Quantum Algorithms Using Clebsch-Gordan Transforms

          @misc{ pirsa_PIRSA:07050006,
            doi = {10.48660/07050006},
            url = {https://pirsa.org/07050006},
            author = {Bacon, Dave},
            keywords = {Quantum Information},
            language = {en},
            title = {Quantum Algorithms Using Clebsch-Gordan Transforms},
            publisher = {Perimeter Institute},
            year = {2007},
            month = {may},
            note = {PIRSA:07050006 see, \url{https://pirsa.org}}
          }
          

Abstract

In nearly every quantum algorithm which exponentially outperforms the best classical algorithm the quantum Fourier transform plays a central role. Recently, however, cracks in the quantum Fourier transform paradigm have begun to emerge. In this talk I will discuss one such development which arises in a new efficient quantum algorithm for the Heisenberg hidden subgroup problem. In particular I will show how considerations of symmetry for this hidden subgroup problem lead naturally to a different transform than the quantum Fourier transform, the Clebsch-Gordan transform over the Heisenberg group. Clebsch-Gordan transforms over finite groups thus appear to be an important new tool for those attempting to find new quantum algorithms. [Part of this work was done in collaboration with Andrew Childs (Caltech) and Wim van Dam (UCSB)]