Classical programmability is enough for quantum circuits universality in approximate sense.pdf

a r X i v : q u a n t - p h / 0 1 0 3 1 1 9 v 1 2 0 M a r 2 0 0 1 Extended Abstract Classical programmability is enough for quantum circuits universality in approximate sense Alexander Yu. Vlasov FRC/IRH, 197101 Mira Street 8, St.–Petersburg, Russia *** It was shown by M. A. Nielsen and I. L Chuang [1], that it is impossible to build strictly universal programmable quantum gate array, that could perform any unitary operation precisely and it was suggested to use prob- abilistic gate arrays instead. In present work is shown, that if to use more physical and weak condition of universality (suggested already in earliest work by D. Deutsch [3]) and to talk about simulation with arbitrary, but finite precision, then it is possible to build universal programmable gate array. But now the same no-go theorem by Nielsen and Chuang [1] will have new interesting consequence — controlling programs for the gate arrays can be considered as pure classical. More detailed design of such deterministic quantum gate arrays universal “in approximate sense” is considered in the paper. 1 Introduction In the paper [1] was discussed conception of programmable quantum gate arrays, i.e., some quantum circuits are acting on a system in form |d;P 〉 ≡ |d〉 ? |P 〉 considered as data register |d〉 and program register (or simply program) |P 〉. Similar with conception of usual classical computer, it was considered, that circuit acts as some fixed unitary transformationU on whole system and different transformations UP of data related only with content P of program register, i.e: U ( |d〉 ? |P 〉 ) = (UP |d〉)? |P ′〉. (1) It should be emphasized, here UP is same for any state of data register |d〉 i.e., depends only on program |P 〉, and states of these two registers are not entangled before and after application of U. In relation with such a definition in [1] was noticed, that if to write the Eq. (1) for two different programs |P 〉 and |Q〉 and corresponding unitary data transformations |UP 〉 and |UQ〉,