Biunitary constructions in quantum information

Abstract

We present an infinite number of construction schemes involving unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on biunitary connections, algebraic objects which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method.

48 pages, Mathematica notebook attached; final version

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• From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics
• Operator dynamics and entanglement in space-time dual Hadamard lattices
• Exactly solvable many-body dynamics from space-time duality
• A covariant Stinespring theorem
• Tensor network decompositions for absolutely maximally entangled states
• Construction of perfect tensors using biunimodular vectors
• Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
• Covariant quantum combinatorics with applications to zero-error communication
• Shaded tangles for the design and verification of quantum circuits
• Geometric constructions of generalized dual-unitary circuits from biunitarity
• Zigzag normalisation for associative $n$-categories
• Planar algebras, quantum information theory and subfactors
• A classical groupoid model for quantum networks
• Entanglement-invertible channels
• A Categorical Model for Classical and Quantum Block Designs