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projects:quantum:category-qc-foundation [2025/04/16 15:37] – kymkiprojects:quantum:category-qc-foundation [2025/04/30 09:26] (current) – [Emmanuel Jeandel, Simon Perdrix & Renaud Vilmart (2020)] kymki
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 +      <div class="tooltip">This post is very close to being completely finished. Not that rambly at all.</div>
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 +
 ===== Categorical Quantum Computing Series: Foundational Papers ===== ===== Categorical Quantum Computing Series: Foundational Papers =====
 +
 +==== Introduction ====
  
 Below is a series of papers that have been setting the direction of category theory applied to quantum computing. The overview provides an easy reading that can give you (the reader) an idea of why you would want to read any of the chosen publications. The papers were selected in terms of general "impact" (not really clearly defined measure, but how formative the texts were for the field). If you have papers that you would like to see in this list, please do contact me. Below is a series of papers that have been setting the direction of category theory applied to quantum computing. The overview provides an easy reading that can give you (the reader) an idea of why you would want to read any of the chosen publications. The papers were selected in terms of general "impact" (not really clearly defined measure, but how formative the texts were for the field). If you have papers that you would like to see in this list, please do contact me.
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  [[https://lmcs.episciences.org/6532/pdf|Completeness of the ZX-Calculus]]  [[https://lmcs.episciences.org/6532/pdf|Completeness of the ZX-Calculus]]
  
-This recent paper resolves a long-standing open question by proving the ZX-calculus is complete for all pure qubit quantum mechanics. The ZX-calculus (introduced by Coecke & Duncan in 2011) was known to be universal but not fully complete (certain true quantum equations had no diagrammatic proof using the original rules). Jeandel et al. provided the first complete axiomatisation: they show that any equality of linear maps (matrices) on qubits can be derived using ZX-diagram rewrite rules. The proof proceeds by first enriching the ZX-calculus with additional rewrite rules to make it complete for the Clifford+T fragment (a universal gate set), and then leveraging a translation to a related graphical language (the ZW-calculus) to reach completeness for arbitrary quantum operations. The result implies that the diagrammatic calculus now fully matches the expressive power of matrix mathematics – if two quantum circuits are mathematically equivalent, the ZX-calculus can prove it. This accomplishment marks a maturation of the categorical approach: the diagrammatic methods are not only convenient and intuitive but also no less powerful than algebraic methods. It has spurred further research and practical tools in quantum circuit optimization, equipping the field with a robust graphical proof technique for quantum computing theory.+This recent paper resolves a long-standing open question by proving the ZX-calculus is complete for all pure qubit quantum mechanics. The ZX-calculus (introduced by Coecke & Duncan in 2011) was known to be universal but not fully complete (certain true quantum equations had no diagrammatic proof using the original rules). Jeandel et al. provided the first complete axiomatisation: they show that any equality of linear maps (matrices) on qubits can be derived using ZX-diagram rewrite rules. The proof proceeds by first enriching the ZX-calculus with additional rewrite rules to make it complete for the Clifford+T fragment (a universal gate set), and then leveraging a translation to a related graphical language (the ZW-calculus) to reach completeness for arbitrary quantum operations. The result implies that the diagrammatic calculus now fully matches the expressive power of matrix mathematics – if two quantum circuits are mathematically equivalent, the ZX-calculus can prove it. This accomplishment marks a maturation of the categorical approach: the diagrammatic methods are not only convenient and intuitive but also no less powerful than algebraic methods. It has generated further research and practical tools in quantum circuit optimization, now with graphical proof technique for quantum computing theory.