[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81844-en":3,"doc-seo-81844-105":31,"detail-sidebar-cat-0-en-105":93},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81844,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","2048-Spin Bulk Acoustic Wave Ising Machine for Number Partitioning and Sudoku","Optical coherent Ising machines face limitations including large physical footprint, high power consumption, poor thermal stability, and high cost. A time-multiplexed Ising machine is presented using propagating wave packets in solid-state delay lines at microwave frequencies, targeting a thermally stable, robust, low-power tabletop design. The system uses two serially connected 20.5 MHz, 707 µs bulk acoustic wave delay lines supporting 2,048 spins with all-to-all connectivity and 15-bit coupling resolution. It finds approximate MAX-CUT solutions in 341 ms and solves number partitioning and Sudoku, outperforming the simulated bifurcation algorithm on harder instances.","arXiv :2607 .02112v1 [ cond-mat .mes-hall ] 2 Jul 2026  \nA 2048-spin bulk acoustic wave Ising machine for number  \npartitioning and Sudoku  \nVenkatesh Vadde 1*, Roman Ovcharov 1 , Victor H. Gonz´alez 1 , Roman Khymyn 1 , Artem Litvinenko 1,2*, Johan ˚Akerman 1,3,4*  \n1 Department of Physics, University of Gothenburg, Gothenburg, Sweden.  \n2 Department of Physics, Oakland University, Rochester, Michigan, USA.  \n3 Center for Science and Innovation in Spintronics, Tohoku University, Sendai, Japan.  \n4 Research Institute of Electrical Communication, Tohoku University, Sendai, Japan.  \n*Corresponding author(s). E-mail(s): [venkatesh.vadde@physics.gu.se](venkatesh.vadde@physics.gu.se) ; [litvinenko@oakland.edu](litvinenko@oakland.edu) ; [johan.akerman@physics.gu.se](johan.akerman@physics.gu.se) ;  \nAbstract  \nOptical coherent Ising machines based on time-multiplexing have demonstrated significant progress in terms of connectivity and spin scalability. However, they are constrained by large physical footprints, high power consumption, poor thermal stability, and high cost. Here, we present a time-multiplexed Ising machine leveraging propagating wave packets in solid-state delay lines at microwave frequencies, enabling thermally stable, robust, low-power, tabletop, and affordable design. We use two serially connected 20.5 MHz, 707 µs bulk acoustic wave delay lines supporting 2,048 spins. Our design provides all-to-all connectivity with 15-bit coupling resolution and finds approximate MAX-CUT solutions in 341 ms, potentially scalable to sub-ms by using higher frequency delay lines. Additionally, we demonstrate solutions to number partitioning and Sudoku problems. Compared with state-of-the-art Coherent Ising machines, our machine exhibits four orders of magnitude higher thermal stability. Against the simulated bifurcation algorithm, our design achieves comparable results on the MAX-CUT problem, while outperforming it on the more complex number-partitioning and Sudoku problems.  \n1 Introduction  \nThe slowing of Moore’s law in terms of advancement in modern computers appears to have reached a plateau, promoting the exploration of physics-based unconventional computational architectures beyond traditional silicon-based technology. This shift is particularly important for addressing nondeterministic polynomial-time hard (NP-hard) and NP-complete problems, which represent classes of highly complex challenges, especially in combinatorial optimization, where solution times increase exponentially with problem size, posing significant difficulties for classical computing approaches. These optimization problems [1, 2] play a crucial role in a wide range of fields, including finance [3], circuit design [4], drug discovery [5], operations [6], and scheduling [7] . Combinatorial optimization problems are known to be mappable onto ground-state search problems of the Ising model using polynomial resources [1] .  \nThe Ising Hamiltonian is given by,  \nH = − XJij si sj −Xhisi (1)  \ni\u003Cj i  \nwhere si = ±1 corresponds to the ith Ising spin, Jij is the coupling term between spins si and sj , and hi is a local bias field. The Ising machines are engineered to find configurations minimizing this Hamiltonian.  \nTo address these computational demands of NP-hard problems, novel approaches have emerged that utilize the physical behavior of systems to perform efficient computation, such as analog Ising machines. A wide range of Ising machines have been developed using diverse physical platforms, including superconducting quantum bits [8], single-electron devices [9], nanomechanical systems [10], stochastic nanomagnets [11], CMOS circuits [12], spin waves [13–15], superparamagnetic tunnel junctions [16], delay-line oscillators [17], and photonic Ising machines [18, 19] . Several commercial products are also available on the market for solving combinatorial problems, including the D-Wave system, which uses superconducting technology, NTT Research’s optical Coherent I","cbCaio0o1ZJFuIum","https://ap.wps.com/l/cbCaio0o1ZJFuIum","pdf",22388380,5,1,13,"English","en",105,"# Abstract\n# Introduction\n## Motivation: NP-hard combinatorial optimization\n## Ising model mapping and Hamiltonian\n## Prior analog Ising machine platforms and limitations","[{\"question\":\"What problem classes does the proposed Ising machine target?\",\"answer\":\"It targets NP-hard combinatorial optimization problems that can be mapped onto Ising ground-state search, demonstrated with MAX-CUT, number partitioning, and Sudoku.\"},{\"question\":\"How is the Ising machine implemented in this work?\",\"answer\":\"It is a time-multiplexed Ising machine using propagating wave packets in solid-state bulk acoustic wave delay lines at microwave frequencies.\"},{\"question\":\"What hardware capabilities are reported for the system?\",\"answer\":\"The design uses two serially connected bulk acoustic wave delay lines (20.5 MHz, 707 µs) supporting 2,048 spins, providing all-to-all connectivity with 15-bit coupling resolution.\"}]","2048-Spin Bulk Acoustic Wave Ising Machine for Number Partitioning and Sudoku | 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problem classes does the proposed Ising machine target?","Question",{"text":77,"@type":78},"It targets NP-hard combinatorial optimization problems that can be mapped onto Ising ground-state search, demonstrated with MAX-CUT, number partitioning, and Sudoku.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How is the Ising machine implemented in this work?",{"text":82,"@type":78},"It is a time-multiplexed Ising machine using propagating wave packets in solid-state bulk acoustic wave delay lines at microwave frequencies.",{"name":84,"@type":75,"acceptedAnswer":85},"What hardware capabilities are reported for the system?",{"text":86,"@type":78},"The design uses two serially connected bulk acoustic wave delay lines (20.5 MHz, 707 µs) supporting 2,048 spins, providing all-to-all connectivity with 15-bit coupling 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