Representation Theory

IBM Q System One describes spin-1/2 via the 2-dimensional representation of SU(2). Pauli matrices control quantum gates in the 127-qubit Eagle processor.

• **Quantum computing:** unitary SU(2) representations describe quantum gates on IBM Q.
• **Molecular spectroscopy:** CO2 orbitals classified by irreducible representations of D_inf_h.
• **Equivariant NN (SE(3)-Transformer):** uses SE(3) representations for molecular property prediction.

Group Representations

IBM Q System One describes spin-1/2 particles via the 2-dimensional representation of SU(2). The Pauli matrices sigma_x, sigma_y, sigma_z are the generators of this representation, controlling real quantum gates.

Irreducible Representations and Schur's Lemma

In quantum chemistry, the molecular orbitals of CO2 are classified by irreducible representations of the symmetry group D_inf_h. This lets chemists predict allowed transitions without solving the Schrodinger equation.

Key ideas

• **Representation rho: G->GL(V):** homomorphism. Character chi(g)=Tr(rho(g)) is the complete invariant.
• **Schur's lemma:** every finite-dim representation of a finite group over C decomposes into irreducibles.
• **Character orthogonality:** <chi_i, chi_j>=delta_{ij}. Number of irreducibles equals number of conjugacy classes.