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We are developing a new lattice QCD code set “Bridge++” aiming at extensible,\nreadable, and portable workbench for QCD simulations, while keeping a high performance at\nthe same time. Bridge++ covers conventional lattice actions and numerical algorithms. The\ncode set is constructed in C++ with an object oriented programming. In this paper we describe\nfundamental ingredients of the code and the current status of development.\n\n1. Introduction\nA perturbative approach to QCD succeeds in the high energy physics such as the deep inelastic\nscattering. On the other hand, it does not apply to the low energy phenomena such as the\nquark confinement and the light hadron spectrum. Some nonperturbative methods are necessary.\nLattice QCD is a SU(3) gauge theory defined on 4-dimensional Euclidean lattice, which provides\na nonperturbative framework of the calculation [1]. The path integral quantization enables\nMonte Carlo simulations.\n\nRapid increase of the processor performance and the integration technology of the large-scale\ncluster system enables us to apply the lattice simulation to wide range of research area: matrix\nelements to test the standard model in LHC and B factory experiments, the phase diagram and\nthe nature of QCD at finite temperature and density, the force among nucleons, and so on. The\nlattice framework is also applied to gauge theories other than QCD, such as a technicolor theory\nto explain the mechanism of the Higgs sector of the standard model.\n\nWhile numerical lattice simulations have become an indispensable tool for theoretical analysis,\nthe programming technique has become more and more involved. The hybrid parallelization with\nMPI and OpenMP is inevitable for the current large-scale system such as K-Computer and Blue\nGene/Q. GPGPUs require an additional development of the code with CUDA or OpenCL. The\n\nACAT2013 IOP Publishing\nJournal of Physics: Conference Series 523 (2014) 012046 doi:10.1088/1742-6596/523/1/012046\n\nContent from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution\nof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.\n\nPublished under licence by IOP Publishing Ltd 1\n\n\n\nFigure 1. Gluon field Uµ and quark ψ on the\nlattice. The plaquette Pµν as well as an interaction\nof fermion operator ψ̄DWψ are also given.\n\nresearch groups have to develop new codes for updated machines. It makes these codes hard\nto be understood for beginners. In particular, it is a serious problem that new ideas or new\nphysical quantities cannot be tested quickly.\n\nIn view of this programming situation, we started a project to develop a new common code set\n“Bridge++” for lattice QCD simulations. We adopt C++ language to make use of the object-\noriented design. Bridge++ has a great deal of readability while keeping sufficient performance\nfor frontier works. The machine dependent part of the code is hidden as much as possible. The\nfirst version of the code was released in July 2012 under the GNU General Public License. It\nis actively being developed to extend functions, brush up its design and the implementation,\nimprove the performance, and provide more documents. The latest version is 1.1.1. The code\nis available from our website [2].\n\nThis paper is organized as follows. In the next section, the core constitution of the lattice\nQCD simulation is briefly summarized. In Sec. 3, the policy of development of Bridge++ project\nis described. Section 4 summarizes our current status of the project and future prospects.\n\n2. Lattice QCD simulations\n2.1. Principle of lattice QCD simulations\nK.G. Wilson proposed to formulate QCD on a discretized Euclidean spacetime, namely on a\nlattice [3]. It enables a nonperturbative analysis of QCD by numerical simulations.\n\nIn lattice QCD, the gauge field Aµ is represented as 3 × 3 complex matrices on the link\nconnecting the nearest neighbor sites such that Uµ(n) = exp(igAµ(n)), where g is the strong\ncoupling constant and n = (nx, ny, nz, nt) is a lattice site (Fig. 1). The number of lattice sites\nis finite in the numerical simulation by the lattice size Lµ, nµ = 1, . . . , Lµ, µ = x, y, z, t. The\nquark field ψ is represented as a complex vector on a lattice site which carries 3 components of\ncolor and 4 components of spinor.\n\nApplying the path integral quantization, an expectation value of observable O is given by a\nfunctional integral\n\n〈O〉 =\n1\n\nZ\n\n∫\nDψDψDUµO(ψ,ψ, Uµ)e−Slat =\n\n1\n\nZ\n\n∫\nDUµO(Uµ) det(D)e−SG , (1)\n\nwhere Z is the partition function and Slat = SG+SF with the gauge part SG and the fermion part\nSF = ψ̄Dψ of the lattice QCD action, respectively. The quark field is integrated by hand and\narrives at the second equality of Eq. (1). The quark propagator D−1 expresses a propagation\nfrom a site to another site. Hadron spectrum can be extracted from the hadron correlation\nfunctions that are composed of quark propagators.\n\nThe expectation value is calculated by generating ensemble of gauge fields with Monte Carlo\nmethod under the probability\n\nP (U) ∝ det(D[U ])e−SG[U ], (2)\n\nACAT2013 IOP Publishing\nJournal of Physics: Conference Series 523 (2014) 012046 doi:10.1088/1742-6596/523/1/012046\n\n2\n\n\n\nwhere U represents a gauge configuration {Uµ(x)}. Once the ensemble of independent\n\nconfigurations U (i) (i = 1, . . . , N) are in hand, an expectation value of the physical observable\nis obtained by\n\n〈O〉 =\n1\n\nN\n\nN∑\ni=1\n\nO[U (i)]. (3)\n\nIt is noted that the expectation value 〈O〉 has not only the statistical error but also several\nsystematic errors. Numerical simulations are inevitably performed at finite lattice spacing and\nfinite volume. It is imperative to control these errors for a precise study.\n\n2.2. Lattice QCD actions\nWe introduce the fundamental form of the gauge and fermion actions. An action on the lattice\nis constructed under the following principles.\n\n• The lattice action must be invariant under the gauge transformation.\n\n• The lattice action must coincide with the continuum theory in the continuum limit, i.e. the\nlattice spacing a→ 0.\n\n• The lattice action should retain symmetries in the continuum theory as much as possible.\n\nThe lattice action is not unique, because there is freedom to add a term which vanishes in\nthe continuum limit. Variety of lattice QCD simulations originate from the type of the lattice\nactions.\n\nThe standard gauge action was given by Wilson, and thus called Wilson (plaquette) action:\n\nSG = −Nc\n\ng2\n\n∑\nn\n\n∑\nµ6=ν\n\nPµν(n), Pµν(n) =\n1\n\nNc\ntr\n[\nUµ(n)Uν(n+ µ̂)U †µ(n+ ν̂)U †ν (n)\n\n]\n, (4)\n\nwhere Nc = 3 is the number of color, n is a site, µ̂ is a unit vector in µ-direction. Pµν(n) is the\nsmallest closed loop made of link variables called plaquette. It corresponds to the field strength.\nThis action agrees with the continuum gauge action up to O(a2) corrections. By adding closed\nloops, such as rectangular loops, improved gauge action is constructed so as to reduce the O(a2)\neffects.\n\nThe lattice fermion action is more complicated. A naive discretization of the continuum\nfermion action results in so-called doubling problem: the propagator possesses undesirable\nadditional poles called doublers corresponding to particles at the edges of the Brillouin zone.\nThere are other ways to circumvent the doubling problem. It provides a type of fermion actions.\nThe simplest idea is to add a second derivative term to eliminate doublers. This leads to the\nWilson fermion action constructed from the Wilson fermion operator DW which connects a site\nm to n such that\n\nDW (m,n) = δm,n − κ\n4∑\n\nµ=1\n\n[\n(1− γµ)Uµ(m)δm+µ̂,n + (1 + γµ)U †µ(m− µ̂)δm−µ̂,n\n\n]\n. (5)\n\nγµ is a 4× 4 matrix acting on the spinor space, and κ is a parameter related to the quark mass\nmq through κ = 1/(2mq + 8). It should be noticed that the second derivative term violates\nthe chiral symmetry on the lattice, though it vanishes in the continuum limit. It has recently\nbecome popular to adopt other fermion operators which respects the chiral symmetry on the\nlattice at the high numerical cost, such as the domain-wall and overlap operators.\n\nACAT2013 IOP Publishing\nJournal of Physics: Conference Series 523 (2014) 012046 doi:10.1088/1742-6596/523/1/012046\n\n3\n\n\n\n2.3. Generating gauge configurations\nThe simplest hybrid Monte Carlo (HMC) algorithm is explained. HMC is a standard algorithm\nto generate gauge field configuration from U (i) to U (i+1).\n\nSince detD is hard to be evaluated, we represent it as an integration of boson fields, called the\npseudo-fermion fields. For two flavors of degenerate quark masses, det(D†D) is to be evaluated\nas\n\nZ =\n\n∫\nDU det(D†D)e−SG =\n\n∫\nDUDφ†Dφe−SG−φ†(D†D)−1φ (6)\n\nWe need to integrate the link variable U and pseudo-fermion φ, whose action contains an inverse\nof the fermion operators, (D†D)−1.\n\nThe new configuration is brought by evolutions of U and its conjugate momentum P . The\nHamiltonian of P and U governs the evolution with respect to the fictitious time that is\nindependent from the original temporal coordinate. During the evolution of P and U , φ is\nkept fixed as external field. P and φ are given randomly at the beginning of the evolution\nequation. In solving the evolution equations numerically, a finite step size provides a systematic\nerror. This error can be compensated by a Metropolis test of the difference of Hamiltonian at\nthe beginning and the end of the evolution.\n\nTo summarize, the gauge configuration is generated in the following steps.\n\n• Generate the pseudo-fermion φ under the probability P ∝ exp[−φ†(D†D)−1φ], this can\nbe realized by Gaussian random number ξ, under the probability P = exp(−ξ†ξ), and\nφ = D−1ξ.\n\n• Set the conjugate momentum P with Gaussian probability distribution.\n\n• Solve the evolution equation for U and P with respect to the time step. To evaluate the\nforce of P , one needs to solve a linear equation for a large sparse matrix, x = (D†D)−1φ.\n\n• After getting a candidate of new configuration {P ′, U ′}, calculate the deference of\nHamiltonian ∆H = H[U ′, P ′, φ] −H[U,P, φ] and accept U ′ as new configuration with the\nprobability max{1, e−∆H}. If rejected, the old configuration is adopted as the new one.\n\nUsing the gauge configurations, expectation values of observables can be computed according to\nEq. (3).\n\n3. Lattice QCD code set Bridge++\n3.1. Bridge++ project\nBridge++ project starts on the following background. There are many lattice QCD codes, such\nas MILC code (in C, USA) [4], CPS++ (C++, USA) [5], Chroma (C++, Europe) [6], and\nLattice QCD Toolkit (Fortran, Japan) [7]. We have realized, however, it is problematic that\nthere has been no genuine code set other than Lattice QCD Toolkit in Japan, although Japan\nis one of the centers of lattice QCD simulations. It is desirable to possess a code set under\nour management, because it can instantly reflect a feedback of modifications, machine specific\ntunings, and other improvements. Another reason is that it has no black box. We can confirm\nand understand the code genuinely. In addition, a new code does not have a phantasm from the\ndevelopment history, which sometimes discourages a beginner.\n\nFor these reasons, we have started a development of our new code set, named Bridge++, in\n2009. We aimed that this general-purpose code set should have the followings features:\n\n• Readability: the code structure is transparent so as to be understandable even for beginners.\n\n• Extensibility: the code is easy to be modified for testing new ideas.\n\n• Portability: the code runs not only on laptop PC but also on supercomputers.\n\n• High-performance: the code has a high performance enough for productive research.\n\nACAT2013 IOP Publishing\nJournal of Physics: Conference Series 523 (2014) 012046 doi:10.1088/1742-6596/523/1/012046\n\n4\n\n\n\nTo achieve these goals, we adopt an object oriented design in C++ programming language. MPI\nis used for distributed memory machines. In Bridge++ code set, we use several simple idioms\nrepeatedly. Such repetition helps beginners to understand the idioms and the code structure.\nThe code is managed with a repository and bug tracking system.\n\nIn July 2012, the first version of Bridge++ was released. It contains fundamental lattice\nactions, linear algebraic algorithms, and various physical quantities. The documents are\ncompiled on wiki as well as TeX based reports. We also provide a HTML document generated\nby doxygen [9] from the comments in the source file.\n\n3.2. Object-oriented programing\nObject-oriented programing (OOP) is employed in Bridge++. It is based on “Objects”, which\nare sets of data fields and methods. A problem is solved using an interaction between objects.\nData field is a characteristic variable of the object, called a member data or a member variable.\nMethod is a function that defines a behavior of the object.\n\nOOP is characterized by the following properties.\n\n• Encapsulation: data is handled through the interface, that defines how to use the object.\n\n• Inheritance: object is expandable by adding new functions.\n\n• Polymorphism: objects with the same kind of behavior can be handled through the same\ninterface.\n\nThese properties are mechanisms to maximize reusability of the code. Using OOP, an interface\nis separated from details of implementation, and localize the latter which is sometimes specific\nto the architecture due to optimization. The polymorphism allows us to implement an algorithm\nwith a set of objects that have the common interface.\n\nThere is a compilation of wisdom to make use of these virtues of OOP. An example is so-\ncalled design patterns [10, 11]. The design pattern is a kind of programing idiom that frequently\nappears as a good solution to certain kind of problems. Their efficiency has been realized after\nthe GoF’s publication [10] that classified such idioms into 23 design patterns. The design pattern\nenables us to use the benefit of OOP easily, as well as to make our code transparent.\n\nIn the following, we introduce two design patterns and show how they are applied to our\nproblems.\n\n3.3. Bridge pattern and linear equation solver\nBridge pattern is a design pattern which decouples an interface from its implementation so that\nthey can switch independently. Bridge++ employs this bridge pattern. It has advantages for\nreadability and further developments.\n\nA typical example of the bridge pattern is an implementation of a solver algorithm and a\nfermion operator. In lattice QCD simulation, the most time consuming part is solving a linear\nequation for the fermion operator. An example is the Wilson fermion operator in Eq. 5, which\ncontains couplings up to the nearest neighbors, and thus is a large sparse complex matrix with\na rank of 3 × 4 × Nx × Ny × Nz × Nt. The propagator is the inverse of the quark operator,\nSq(m,n) = D−1(m,n). It is obtained by solving a linear equation∑\n\nl\n\nD(m, l)Sq(l, n) = δm,n, (7)\n\nwhere color and spin indexes are omitted. There are numbers of fermion operators and solver\nalgorithms. The best solver algorithm depends on a type of the fermion operator. The solver\nalgorithm should be implemented so as to be applied to any kind of fermion operators, and\nsimultaneously solver algorithms must be changed easily.\n\nACAT2013 IOP Publishing\nJournal of Physics: Conference Series 523 (2014) 012046 doi:10.1088/1742-6596/523/1/012046\n\n5\n\n\n\nFigure 2. Class diagram of the Bridge\npattern applied to a linear solvers.\n\nFigure 3. Class diagram of the Composite\npattern applied to the HMC integrators.\n\nFigure 2 shows the class diagram of the Bridge pattern based on UML (Unified Modeling\nLanguage). The interface of the fermion operator is Fopr. mult() is a function that applies\nthe fermion operator to a given vector and returns the resultant vector. The practical\nimplementation of mult() is given in a subclass of Fopr, such as Fopr Wilson for the Wilson\nfermion and Fopr Overlap for the overlap fermion. Similarly the base class Solver defines the\ninterface of the solver algorithms. solve() function returns the solution of a linear equation\nfor a given source vector. Again the practical implementation is given in subclasses, Solver CG\n\nand Solver BiCGStab. Bridge++ users can select any combination such as Fopr Wilson with\nSolver CG.\n\n3.4. Composite pattern and HMC integrator\nComposite pattern is a design pattern that treats a group of objects as a single instance of an\nobject. It is convenient to implement a tree structure or nested objects. In Bridge++, it is used\nin the HMC algorithm.\n\nHMC evolves {P,U} according to the Hamiltonian equation discretized by a leapfrog\nintegrator, (\n\nP (τ)\nU(τ)\n\n)\n= V (τ)\n\n(\nP (0)\nU(0)\n\n)\n, V (τ) =\n\n[\nVP\n\n(\n∆τ\n\n2\n\n)\nVU (∆τ)VP\n\n(\n∆τ\n\n2\n\n)]N\n, (8)\n\nwhere N is the total number of the leapfrog steps and ∆τ = τ/N . V describes the evolution,\n\nVP (∆τ) · P (τ) = P (τ) + ∆τF (τ), VU (∆τ) · U(τ) = exp(iP∆τ)U(τ). (9)\n\nF (τ) = δSlat/δU is the force of the conjugate momentum.\nInstead of the simple integrator, multi-time step integrator can be adopted. It varies the\n\nnumber of time steps for the action terms, according to the contribution to the force. The\nintegrator is\n\nV =\n\n[\nV\n\n(F )\nP\n\n(\n∆τ\n\n2\n\n)\nV1(∆τ)V\n\n(F )\nP\n\n(\n∆τ\n\n2\n\n)]N\n, V1 =\n\n[\nV\n\n(G)\nP\n\n(\n∆τ\n\n2N1\n\n)\nVU (∆τ/N1)V\n\n(G)\nP\n\n(\n∆τ\n\n2N1\n\n)]N1\n\n, (10)\n\nwhere V\n(G)\nP and V\n\n(F )\nP are the integrator of P with the gauge and fermion forces, respectively.\n\nAdjusting N1 by monitoring the size of the forces enables us to reduce the step size error,\nkeeping the total numerical cost fixed. In addition, the leapfrog integrator can be replaced\nwith an improved integrator, such as Omelyan integrator, which reduces the finite step size\ncorrections.\n\nFigure 3 shows the class diagram of the Composite pattern applied to the integrator. The\nbase class of the integrator is defined as Integrator, which has a virtual method evolve().\nIts subclass Update U plays the role of VU . The evolution operators V and V1 are implemented\nby Leapfrog or Omelyan. An object of these classes may have other objects of subclasses of\nIntegrator. This relation is represented in Figure 3 as lines with diamond. In analogy to the\nfile system, V play a role of the directory, and VP and VU are files in the directory.\n\nACAT2013 IOP Publishing\nJournal of Physics: Conference Series 523 (2014) 012046 doi:10.1088/1742-6596/523/1/012046\n\n6\n\n\n\nTable 1. Features and our development status of the fermion actions. In the last column,\n‘public’ means that is included in the public version and ‘testing’ has already been implemented\nand being confirmed.\n\nAction chiral symmetry numerical cost Status\n\nWilson violated middle public\nClover violated middle public\nTwisted mass violated middle testing\nStaggered partially remains low testing\nDomain-wall almost exact high testing\nOverlap exact very high testing\n\n3.5. Parallelization\nFor a current computational environment, parallelization is inevitable. Large-scale computers\nconsist of nodes which have own memory devices and communicate with each other via high\nspeed network. In addition, each node has many processor cores sharing memory.\n\nBridge++ works on one node as well as parallel nodes. In Bridge++, we adopt MPI (Message\nPassing Interface) for a distributed memory parallelization, i.e. for parallel nodes. An inter-\nnode communication, such as broadcast and one-to-one data transfer, is performed through the\nfunctions of the Communicator class, which wraps API functions of MPI. If a machine-specific\nefficient library is available, the implementation of Communicator class is replaced with the\none that uses the library. For an environment without MPI, so-called stub implementation of\nCommunicator class is provided.\n\nIt is noted that a shared memory parallelization of Bridge++ is now under development.\nHybrid parallelization with MPI and multi-threading has been considered. Two multi-thread\nlibraries, Pthread and OpenMP, are compared based on a working implementation of the Wilson\nfermion operator. While Pthread can accomplish better performance, it is not easy to apply it\nto the entire code set. OpenMP has been selected as a primary method to multi-threading and\nperforming performance tuning toward incorporating in the next release.\n\n3.6. I/O format\nIn Bridge++, simulation parameters are given by ASCII files in YAML format. The parameters\nare held in Parameter object. It passes the parameters to each object. Extraction of values\nfrom files is handled by a parameter manager class. For the I/O of field objects, several formats\nare available including ILDG standard format for the gauge configuration. ILDG (International\nLattice Data Grid) is an activity to promote sharing configuration data and standardizing data\nformat and description of metadata [8]. The standard output is classified with a verbose level.\nAt present 4 levels are prepared and the output level is selected for each object, such as a linear\nsolver or an observable. There is a extra mode to generate a message with pragma for ILDG\nmetadata generation.\n\n4. Current status and future prospects\nThe current public version contains major algorithms and observables, though the fermion\nactions are limited to Wilson and clover fermions. Other fermion actions have been implemented\nand now being tested. Our status is summarized in Table 1, together with the features of each\naction. These fermion action can be improved by smearing link variable. The smearing has been\nimplemented in Bridge++.\n\nACAT2013 IOP Publishing\nJournal of Physics: Conference Series 523 (2014) 012046 doi:10.1088/1742-6596/523/1/012046\n\n7\n\n\n\nFor linear algorithms, major iterative algorithms, such as CG, BiCGStab, GMRES, are\navailable. Multi-shift solver with CG algorithm and eigensolver with Implicitly restarted Lanczos\nalgorithm are also ready.\n\nThe status of performance is as follows. As a typical example, we quote the sustained\nperformance for the HMC with clover fermions. On Hitachi SR16000, the rate to the peak\nperformance is about 5% on 1 node (32 cores). On Blue Gene/Q, the public version runs with 2-\n3% on 32-nodes, while the latest code under development has improved the most time consuming\nsolver part to more than 10%.\n\nToward the next public release, we are now incorporating the multi-threading to Bridge++.\nPerformance tuning is now in progress. Active investigation to make use of GPGPU is also\nunderway.\n\nAcknowledgments\nIn addition to the authors of this paper, this project has been contributed by many of\nour colleagues, as listed in a document enclosed in the source code. This project is\nsupported by H20 Grant-in-Aid for Scientific Research on Innovative Areas ‘Research on the\nEmergence of Hierarchical Structure of Matter by Bridging Particle, Nuclear and Astrophysics\nin Computational Science’, Joint Institute for Computational Fundamental Science and HPCI\nStrategic Program Field 5 ‘The origin of matter and the universe’. The code was developed\nand tested on Hitachi SR16000 and IBM System Blue Gene/Q at KEK under a support of its\nLarge-scale Simulation Program (No.12/13-15), Hitachi SR16000 at YITP in Kyoto University,\nK-computer at RIKEN Advanced Institute for Computational Science, HA-PACS at University\nof Tsukuba under a support for its Interdisciplinary Computational Science Program (No.13a-\n25) and FX10 at University of Tokyo. This work is supported in part by the Grand-in-Aid for\nScientific Research of the Japan (Nos.20105005, 24540250, 25400284).\n\nReferences\n[1] There are many textbooks of lattice QCD simulations, for example, Montvay I and Münster G 1994 Quantum\n\nFields on a Lattice (Cambridge: Cambridge Univ. Press); DeGrand T and DeTar C 2006 Lattice Methods\nfor Quantum Chromodynamics (Singapore: World Scientific Pub.).\n\n[2] Bridge++ website, http://bridge.kek.jp/Lattice-code/.\n[3] Wilson K G 1974 Confinement of Quarks Phys. Rev. D 10 2445.\n[4] http://www.physics.utah.edu/~detar/milc/\n\n[5] http://qcdoc.phys.columbia.edu/cps.html\n\n[6] http://usqcd.jlab.org/usqcd-docs/chroma/\n\n[7] http://nio-mon.riise.hiroshima-u.ac.jp/~LTK/LTKf90.html\n\n[8] http://www.usqcd.org/ildg/\n\n[9] http://www.stack.nl/~dimitri/doxygen/index.html\n\n[10] Gamma E, Helm R, Johnson R, and Vlissides J 1995 Design Patterns: Elements of Reusable Object-Oriented\nSoftware. (Boston: Addison-Wesley).\n\n[11] Trott A and Shalloway J R 2004 Design Patterns Explained: A New Perspective on Object-Oriented Design,\n2nd ed. (Boston: Addison-Wesley Professional)\n\nACAT2013 IOP Publishing\nJournal of Physics: Conference Series 523 (2014) 012046 doi:10.1088/1742-6596/523/1/012046\n\n8"},"fulltext":true,"key":"eae9ffdb3bc6ad098e068d9975ef37ee","url":"https://inspirehep.net/files/eae9ffdb3bc6ad098e068d9975ef37ee"}],"publication_info":[{"journal_volume":"523","artid":"012046","conference_record":{"$ref":"https://inspirehep.net/api/conferences/1123949"},"year":2014,"journal_record":{"$ref":"https://inspirehep.net/api/journals/1212279"},"journal_title":"J.Phys.Conf.Ser.","parent_record":{"$ref":"https://inspirehep.net/api/literature/1299844"},"cnum":"C13-05-16"}],"citation_count_without_self_citations":29,"authors":[{"raw_affiliations":[{"value":"Theory Center - IPNS - High Energy Accelerator Research Organization (KEK) - Tsukuba 305-0810 - Japan"}],"full_name_unicode_normalized":"ueda, s.","full_name":"Ueda, S.","record":{"$ref":"https://inspirehep.net/api/authors/1959944"},"last_name":"Ueda","ids":[{"schema":"INSPIRE BAI","value":"S.Ueda.2"}],"affiliations":[{"record":{"$ref":"https://inspirehep.net/api/institutions/902916"},"value":"KEK, Tsukuba"}],"signature_block":"UADs","uuid":"639e7e4b-6538-4aaa-9891-296740bf6683","first_name":"S.","recid":1959944},{"raw_affiliations":[{"value":"Yukawa Institute for Theoretical Physics - 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Under the line-graph transformation, event-graph nodes correspond to observable isospins whose interactions through bidirectional edges provide a natural substrate for learning from event datasets, while the quantum states associated with the underlying participant nodes remain latent and inaccessible to direct observation. Within this framework we formulate a compact U(1) lattice gauge theory (LGT) on the event graph that leads to a Kogut-Susskind Hamiltonian (KSH) in the form of an XY-type spin model governing the dynamics of sparse anomalous-event isospins immersed in a bath of many nominal events. The proposed framework establishes a mathematical foundation for quantum-inspired graph-based anomaly detection and provides a principled bridge between graph learning, LGT, and quantum information.","abstract_source_suggest":{"input":"arXiv"}}],"primary_arxiv_category":["quant-ph"],"titles":[{"source":"arXiv","title":"A quantum framework for event graphs"}],"facet_author_name":["2585851_Robert P. Erickson"],"core":true,"license":[{"license":"arXiv nonexclusive-distrib 1.0","material":"preprint","url":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/"}],"_oai":{"sets":["Literature"],"id":"oai:inspirehep.net:3188207","updated":"2026-08-08T02:37:38.710786"},"curated":false,"arxiv_eprints":[{"categories":["quant-ph"],"value":"2608.06058"}]}}],"total":7190},"links":{"self":"https://inspirehep.net/api/literature/?q=refersto%20recid%2089145&size=10&page=1","next":"https://inspirehep.net/api/literature/?q=refersto%20recid%2089145&size=10&page=2","bibtex":"https://inspirehep.net/api/literature/?q=refersto%20recid%2089145&size=10&page=1&format=bibtex","latex-eu":"https://inspirehep.net/api/literature/?q=refersto%20recid%2089145&size=10&page=1&format=latex-eu","latex-us":"https://inspirehep.net/api/literature/?q=refersto%20recid%2089145&size=10&page=1&format=latex-us","json":"https://inspirehep.net/api/literature/?q=refersto%20recid%2089145&size=10&page=1&format=json","json-expanded":"https://inspirehep.net/api/literature/?q=refersto%20recid%2089145&size=10&page=1&format=json-expanded","cv":"https://inspirehep.net/api/literature/?q=refersto%20recid%2089145&size=10&page=1&format=cv"}}