{"title":"Logical paradoxes in quantum computation","authors":"Nadish de Silva","doi":"10.1145/3209108.3209123","DOIUrl":"https://doi.org/10.1145/3209108.3209123","url":null,"abstract":"The precise features of quantum theory enabling quantum computational power are unclear. Contextuality---the denial of a notion of classical physical reality---has emerged as a promising hypothesis: e.g. Howard et al. showed that the magic states needed to practically achieve quantum computation are contextual. Strong contextuality, as defined by Abramsky-Brandenburger, is an extremal form of contextuality describing systems that exhibit logically paradoxical behaviour. After introducing number-theoretic techniques for constructing exotic quantum paradoxes, we present large families of strongly contextual magic states that are computationally optimal in the sense that they enable universal quantum computation via deterministic injection of gates of the Clifford hierarchy. We thereby bolster a refinement of the resource theory of contextuality that emphasises the computational power of logical paradoxes.","PeriodicalId":389131,"journal":{"name":"Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115724121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics","authors":"E. Jeandel, S. Perdrix, R. Vilmart","doi":"10.1145/3209108.3209131","DOIUrl":"https://doi.org/10.1145/3209108.3209131","url":null,"abstract":"We introduce the first complete and approximately universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics -- also called the π/4-fragment of the ZX-Calculus -- was one of the main open questions in categorical quantum mechanics. We prove the completeness of this fragment using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that the π/4-fragment of the ZX-Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.","PeriodicalId":389131,"journal":{"name":"Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130035083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the number of types in sparse graphs","authors":"Michal Pilipczuk, S. Siebertz, Szymon Toruńczyk","doi":"10.1145/3209108.3209178","DOIUrl":"https://doi.org/10.1145/3209108.3209178","url":null,"abstract":"We prove that for every class of graphs ℒ which is nowhere dense, as defined by Nešetřil and Ossona de Mendez [28, 29], and for every first order formula φ(x, y), whenever one draws a graph G ∈ ℒ and a subset of its nodes A, the number of subsets of A|y| which are of the form {u ∈ A|y|: G |= φ(ū, v)} for some valuation ū of x in G is bounded by O(|A||x|ε), for every ε > 0. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler [1] stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices).","PeriodicalId":389131,"journal":{"name":"Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"43 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120900158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Neil Ghani, Jules Hedges, Viktor Winschel, Philipp Zahn
{"title":"Compositional Game Theory","authors":"Neil Ghani, Jules Hedges, Viktor Winschel, Philipp Zahn","doi":"10.1145/3209108.3209165","DOIUrl":"https://doi.org/10.1145/3209108.3209165","url":null,"abstract":"We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which standard economic tools are not practical. An open game represents a game played relative to an arbitrary environment and to this end we introduce the concept of coutility, which is the utility generated by an open game and returned to its environment. Open games are the morphisms of a symmetric monoidal category and can therefore be composed by categorical composition into sequential move games and by monoidal products into simultaneous move games. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. We show that a variety of games can be faithfully represented as open games in the sense of having the same Nash equilibria and off-equilibrium best responses.","PeriodicalId":389131,"journal":{"name":"Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125660441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}