Solutions of Heat Equation on Unitary Groups: Relation Between ϵ-nets and Approximate Unitary t-designs

Explore a 57-minute lecture by Oskar Słowik from the Center for Theoretical Physics, PAS, examining the fundamental solutions of heat equations on unitary groups and their role in establishing improved relationships between ϵ-nets and approximate unitary t-designs. Delve into these crucial concepts that are widely used in quantum computation and information, and discover how their quantitative relations impact applications like inverse-free Solovay-Kitaev theorems and random quantum circuits. Learn about the significant improvement in bounds for δ-approximate t-designs forming ϵ-nets, achieved through polynomial approximations to the Dirac delta using heat kernels on the projective unitary group PU(d). The lecture also covers potential applications of these findings in quantum circuit overheads, quantum complexity, and black hole physics, presented as part of the Quantum Information and Quantum Computing Seminars at CTP PAS.