A computational interpretation of compact closed categories: reversible programming with negative and fractional types

Chen, Chao-Hong; Sabry, Amr

doi:10.1145/3434290

Cited by 8 publications

(2 citation statements)

References 48 publications

(48 reference statements)

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“…Π is a reversible combinator language introduced in [Bowman et al 2011;James and Sabry 2012] to study strongly typed reversible classical programming. Many extensions exist, such as partiality and iteration [Bowman et al 2011;James and Sabry 2012], fractional types [Chen et al 2020;Chen and Sabry 2021], negative types [Chen and Sabry 2021], and higher combinators [Carette and Sabry 2016;Kaarsgaard and Veltri 2019]. This section introduces a quantum extension to Π, and shows it to be approximately universal for unitaries, the canonical model of pure quantum computation (without measurement).…”

Section: Three Generations Of Yuppiementioning

confidence: 99%

Quantum information effects

Heunen

Kaarsgaard

2022

Proc. ACM Program. Lang.

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We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate (noniterative) quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.

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Section: Three Generations Of Yuppiementioning

confidence: 99%

Quantum information effects

Heunen

Kaarsgaard

2022

Proc. ACM Program. Lang.

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“…Π is a reversible combinator language introduced in [4,19] to study strongly typed reversible classical programming. Many extensions exist, such as partiality and iteration [4,19], fractional types [6,7], negative types [7], and higher combinators [5,22]. This section introduces a quantum extension to Π, and shows it to be approximately universal for unitaries, the canonical model of pure quantum computation (without measurement).…”

Section: Three Generations Of Yuppiementioning

confidence: 99%

Quantum Information Effects

Heunen,

Kaarsgaard

2021

Preprint

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We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including measurement. We provide universal categorical constructions that semantically interpret this arrow metalanguage with choice, starting with any rig groupoid interpreting the reversible base language. Several properties of quantum measurement follow in general, and we translate quantum flow charts into our language. The semantic constructions turn the category of unitaries between Hilbert spaces into the category of completely positive trace-preserving maps, and they turn the category of bijections between finite sets into the category of functions with chosen garbage. Thus they capture the fundamental theorems of classical and quantum reversible computing of Toffoli and Stinespring.

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Reconciling Partial and Local Invertibility

Ågren Thuné,

Matsuda,

Wang

2024

Lecture Notes in Computer Science

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Invertible programming languages specify transformations to be run in two directions, such as compression/decompression or encryption/decryption. Two key concepts in invertible programming languages are partial invertibility and local invertibility. Partial invertibility lets invertible code be parameterized by the results of non-invertible code, whereas local invertibility requires all code to be invertible. The former allows for more flexible programming, while the latter has connections to domains such as low-energy computing and quantum computing. We find that existing approaches lack a satisfying treatment of partial invertibility, leaving the connection to local invertibility unclear.In this paper, we identify four core constructs for partially invertible programming, and show how to give them a locally invertible interpretation. We show the expressiveness of the constructs by designing the functional invertible language Kalpis, and show how to give them a locally invertible semantics using the novel arrow combinator language $$\textsc {rrArr}$$ R R A R R —the key idea is viewing partial invertibility as an invertible effect. By formalizing the two systems and giving Kalpis semantics by translation to $$\textsc {rrArr}$$ R R A R R , we reconcile partial and local invertibility, solving an open problem in the field. All formal developments are mechanized in Agda.

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A computational interpretation of compact closed categories: reversible programming with negative and fractional types

Cited by 8 publications

References 48 publications

Quantum information effects

Quantum information effects

Quantum Information Effects

Reconciling Partial and Local Invertibility

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