With a Few Square Roots, Quantum Computing is as Easy as Pi
summary
Extending rig groupoids with two distinguished maps and three coherence equations yields a computationally universal model of unitary quantum computing: one map is an 8th root of unity, the other a square root of the symmetry on 1+1, constrained by a nondegeneracy axiom tied to the Euler decomposition of Hadamard. The resulting language √Π represents Clifford, Clifford+T (≤2 qubits) and Gaussian Clifford+T circuits and gives equational soundness and completeness results, with a mechanized Agda formalization.
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chapters (10)
- 0:00Introduction
- 1:28Algebraic closure and quantum computing
- 3:13Classical vs quantum
- 4:36Square roots for universality
- 6:00Extending Pi with square roots
- 7:29Square of Pi models
- 8:32Quantum gates in Square of Pi
- 9:33Equational theories for circuits
- 11:43Summary of results
- 13:36Question and answer session
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