Quantum information effects

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.

[1]  Thorsten Altenkirch,et al.  Structuring quantum effects: superoperators as arrows , 2006, Math. Struct. Comput. Sci..

[2]  G. M. Kelly.  Coherence theorems for lax algebras and for distributive laws , 1974 .

[3]  Abraham Westerbaan.  Quantum Programs as Kleisli Maps , 2016, QPL.

[4]  Maximilian Baader,et al.  Silq: a high-level quantum language with safe uncomputation and intuitive semantics , 2020, PLDI.

[5]  Philip Wadler.  XQuery: A Typed Functional Language for Querying XML , 2002, Advanced Functional Programming.

[6]  Roshan P. James.  Theseus : A High Level Language for Reversible Computing , 2014 .

[7]  Benoît Valiron,et al.  Quipper: a scalable quantum programming language , 2013, PLDI.

[8]  Jacques Carette,et al.  Computing with Semirings and Weak Rig Groupoids , 2016, ESOP.

[9]  Jacques Carette,et al.  Fractional Types , 2020, RC.

[10]  Tom Leinster,et al.  Basic Category Theory , 2014, 1612.09375.

[11]  Nicholas Gauguin Houghton-Larsen.  A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing , 2021, ArXiv.

[12]  Neil Ghani,et al.  Categorical Foundations of Gradient-Based Learning , 2021, ArXiv.

[13]  Peter Selinger,et al.  Towards a quantum programming language , 2004, Mathematical Structures in Computer Science.

[14]  Juliana Kaizer Vizzotto,et al.  The Arrow Calculus as a Quantum Programming Language , 2009, WoLLIC.

[15]  David I. Spivak,et al.  Backprop as Functor: A compositional perspective on supervised learning , 2017, 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[16]  Thorsten Altenkirch,et al.  The Quantum IO Monad , 2006 .

[17]  Rolf Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[18]  C. Heunen,et al.  Categories for Quantum Theory , 2019 .

[19]  Sam Staton,et al.  Universal Properties in Quantum Theory , 2019, QPL.

[20]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[21]  Ross Paterson,et al.  A new notation for arrows , 2001, ICFP '01.

[22]  I. Chuang,et al.  Quantum Computation and Quantum Information: Introduction to the Tenth Anniversary Edition , 2010 .

[23]  John Hughes.  Programming with Arrows , 2004, Advanced Functional Programming.

[24]  Edmund Robinson,et al.  Premonoidal categories and notions of computation , 1997, Mathematical Structures in Computer Science.

[25]  Juliana Kaizer Vizzotto,et al.  From Symmetric Pattern-Matching to Quantum Control , 2018, FoSSaCS.

[26]  Bart Jacobs,et al.  Under Consideration for Publication in J. Functional Programming Categorical Semantics for Arrows , 2022 .

[27]  Jennifer Paykin,et al.  QWIRE: a core language for quantum circuits , 2017, POPL.

[28]  Jacques Carette,et al.  Fractional Types , 2020, RC.

[29]  Robin Kaarsgaard,et al.  Bennett and Stinespring, Together at Last , 2021, QPL.

[30]  Kenta Cho,et al.  Von Neumann Algebras form a Model for the Quantum Lambda Calculus , 2016, ArXiv.

[31]  Noson S. Yanofsky,et al.  Quantum Computing for Computer Scientists , 2008 .

[32]  Claudio Hermida,et al.  Monoidal indeterminates and categories of possible worlds , 2012, Theor. Comput. Sci..

[33]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[34]  Robin Kaarsgaard,et al.  En Garde! Unguarded Iteration for Reversible Computation in the Delay Monad , 2019, MPC.

[35]  Amr Sabry,et al.  A computational interpretation of compact closed categories: reversible programming with negative and fractional types , 2021, Proc. ACM Program. Lang..

[36]  Tommaso Toffoli,et al.  Reversible Computing , 1980, ICALP.

[37]  Thorsten Altenkirch,et al.  From Reversible to Irreversible Computations , 2008, Electron. Notes Theor. Comput. Sci..

[38]  Sam Staton,et al.  Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory , 2017, Log. Methods Comput. Sci..

[39]  Sam Staton,et al.  Quantum channels as a categorical completion , 2019, 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[40]  Amr Sabry,et al.  Information effects , 2012, POPL '12.

[41]  Peter Sestoft,et al.  Partial evaluation and automatic program generation , 1993, Prentice Hall international series in computer science.

[42]  M. Laplaza,et al.  Coherence for distributivity , 1972 .

[43]  Norman Ramsey,et al.  Advanced Functional Programming , 2005 .

[44]  John Hatcliff,et al.  An Introduction to Online and Offline Partial Evaluation using a Simple Flowchart Language , 1998, Partial Evaluation.

[45]  Simon Perdrix,et al.  Quantum Programming with Inductive Datatypes: Causality and Affine Type Theory , 2019, FoSSaCS.