[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83116-en":3,"doc-seo-83116-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83116,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Quantum Combinatorial Games: Can You Beat a Quantum Opponent?","A combinatorial game, such as tic-tac-toe, checkers, or chess, is extended to a quantum setting through a practical quantization framework designed for implementability on quantum computers. The work analyzes when a quantum opponent can outperform a classical one, revisiting and reformulating classical combinatorial game theory and applying Zermelo’s theorem in the quantum regime. Results show quantum mechanics yields no advantage against a classical player using a perfect non-losing strategy. In contrast, with realistic classical mistakes, the quantum opponent can amplify errors to increase winning chances, with broader applicability to finite deterministic adversarial models of perfect information.","arXiv :2607 .06550v 1 [ cs .GT] 7 Jul 2026  \nQuantum combinatorial games  \nCan you beat a quantum opponent?  \nDieks Scholten 1 , Bram Westerbaan2 , and Simona Samardjiska 1  \n1 Radboud University, Nijmegen, Netherlands  \n[dieks.scholten@ru.nl](dieks.scholten@ru.nl)  \n[simonas@cs.ru.nl](simonas@cs.ru.nl)  \n[2](2 bram@westerbaan.name)[ bram@westerbaan.name](2 bram@westerbaan.name)  \nAbstract. A combinatorial game is a deterministic game with no hidden information played between two opponents such as tic-tac-toe, checkers or chess. In this paper we extend combinatorial games to the quantum setting, by first revisiting and reformulating existing theory of classical combinatorial games.  \nWe investigate in which case a quantum opponent has an advantage over a classical one. Surprisingly, our instantiation of Zermelo’s classical theorem in the quantum setting shows that the effects of quantum mechanics do not convey an advantage against a classical player that plays a perfect classical strategy. Ina more realistic scenario, when the classical player makes mistakes, we show how the quantum opponent can amplify the mistake to increase their chance of winning. Our theory has application beyond themere playing of board games and can be used as a tool in finite deterministic adversarial models with perfect information.  \nKeywords: quantum games, combinatorial game theory, Zermelo’s theorem  \n1 Introduction  \nA common characteristic of a game of tic-tac-toe, checkers, chess, abalone or go is that they are played between two opponents, they are deterministic and don’t involve any hidden information. In the literature, they are known as combinatorial games. In this paper we ask ourselves what a match of a combinatorial game might look like on a quantum computer, and whether one can have strategies that guarantee a certain outcome like winning, or not losing (e.g. tic-tac-toe) . While the rules for any of the above mentioned classical, well-known combinatorial games are set in stone, different generalizations are possible for a quantum variant. Indeed, several quantum variants of tictac-toe have already been proposed for different purposes [Gof06,WW23,SDBP19,NN12,CFM+22], and also for chess [Can19] and go [Ran16,SPC+22,QGJ+20] . None of them, however, quite fit the following reasonable requirements:  \n1. The quantum variant should be actually practically playable on a quantum computer; e.g. measurement at the end.  \n2. The quantum variant should be an extension of the regular, classical game in the sense that a strategy for the regular game (a set of instructions on how to react to each move of the opponent) should be playable in the quantum variant.  \n3. The player in the quantum variant should not be arbitrarily restricted to, say, make superpositions of only two moves.  \nWe propose a general framework for quantization of combinatorial games that meets all these realistic requirements and applies to a broad class of combinatorial games with ties, including tic-tac-toe,  \nThis research has been supported by the Dutch government through the NWO Quantum Technology grant NGF.1623.23.020.  \n2 Dieks Scholten, Bram Westerbaan, and Simona Samardjiska  \nchess and hex. After having defined the exact rules of our quantum variant of a combinatorial game, we turn to the question of winning strategies. For example, it is well known that there is no winning strategy for tic-tac-toe, but there is a non-losing one. Our main result implies that, surprisingly, in our variant of quantum tic-tac-toe a ‘quantum player’, a player that takes full advantage of the moves permitted under this quantum variant, can still not win against a regular ‘classical player’playing the classical non-losing strategy. A similar result holds for quantum combinatorial games in general, see Section 3 .  \nOne might conclude from this that playing on a quantum computer (under our rules) is not really interesting as the classical best strategies still apply. This might well be the case","cbCaifoZbRRoaTHg","https://ap.wps.com/l/cbCaifoZbRRoaTHg","pdf",398426,1,19,"English","en",105,"# Introduction\n## Motivation and requirements for practical quantization\n## Related work in quantized combinatorial games\n# Quantum variant and winning strategies\n## Revisited classical theory and main results\n## Perfect classical strategies vs quantum play\n# Applications and further examples\n## Translation to deterministic adversarial models","[{\"question\":\"What problem does the paper study about quantum combinatorial games?\",\"answer\":\"It studies when a quantum opponent can gain an advantage over a classical opponent in games modeled as deterministic combinatorial games with perfect information.\"},{\"question\":\"Does quantum play provide an advantage against a perfectly playing classical opponent?\",\"answer\":\"No. Instantiating Zermelo’s classical theorem in the quantum setting shows that quantum mechanics does not grant an advantage against a classical player who follows a perfect non-losing strategy.\"},{\"question\":\"How can the quantum opponent win more often when the classical player makes mistakes?\",\"answer\":\"When the classical player makes errors, the paper shows the quantum opponent can amplify those mistakes, increasing the quantum player’s probability of winning.\"}]",1784185377,48,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"quantum-combinatorial-games-can-you-beat-a-quantum-opponent","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/quantum-combinatorial-games-can-you-beat-a-quantum-opponent/83116/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper study about quantum combinatorial games?","Question",{"text":75,"@type":76},"It studies when a quantum opponent can gain an advantage over a classical opponent in games modeled as deterministic combinatorial games with perfect information.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Does quantum play provide an advantage against a perfectly playing classical opponent?",{"text":80,"@type":76},"No. Instantiating Zermelo’s classical theorem in the quantum setting shows that quantum mechanics does not grant an advantage against a classical player who follows a perfect non-losing strategy.",{"name":82,"@type":73,"acceptedAnswer":83},"How can the quantum opponent win more often when the classical player makes mistakes?",{"text":84,"@type":76},"When the classical player makes errors, the paper shows the quantum opponent can amplify those mistakes, increasing the quantum player’s probability of 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