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On Circuit Description Languages, Indexed Monads, and Resource Analysis (Video, POPL 2026) Ken Sakayori, Andrea Colledan, Ugo Dal Lago (University of Tokyo, Japan; University of Bologna, Italy / Centre Inria d’Université Côte d’Azur, France; University of Bologna, Italy / Centre Inria d’Université Côte d’Azur, France) Abstract: In this paper, a monad-based denotational model is introduced and shown adequate for the Proto-Quipper family of calculi, themselves being idealized versions of the Quipper programming language. The use of a monadic approach allows us to separate the value to which a term reduces from the circuit that the term itself produces as a side effect. In turn, this enables the denotational interpretation and validation of rich type systems in which the size of the produced circuit can be controlled. Notably, the proposed semantic framework, through the novel concept of circuit algebra, suggests forms of effect typing guaranteeing quantitative properties about the resulting circuit, even in presence of optimizations. Article: https://doi.org/10.1145/3776666 ORCID: https://orcid.org/0000-0003-3238-9279, https://orcid.org/0000-0002-0049-0391, https://orcid.org/0000-0001-9200-070X Video Tags: Circuit Description Language, Quipper, doi:10.1145/3776666, orcid:0000-0003-3238-9279, orcid:0000-0002-0049-0391, orcid:0000-0001-9200-070X Presentation at the POPL 2026 conference, Jan 11-17, 2026, https://popl26.sigplan.org/ Sponsored by ACM SIGPLAN.