Engineering of arbitraryU(N)transformations by quantum Householder reflections

Abstract

We propose a simple physical implementation of the quantum Householder reflection ͑QHR͒ M͑v͒ = I −2͉v͗͘v͉ in a quantum system of N degenerate states ͑forming a qunit͒ coupled simultaneously to an ancillary ͑excited͒ state by N resonant or nearly resonant pulsed external fields. We also introduce the generalized QHR M͑v ; ͒ = I + ͑e i −1͉͒v͗͘v͉, 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 U͑N͒ transformation can be factorized ͑and thereby produced͒ by either N − 1 standard QHRs and an N-dimensional phase gate, or N −1 generalized QHRs and a one-dimensional phase gate. Viewed mathematically, these QHR factorizations provide parametrizations of the U͑N͒ group. As an example, we propose a recipe for constructing the quantum Fourier transform ͑QFT͒ 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.

Quantum fuel with multilevel atomic coherence for ultrahigh specific work in a photonic Carnot engine

“…An arbitrary N dimensional unitary matrix U(N ) can be decomposed into so called N generalized quantum Householder reflection (QHR) matrices or N − 1 standard QHRs and a phase gate [28,29]. A generalized QHR is defined by M(ν; φ) = I + (e iφ − 1)|ν ν| (15) where I is the identity operator and the |ν is the normalized column vector with dimension N , same with the number of the pulses, and φ is an arbitrary phase factor.…”

“…The preceding discussion is applicable to the case of mixed states as well, for which the normalized vectors of generalized QHRs are defined as [28] …”

“…The initial and final states would then possess different spectral decompositions so that they cannot be unitarily connected. A resolution to this is suggested to exploit decoherence channels such as spontaneous emission or pure dephasing, in combination with the unitary transformation [28,29]. We will not follow this route but use an alternative, which allows for a fully unitary procedure to generate desired coherences.…”

Population trapping in three-state quantum loops revealed by Householder reflections

“…As we will show, the desired result can be produced by a Householder reflection acting upon the RWA Hamiltonian [19].…”

“…Such matrix manipulations, commonplace in works dealing with linear algebra [17], have recently been applied to quantum-state manipulations [18][19][20][21].…”