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Lang."],"published-print":{"date-parts":[[2024,8,15]]},"abstract":"<jats:p>\n                    We construct a computationally universal quantum programming language\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                        <mml:mtext>Quantum<\/mml:mtext>\n                        <mml:mtext>\u03a0<\/mml:mtext>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    from two copies of\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                        <mml:mtext>\u03a0<\/mml:mtext>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    , the internal language of rig groupoids. The first step constructs a pure (measurement-free) term language by interpreting each copy of\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                        <mml:mtext>\u03a0<\/mml:mtext>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    in a generalisation of the category\n                    <jats:bold>Unitary<\/jats:bold>\n                    in which every morphism is \u201crotated\u201d by a particular angle, and the two copies are amalgamated using a free categorical construction expressed as a computational effect. The amalgamated language only exhibits quantum behaviour for specific values of the rotation angles, a property which is enforced by imposing a small number of equations on the resulting category. 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