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Physical Review A

Synthesis of arbitrary unitary transformations of collective states of trapped ions by quantum Householder reflections

Peter A. Ivanov, Nikolay V. Vitanov

Published Jan 30, 2008• 35 citations
Publisher

Abstract

We propose an efficient technique for creation and manipulation of entangled states and for construction of arbitrary many-particle unitary transformations in a linear ion trap. We show that, under suitable conditions on the laser pulse areas and detunings, the propagator within the single-excitation manifold has the form of a Householder reflection (HR). We demonstrate that a single HR is sufficient to create an entangled $W$-type state or to transfer one entangled state into another. An arbitrary $N$-dimensional unitary transformation can be constructed by at most $N\ensuremath{-}1$ HRs, i.e., significantly faster than $O({N}^{2})$ steps in previous proposals, which makes this technique less sensitive to heating and other types of decoherence. We present examples of synthesis of quantum Fourier transforms for four, five, and six ions, which are constructed using only two, three, and four HRs, respectively.

DOI
10.1103/physreva.77.012335

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We propose an efficient technique for creation and manipulation of entangled states and for construction of arbitrary many-particle unitary transformations in a linear ion trap. We show that, under suitable conditions on the laser pulse areas and detunings, the propagator within the single-excitation manifold has the form of a Householder reflection (HR). We demonstrate that a single HR is sufficient to create an entangled WWW-type state or to transfer one entangled state into another. An arbitrary NNN-dimensional unitary transformation can be constructed by at most N\ensuremath−1N\ensuremath{-}1N\ensuremath−1 HRs, i.e., significantly faster than O(N2)O({N}^{2})O(N2) steps in previous proposals, which makes this technique less sensitive to heating and other types of decoherence. We present examples of synthesis of quantum Fourier transforms for four, five, and six ions, which are constructed using only two, three, and four HRs, respectively.
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 NNN degenerate states (forming a qunit) coupled simultaneously to an ancillary (excited) state by NNN 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 NNN-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)\mathrm{U}(N)U(N) transformation can be factorized (and thereby produced) by either $N\ensuremath{-}1$ standard QHRs and an NNN-dimensional phase gate, or $N\ensuremath{-}1$ generalized QHRs and a one-dimensional phase gate. Viewed mathematically, these QHR factorizations provide parametrizations of the U(N)\mathrm{U}(N)U(N) group. As an example, we propose a recipe for constructing the quantum Fourier transform (QFT) by at most NNN interaction steps. For example, the QFT requires a single QHR for N=2N=2N=2, and only two QHRs for N=3N=3N=3 and 4.
We derive an adiabatic technique that implements the most general SU(ddd) transformation in a quantum system of ddd degenerate states, featuring a qudit. This technique is based on the factorization of the SU(ddd) transformation into ddd generalized quantum Householder reflections, each of which is implemented by a two-shot stimulated Raman adiabatic passage with appropriate static phases. The energy of the lasers needed to synthesize a single Householder reflection is shown to be remarkably constant as a function of ddd. This technique is directly applicable to a linear trapped ion system with d+1d+1d+1 ions. We implement the quantum Fourier transform numerically in a qudit with d=4d=4d=4 (defined as a quartit) as an example.
Three experimentally feasible schemes are proposed for generating multipartite entangled states of cold trapped ions.In the first scheme,a GHZ state can be simply synthesized in a single step. The second scheme is to prepare fully symmetric W states with two steps.The third scheme describes the sequential generation of linear cluster states.In addition,a method to create a larger cluster state was also presented by fuzing the end qubits of two smaller cluster states in order to decrease the distortion caused by decoherence,which can speed up the generation of large cluster states.The above schemes involve the collective interaction between multiple ions with the vibrational ground state and a large detuned laser. Compared with previous schemes,the above schemes have four advantages.Firstly,they do not require the vibrational mode of ions as memory.Secondly,they do not need to address the ions individually.Thirdly, they are free from the effects of phase fluctuation of laser field.Finally,they need less operations.
Nature provides us with a restricted set of microscopic interactions. The question is whether we can synthesize out of these fundamental interactions an arbitrary unitary operator. In this paper we present a constructive algorithm for realization of any unitary operator which acts on a (truncated) Hilbert space of a single bosonic mode. The algorithm itself is not unitary because it involves a conditional measurement. However, it does yield a constant probability of the conditional measurement which does not depend on the input state of the bosonic system. We consider a physical implementation of unitary transformations acting on one-dimensional vibrational states of a trapped ion. As an example we present an algorithm which realizes the discrete Fourier transform.
Several scenarios are proposed and analyzed for engineering of arbitrary preselected SU(3) transformations of laser-driven atomic qutrits. Two of the most natural implementations of qutrits are considered, in which the three qutrit states are coupled to each other via two-photon transitions through either (i) a common state in a tripod linkage pattern or (ii) two upper states in an M-shaped linkage chain. The SU(3) transformation for the tripod qutrit can be realized by 3 Givens SU(2) rotations, which require 9 consecutive interaction steps. Alternatively, because under certain conditions the propagator of the tripod system reduces to the Householder reflection operator, any SU(3) transformation can be constructed physically by 3 Householder reflections, each of which is implemented in a single interaction step. As an example, the discrete Fourier transform can be synthesized by 7 consecutive interaction steps with Givens rotations or, alternatively, by only a single Householder reflection and a phase gate. For the M-qutrit, the propagator is given by coupled Householder reflections and it cannot be reduced to Givens rotations or independent Householder reflections. By using these coupled Householder reflections it is shown that an arbitrary SU(3) transformation of the M qutrit can be realized with just two fields in at most 3 interaction steps; the discrete Fourier transform, in particular, requires only 2 interaction steps.