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International Technical Conference on Circuits/Systems, Computers and Communications (ITC-CSCC)

Modeling Shor's Algorithm with Colored Petri Net

Atsushi Ohta, Kohkichi Tsuji

Published Jul 5, 2022• 0 citations
Publisher

Abstract

Petri net is a mathematical tool for concurrent systems. This paper suggests colored Petri net to model measurement of part of registers in quantum algorithm. We construct a colored Petri net model of Shor's algorithm using CPN Tools.
DOI
10.1109/itc-cscc55581.2022.9895042

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The development of quantum software applications requires models capable to describe quantum features (superposition, entanglement, reversibility etc.) as well as the software algorithm requirements. The newly proposed Quantum Petri Nets (QPNs) have the structure of the classical Petri Nets (PNs), but their tokens represent superposed quantum states fulfilling the Hilbert space features. Instead of the integer numerical values of the PN marking, QPNs use quantum marking describing an extreme huge space. New conditions for the transition enabling and new mappings for the transition executions are conceived. A PN based language is used for the structure analysis. Replacing the classical tokens with the probability distribution of the quantum vectors, the QPN simulation is possible as well as the construction of a reachability graph.
Classical Petri nets provide a canonical model of concurrency, with unfolding semantics linking nets, occurrence nets, and event structures. No comparable framework exists for quantum concurrency: existing ''quantum Petri nets'' lack rigorous concurrent and sound quantum semantics, analysis tools, and unfolding theory. We introduce Quantum Petri Nets (QPNs), Petri nets equipped with a quantum valuation compatible with the quantum event structure semantics of Clairambault, De Visme, and Winskel (2019). Our contributions are: (i) a local definition of Quantum Occurrence Nets (LQONs) compatible with quantum event structures, (ii) a construction of QPNs with a well-defined unfolding semantics, (iii) a compositional framework for QPNs. This establishes a semantically well grounded model of quantum concurrency, bridging Petri net theory and quantum programming.
In this paper, we propose a novel quantum solution to address the problem of reachability in bounded quantum Petri nets (QPNs), an advanced modeling framework that combines the classical Petri net model with quantum mechanical principles. The proposed approach exploits quantum parallelism to construct a superposition over all reachable markings from the initial marking. Grover's amplitude amplification algorithm is then applied to efficiently identify a desired target marking, while ancillary q-tokens used solely for transition control are excluded from the search space, significantly reducing its size. Theoretical analysis shows that our approach achieves a quadratic speed up over classical exhaustive algorithms. Experimental results also confirm the correctness and feasibility of the proposed quantum algorithm for solving the bounded reachability problem in Quantum Petri Nets.
Classical Petri nets provide a canonical model of concurrency, with unfolding semantics linking nets, occurrence nets, and event structures. No comparable framework exists for quantum concurrency: existing ''quantum Petri nets'' lack rigorous concurrent and sound quantum semantics, analysis tools, and unfolding theory. We introduce Quantum Petri Nets (QPNs), Petri nets equipped with a quantum valuation compatible with the quantum event structure semantics of Clairambault, De Visme, and Winskel (2019). Our contributions are: (i) a local definition of Quantum Occurrence Nets (LQONs) compatible with quantum event structures, (ii) a construction of QPNs with a well-defined unfolding semantics, (iii) a compositional framework for QPNs. This establishes a semantically well grounded model of quantum concurrency, bridging Petri net theory and quantum programming.