Physical Review A

Arbitrary qudit gates by adiabatic passage

Benjamin Rousseaux, S. Guérin, Nikolay V. Vitanov

Published Mar 25, 2013• 67 citations

Abstract

We derive an adiabatic technique that implements the most general SU($d$) transformation in a quantum system of $d$ degenerate states, featuring a qudit. This technique is based on the factorization of the SU($d$) transformation into $d$ 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 $d$. This technique is directly applicable to a linear trapped ion system with $d+1$ ions. We implement the quantum Fourier transform numerically in a qudit with $d=4$ (defined as a quartit) as an example.

DOI: 10.1103/physreva.87.032328

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Physical Review A

Arbitrary qudit gates by adiabatic passage

Benjamin Rousseaux, S. Guérin, Nikolay V. Vitanov

Published Mar 25, 2013• 67 citations

Publisher

Abstract

We derive an adiabatic technique that implements the most general SU(

d

) transformation in a quantum system of

d

degenerate states, featuring a qudit. This technique is based on the factorization of the SU(

d

) transformation into

d

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

d

. This technique is directly applicable to a linear trapped ion system with

d+1

ions. We implement the quantum Fourier transform numerically in a qudit with

d=4

(defined as a quartit) as an example.

DOI: 10.1103/physreva.87.032328

Related papers

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

Peter A. Ivanov, Elica S. Kyoseva, Nikolay V. Vitanov

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Nikolay V. Vitanov

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.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.

Physical Review AMar 27, 201255 citations

Designing single-qutrit quantum gates via tripod adiabatic passage

M. Amniat-Talab, Sare Golkar, S. Nasser

In this paper, we use stimulated Raman adiabatic passage technique to implement single-qutrit quantum gates in tripod systems. It is shown by using the Morris-Shore (MS) transformation, the six-state problem with 5 pulsed fields can be reduced to a basis that decouples two states from the others. This imposes three pulses not connected to the initial condition with have the same shape. Using this method, the six-state penta-pod system is reduced to a tripod system. We can design single-qutrit quantum gates by ignoring the fragile dynamical phase, and by suitable design of Rabi frequencies of the effective HamiltonianIn this paper, we use stimulated Raman adiabatic passage technique to implement single-qutrit quantum gates in tripod systems. It is shown by using the Morris-Shore (MS) transformation, the six-state problem with 5 pulsed fields can be reduced to a basis that decouples two states from the others. This imposes three pulses not connected to the initial condition with have the same shape. Using this method, the six-state penta-pod system is reduced to a tripod system. We can design single-qutrit quantum gates by ignoring the fragile dynamical phase, and by suitable design of Rabi frequencies of the effective Hamiltonian

Apr 1, 20140 citations

We derive an adiabatic technique that implements the most general SU(

d

) transformation in a quantum system of

d

degenerate states, featuring a qudit. This technique is based on the factorization of the SU(

d

) transformation into

d

d

. This technique is directly applicable to a linear trapped ion system with

d+1

ions. We implement the quantum Fourier transform numerically in a qudit with

d=4

(defined as a quartit) as an example.

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.

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.