Quantum computing and Zeta functions

          @misc{ pirsa_PIRSA:06120001,
            doi = {},
            url = {https://pirsa.org/06120001},
            author = {van Dam, Wim},
            keywords = {},
            language = {en},
            title = {Quantum computing and Zeta functions},
            publisher = {Perimeter Institute},
            year = {2006},
            month = {dec},
            note = {PIRSA:06120001 see, \url{https://pirsa.org}}
          }
          

Abstract

In this talk I describe a possible connection between quantum computing and Zeta functions of finite field equations that is inspired by the \'spectral approach\' to the Riemann conjecture. This time the assumption is that the zeros of such Zeta functions correspond to the eigenvalues of finite dimensional unitary operators of quantum mechanical systems. To model the desired quantum systems I use the notion of universal, efficient quantum computation. Using eigenvalue estimation, such quantum systems should be able to approximately count the number of solutions of the specific finite field equations with an accuracy that does not appear to be feasible classically. For certain equations (Fermat hypersurfaces) one can indeed model their Zeta functions with efficient quantum algorithms, which gives some evidence in favor of the proposal. In the case of equations that define elliptic curves, the corresponding unitary transformation is an SU(2) matrix. Hence for random elliptic curves one expects to see the kind of statistics predicted by random matrix theory. In the last part of the talk I discuss to which degree this expectation does indeed hold. Reference: arXiv:quant-ph/0405081