Engineering of arbitrary U(N) transformations by quantum Householder reflections

TL;DR

This paper introduces a method to implement any U(N) quantum transformation using quantum Householder reflections in a system of degenerate states, enabling efficient quantum gate construction.

Contribution

The authors develop a physical implementation of quantum Householder reflections and demonstrate how to construct arbitrary U(N) transformations with minimal steps.

Findings

U(N) transformations can be factorized using QHRs and phase gates.

The method reduces the number of steps needed for quantum Fourier transform.

Efficient construction of quantum gates for qunits using QHRs.

Abstract

We propose a simple physical implementation of the quantum Householder reflection (QHR) M(v)=I-2|v><v| 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 M(v;k)=I+(exp{ik}-1)|v><v|, 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 U(N) transformation can be factorized (and thereby produced) 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 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, QFT requires a single QHR for N=2, and only two QHRs for N=3 and 4.