Engineering of arbitraryU(N)transformations by quantum Householder reflections

Abstract

We propose a simple physical implementation of the quantum Householder reflection (QHR) $\mathbf{M}(v)=\mathbf{I}\ensuremath{-}2\ensuremath{\mid}v⟩⟨v\ensuremath{\mid}$ in a quantum system of $N$ degenerate states (forming a qunit) coupled simultaneously to an ancillary (excited) state by $N$ resonant or nearly resonant pulsed external fields. We also introduce the generalized QHR $\mathbf{M}(v;\ensuremath{\varphi})=\mathbf{I}+({e}^{i\ensuremath{\varphi}}\ensuremath{-}1)\ensuremath{\mid}v⟩⟨v\ensuremath{\mid}$, which can be produced in the same $N$-pod system when the fields are appropriately detuned from resonance with the excited state. We use these two operators as building blocks in constructing arbitrary preselected unitary transformations. We show that the most general $\mathrm{U}(N)$ transformation can be factorized (and thereby produced) by either $N\ensuremath{-}1$ standard QHRs and an $N$-dimensional phase gate, or $N\ensuremath{-}1$ generalized QHRs and a one-dimensional phase gate. Viewed mathematically, these QHR factorizations provide parametrizations of the $\mathrm{U}(N)$ group. As an example, we propose a recipe for constructing the quantum Fourier transform (QFT) by at most $N$ interaction steps. For example, the QFT requires a single QHR for $N=2$, and only two QHRs for $N=3$ and 4.