K.; Tamura, R.Determination of Stable Proton Configurations by Black-BoxOptimization Using an Ising Machine. J. Phys. Chem. C 2025, 129(5), 2332−2340.(22) Koizumi, A.; Tamura, R.; Geng, W. T. submitted for publication.(23) Couzinié, Y.; Seki, Y.; Nishiya, Y.; Nishi, H.; Kosugi, T.; Tanaka,S.; Matsushita, Y. Machine Learning Supported Annealing forPrediction of Grand Canonical Crystal Structures. J. Phys. Soc. Jpn.2025, 94 (4), 044802.(24) Xu, Z.; Shang, W.; Kim, S.; Lee, E.; Luo, T. Quantum Annealing-Assisted Lattice Optimization. Npj Comput. Mater. 2025, 11 (1), 4.(25) Liang, C.; Das, D.; Guo, J.; Tamura, R.; Mao, Z.; Tsuda, K.Predicting Symmetric Structures of Large Crystals with GPU-BasedIsing Machines. Commun. Phys. 2025, 8 (1), 477.(26) Kim, S.; Suh, I.-S.; Humble, T. S.; Beck, T.; Lee, E.; Luo, T.Harnessing Quantum Computing for Energy Materials: Opportunitiesand Challenges. ACS Energy Lett. 2026, 11 (2), 1182−1192.(27) Yamahara, K.; Jacobson, C. P.; Visco, S. J.; De Jonghe, L. C.Influence of Powders on Ionic Conductivity of PolycrystallineZirconias. ECS Proc. 2003, 2003-07 (1), 187.(28) Lee, H. B.; Prinz, F. B.; Cai, W. Atomistic Simulations of SurfaceSegregation of Defects in Solid Oxide Electrolytes. Acta Mater. 2010, 58(6), 2197−2206.(29) Rendle, S. FactorizationMachines with libFM. ACM Trans. Intell.Syst. Technol. 2012, 3 (3), 1−22.(30) Fixstars Amplify AE. https://amplify.fixstars.com/en/ (accessedJanuary 11, 2026).The Journal of Physical Chemistry Letters pubs.acs.org/JPCL Letterhttps://doi.org/10.1021/acs.jpclett.6c02157J. Phys. Chem. Lett. 2026, 17, 10758−1076210762Downloaded from pubs.​acs.​org/​jpclcd/​article-pdf/​17/​37/​10758/​68334494/​acs.​jpclett.​6c02157.​pdf by NATL INST FOR MATLS SCIENCE (NIMS) user on 01 November 2026https://doi.org/10.1039/D4MA00690Ahttps://doi.org/10.1039/D4MA00690Ahttps://doi.org/10.1039/D4MA00690Ahttps://doi.org/10.1111/j.1151-2916.1993.tb03645.xhttps://doi.org/10.1016/j.actamat.2003.08.004https://doi.org/10.1039/b915141ahttps://doi.org/10.1039/b915141ahttps://doi.org/10.1039/b915141ahttps://doi.org/10.1016/S0013-4686(00)00330-3https://doi.org/10.1016/S0013-4686(00)00330-3https://doi.org/10.1016/S1364-0321(02)00014-Xhttps://doi.org/10.1016/S1364-0321(02)00014-Xhttps://doi.org/10.1111/j.1151-2916.1964.tb14368.xhttps://doi.org/10.1111/j.1151-2916.1964.tb14368.xhttps://doi.org/10.1007/s10832-013-9857-1https://doi.org/10.1007/s10832-013-9857-1https://doi.org/10.1149/05701.2437ecsthttps://doi.org/10.1149/05701.2437ecsthttps://doi.org/10.1149/07801.2815ecsthttps://doi.org/10.1149/07801.2815ecsthttps://doi.org/10.1149/07801.2815ecsthttps://doi.org/10.1021/acs.jpcc.5c06852?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jpcc.5c06852?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jpcc.5c06852?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1038/s41570-018-0121https://doi.org/10.1038/s41570-018-0121https://doi.org/10.1038/s41524-021-00670-xhttps://doi.org/10.1038/s41524-021-00670-xhttps://doi.org/10.1038/s41524-021-00670-xhttps://doi.org/10.1021/acs.jpclett.3c01260?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jpclett.3c01260?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jpclett.3c01260?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.accounts.0c00713?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.accounts.0c00713?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1103/PhysRevMaterials.2.013803https://doi.org/10.1162/neco_a_01767https://doi.org/10.1162/neco_a_01767https://doi.org/10.1103/PhysRevResearch.2.013319https://doi.org/10.1103/PhysRevResearch.2.013319https://doi.org/10.1063/5.0275450https://doi.org/10.1063/5.0275450https://doi.org/10.1038/s42254-022-00440-8https://doi.org/10.1038/s42254-022-00440-8https://doi.org/10.1021/acs.jpcc.4c07104?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acs.jpcc.4c07104?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.7566/JPSJ.94.044802https://doi.org/10.7566/JPSJ.94.044802https://doi.org/10.1038/s41524-024-01505-1https://doi.org/10.1038/s41524-024-01505-1https://doi.org/10.1038/s42005-025-02380-yhttps://doi.org/10.1038/s42005-025-02380-yhttps://doi.org/10.1021/acsenergylett.5c04009?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1021/acsenergylett.5c04009?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-ashttps://doi.org/10.1149/200307.0187PVhttps://doi.org/10.1149/200307.0187PVhttps://doi.org/10.1016/j.actamat.2009.12.005https://doi.org/10.1016/j.actamat.2009.12.005https://doi.org/10.1145/2168752.2168771https://amplify.fixstars.com/en/pubs.acs.org/JPCL?ref=pdfhttps://doi.org/10.1021/acs.jpclett.6c02157?urlappend=%3Fref%3DPDF&jav=VoR&rel=cite-as