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Trace monoids with idempotent generators and measure-only quantum automata

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Abstract

In this paper, we analyze a model of 1-way quantum automaton where only measurements are allowed ( MON -1qfa). The automaton works on a compatibility alphabet \((\Sigma, E)\) of observables and its probabilistic behavior is a formal series on the free partially commutative monoid \(\hbox{FI}(\Sigma, E)\) with idempotent generators. We prove some properties of this class of formal series and we apply the results to analyze the class \({\bf LMO}(\Sigma, E)\) of languages recognized by MON -1qfa’s with isolated cut point. In particular, we prove that \({\bf LMO}(\Sigma, E)\) is a boolean algebra of recognizable languages with finite variation, and that \({\bf LMO}(\Sigma, E)\) is properly contained in the recognizable languages, with the exception of the trivial case of complete commutativity.

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Acknowledgements

The authors wish to thank the anonymous referees for useful comments and remarks. Partially supported by MURST, under the project “PRIN: Aspetti matematici e applicazioni emergenti degli automi e dei linguaggi formali: metodi probabilistici e combinatori in ambito di linguaggi formali”.

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  1. Dipartimento di Scienze dell’Informazione, Università degli Studi di Milano, via Comelico 39, 20135, Milano, Italy

    Alberto Bertoni, Carlo Mereghetti & Beatrice Palano

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  1. Alberto Bertoni
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  2. Carlo Mereghetti
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  3. Beatrice Palano
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Correspondence to Alberto Bertoni.

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Bertoni, A., Mereghetti, C. & Palano, B. Trace monoids with idempotent generators and measure-only quantum automata. Nat Comput 9, 383–395 (2010). https://doi.org/10.1007/s11047-009-9154-8

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