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The abelian Hidden Subgroup Problem (HSP) is extremely general, and manyproblems with known quantum exponential speed-up (such as integersfactorisation, the discrete logarithm and Simon's problem) can be seen asspecific instances of it. The traditional presentation of the quantum protocolfor the abelian HSP is low-level, and relies heavily on the the interplaybetween classical group theory and complex vector spaces. Instead, we give ahigh-level diagrammatic presentation which showcases the quantum structurestruly at play. Specifically, we provide the first fully diagrammatic proof ofcorrectness for the abelian HSP protocol, showing that strongly complementaryobservables are the key ingredient to its success. Being fully diagrammatic,our proof extends beyond the traditional case of finite-dimensional quantumtheory: for example, we can use it to show that Simon's problem can beefficiently solved in real quantum theory, and to obtain a protocol that solvesthe HSP for certain infinite abelian groups.

Fully graphical treatment of the quantum algorithm for the Hidden Subgroup Problem