Implementing quantum Fourier transform using three qubits

Mouhcine Yachia, Radouan Hab-arriha and Ahmed Jellal***a.jellal@ucd.ac.maa,b

aLaboratory of Theoretical Physics, Faculty of Sciences, Chouaïb Doukkali
University,

PO Box 20, 24000 El Jadida, Morocco

bCanadian Quantum Research Center, 204-3002 32 Ave Vernon,
BC V1T 2L7, Canada

Using the circulant symmetry of a Hamiltonian describing three qubits, we
realize the quantum Fourier transform. This symmetry allows us to construct a
set of eigenvectors independently on the magnitude of physical parameters
involved in the Hamiltonian and as a result the entanglement will be
maintained. The realization will be leaned on trapped ions and the gate
implementation requires an adiabatic transition from each spin product state
to Fourier modes. The fidelity was numerically calculated and the results show
important values. Finally, we discuss the acceleration of the gate by using
the counter-driving field.

PACS numbers: 03.65.Fd, 03.65.Ge, 03.65.Ud, 03.67.Hk
Keywords: Three qubits, circulant symmetry, entanglement, quantum Fourier
transform, adiabatic transition, counter-driving-field.

##  1 Introduction

Paul Benioff suggested a quantum mechanical model of the Turing machine [1] in
1980, launching a new field of study known as quantum computers (QCs). Richard
Feynman demonstrated in 1982 that QCs may be used to mimic complicated systems
(living cells, city traffic, the human brain, and the universe, $\cdots$) [2].
Seth Lloyd demonstrated four decades later that the basic units of a QC are an
array of nuclear magnetic resonance spins [3]. QC is a device that uses
quantum mechanics to process data while maintaining quantum coherence [4]. As
a result, QC can tackle the most difficult computational problems that even
today’s most powerful supercomputers cannot solve [5]. As an example, Shor’s
quantum algorithm (SQA) for factoring large numbers[6] is the most seminal
motivation behind the development of QCs [5]. It is well-known in quantum
computing that the physical implementation of SQA requires a gate of
particular importance called quantum Fourier transform (QFT)[7]. QFT is the
quantum implementation of the discrete Fourier transform over the amplitudes
of a wavefunction, which acts on a vector $|x\rangle\in\mathbb{C}^{N}$ as
$|x\rangle\xrightarrow{\text{QFT}}|y\rangle=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}e^{2\pi
ixj/N}|j\rangle$ [5]. QFT is widely used as a key subroutine in several
factoring algorithms like for instance quantum amplitude estimation [8] and
quantum counting [9].

The circulant matrices have been thoroughly investigated in [10, 11], and the
set of invertible circulant matrices forms a group in terms of matrix
multiplication. Geometry, linear coding, and graph theory, all use such
matrices, one may see [12, 13, 14, 15, 16, 17]. Also for more than 47 years,
vibration analysis has focused on systems with circulant symmetry. Many of
these contributions were motivated by vibration studies of general
rotationally periodic systems [15], bladed disks, planetary gear systems,
rings, circular plates, disk spindle systems, centrifugal pendulum vibration
absorbers, space antennae, and microelectromechanical system frequency
filters.

The circulant Hamiltonians are addressed [18] in terms of the physical
implementation of logical QFT, which related to the fact that the
eigenspectrum of circulant matrix is spanned by the Fourier modes[19, 10].
Ivanov and Vitanov [20] recently proposed a Hamiltonian based on two spins
emerging in a magnetic field, creating Rabi oscillations, and obtained
circulant symmetry by modifying the coupling strength of the spin-spin
interaction. As a result, they demonstrated that the eigenvectors are
independent of the magnitude of the physical parameters, implying entanglement
protection, and the resulting system can subsequently be employed as a logical
QFT. Wu and his colleagues [21] presented two approaches for implementing
quantum phase gates based on adiabatic passing and phase control of the
driving fields. They studied the experimental feasibility, gate fidelity and
docoherence effect for both schemes. Moreover, the shortcuts to adiabaticity
have become instrumental in preparing and driving internal and motional states
in atomic, molecular and solid-state physics, for a complete review we refer
to [22].

Motivated by the results developed in [20], we study a Hamiltonian describing
three spins in a magnetic field, coupled via linear and non-linear
interactions. The last coupling generally arises when the interaction medium
is non-linear [23, 24, 25] and remains an important ingredient in generating a
circulant Hamiltonian. To build a logical QFT, we propose two schemes based on
the physical parameter choices. The eigenvectors of our circulant Hamiltonian
do not depend on the parameters, which protects entanglement during the gate
implementation as long as the circulant symmetry is maintained. We use an
energy offset Hamiltonian
${H}_{0}(t)=\sum\limits_{j=1}^{3}\Delta_{j}(t){\sigma}_{j}^{z}$ to break the
circulant symmetry during the transition. We adjust the physical parameters
and the detuning $\Delta_{j}(t)$ to realize the circulant symmetry at the end
of the transition $t_{f}$. We sinusoidally modulate the physical parameters in
time to show that it is possible to adiabatically obtain a superposition of
the quantum Fourier modes from any initial state with high fidelity. As the
adiabatic evolution is robust but still limited by the non-adiabatic
transition, we introduce a counter-driving Hamiltonian ${H}_{\sf CD}(t)$[26]
to suppress these transitions. Under suitable conditions of the physical
parameters, we determine the eigenvectors associated with the total
Hamiltonian. These allow us to combine our gate with the short-cut to
adiabacity scheme in order to accelerate the gate. Based on the proposal
described in [20], we suggest a way to physically implement the gate.

The outlines of our paper are summarized as follows. In Sec. 2, we propose a
Hamiltonian describing three qubits with different interactions in addition to
Rabi oscillations and show how to obtain the circulant symmetry. The adiabatic
transition technique is performed to end up with the Fourier modes in Sec. 3.
The physical implementation of the obtained QFT gate will be discussed in Sec.
4. The non-degeneracy of frequencies, the fidelities of our gate and the
creation of entangled state will be numerically analyzed by sinusoidally
varying the physical parameters in Sec. 5. We discuss the shortcut to
adiabacity scheme combined with our gate in Sec. 6. Finally, we close by
concluding our work.

##  2 Theoretical model

We consider three spins that emerged in a magnetic field to achieve our task,
forming a three-qubit gate system as presented in Figure 1.

Figure 1: (color online) The schematic presents the interactions of strengths
$J_{j}$ between three coupled qubits emerging in a magnetic field with the
Rabi frequencies $\Omega_{j}$.

It can be described by a Hamiltonian involving two types of interaction, such
as

$\displaystyle{H}=$ $\displaystyle
J_{1}(\sigma_{1}^{+}e^{-i\phi_{12}}+\sigma_{1}^{-}e^{i\phi_{12}})(\sigma_{2}^{+}e^{-i\phi_{21}}+\sigma_{2}^{-}e^{i\phi_{21}})+J_{2}(\sigma_{2}^{+}e^{-i\phi_{23}}+\sigma_{2}^{-}e^{i\phi_{23}})(\sigma_{3}^{+}e^{-i\phi_{32}}+\sigma_{3}^{-}e^{i\phi_{32}})$
$\displaystyle+J_{3}(\sigma_{1}^{+}e^{-i\phi_{13}}+\sigma_{1}^{-}e^{i\phi_{13}})(\sigma_{3}^{+}e^{-i\phi_{31}}+\sigma_{3}^{-}e^{i\phi_{31}})+\Omega_{1}(\sigma_{1}^{+}e^{i\theta_{1}}+\sigma_{1}^{-}e^{-i\theta_{1}})+\Omega_{2}(\sigma_{2}^{+}e^{i\theta_{2}}+\sigma_{2}^{-}e^{-i\theta_{2}})$
$\displaystyle+\Omega_{3}(\sigma_{3}^{+}e^{i\theta_{3}}+\sigma_{3}^{-}e^{-i\theta_{3}})+J(\sigma_{1}^{+}e^{-i\phi_{1}}+\sigma_{1}^{-}e^{i\phi_{1}})(\sigma_{2}^{+}e^{-i\phi_{2}}+\sigma_{2}^{-}e^{i\phi_{2}})(\sigma_{3}^{+}e^{-i\phi_{3}}+\sigma_{3}^{-}e^{i\phi_{3}})$
(1)

where $\sigma_{j}^{+}=\ket{\uparrow_{j}}\bra{\downarrow_{j}}$ and
$\sigma_{j}^{-}=\ket{\downarrow_{j}}\bra{\uparrow_{j}}$ stand for spin flip
operators, $\ket{\uparrow_{j}}$ and $\ket{\downarrow_{j}}$ being the qubit
states of the $j^{th}$ spin, with $j=1,2,3$. The first term describes the
interaction between spins $1$ and $2$ of phases $\phi_{12}$ and $\phi_{21}$,
whereas the second term covers the interaction between spins $2$ and $3$ of
phases $\phi_{23}$ and $\phi_{32}$, and the third term describes the
interaction between spins $1$ and $3$ of phases $\phi_{13}$ and $\phi_{31}$.
The terms, including the Rabi frequencies $\Omega_{j}$, are the single-qubit
transitions with phases $\theta_{j}$. The last term describes the coupling
between the three spins with phases $\phi_{j}$ [25]. Note that, our
Hamiltonian can be considered as a three-qubit generalization of the Ivanov
and Vitanov model for two qubits [20], thanks to the trilinear interaction
term.

It is convenient for our task to consider the matrix form of the Hamiltonian
(2) and then, in the basis
$\mathcal{B}_{c}=\left\\{\mid\downarrow\downarrow\downarrow\rangle,\mid\downarrow\downarrow\uparrow\rangle,\mid\downarrow\uparrow\downarrow\rangle,\mid\downarrow\uparrow\uparrow\rangle,\mid\uparrow\downarrow\downarrow\rangle,\mid\uparrow\downarrow\uparrow\rangle,\mid\uparrow\uparrow\downarrow\rangle,\mid\uparrow\uparrow\uparrow\rangle\right\\}$,
we have

$\displaystyle{H}=\begin{pmatrix}0&\Omega_{3}e^{i\theta_{3}}&\Omega_{2}e^{i\theta_{2}}&J_{2}e^{-i\xi_{1}}&\Omega_{1}e^{i\theta_{1}}&J_{3}e^{-i\xi_{2}}&J_{1}e^{-i\xi_{3}}&Je^{-i\xi_{7}}\\\
\Omega_{3}e^{-i\theta_{3}}&0&J_{2}e^{-i\xi_{4}}&\Omega_{2}e^{i\theta_{2}}&J_{3}e^{-i\xi_{5}}&\Omega_{1}e^{i\theta_{1}}&Je^{-i\xi_{8}}&J_{1}e^{-i\xi_{3}}\\\
\Omega_{2}e^{-i\theta_{2}}&J_{2}e^{i\xi_{4}}&0&\Omega_{3}e^{i\theta_{3}}&J_{1}e^{-i\xi_{6}}&Je^{-i\xi_{9}}&\Omega_{1}e^{i\theta_{1}}&J_{3}e^{-i\xi_{2}}\\\
J_{2}e^{i\xi_{1}}&\Omega_{2}e^{-i\theta_{2}}&\Omega_{3}e^{-i\theta_{3}}&0&Je^{-i\xi_{10}}&J_{1}e^{-i\xi_{6}}&J_{3}e^{-i\xi_{5}}&\Omega_{1}e^{i\theta_{1}}\\\
\Omega_{1}e^{-i\theta_{1}}&J_{3}e^{i\xi_{5}}&J_{1}e^{i\xi_{6}}&Je^{i\xi_{10}}&0&\Omega_{3}e^{i\theta_{3}}&\Omega_{2}e^{i\theta_{2}}&J_{2}e^{-i\xi_{1}}\\\
J_{3}e^{i\xi_{2}}&\Omega_{1}e^{-i\theta_{1}}&Je^{i\xi_{9}}&J_{1}e^{i\xi_{6}}&\Omega_{3}e^{-i\theta_{3}}&0&J_{2}e^{-i\xi_{4}}&\Omega_{2}e^{i\theta_{2}}\\\
J_{1}e^{i\xi_{3}}&Je^{i\xi_{8}}&\Omega_{1}e^{-i\theta_{1}}&J_{3}e^{i\xi_{5}}&\Omega_{2}e^{-i\theta_{2}}&J_{2}e^{i\xi_{4}}&0&\Omega_{3}e^{i\theta_{3}}\\\
Je^{i\xi_{7}}&J_{1}e^{i\xi_{3}}&J_{3}e^{i\xi_{2}}&\Omega_{1}e^{-i\theta_{1}}&J_{2}e^{i\xi_{1}}&\Omega_{2}e^{-i\theta_{2}}&\Omega_{3}e^{-i\theta_{3}}&0\end{pmatrix}$
(2)

where the ten angles $\xi_{1}=\phi_{23}+\phi_{32}$,
$\xi_{2}=\phi_{13}+\phi_{31}$, $\xi_{3}=\phi_{12}+\phi_{21}$,
$\xi_{4}=\phi_{23}-\phi_{32},$ $\xi_{5}=\phi_{13}-\phi_{31}$,
$\xi_{6}=\phi_{12}-\phi_{21},$ $\xi_{7}=\phi_{1}+\phi_{2}+\phi_{3},$
$\xi_{8}=\phi_{1}+\phi_{2}-\phi_{3},$ $\xi_{9}=\phi_{1}-\phi_{2}+\phi_{3},$
$\xi_{10}=\phi_{1}-\phi_{2}-\phi_{3}$ have been introduced.

In what follows, we are going to find conditions on the involved parameters to
end up with the Hamiltonian (2) as a circulant matrix [19, 10]. The benefit of
the circulant matrix is that its eigenvectors are the vectors of columns of
the discrete quantum Fourier transform, and therefore they do not depend on
the elements of the circulant matrix. The eigenvectors of our circulant matrix
can be mapped into the spin basis $\mathcal{B}_{c}$ as

$\displaystyle\ket{\psi_{0}}=\dfrac{1}{2\sqrt{2}}(\ket{\downarrow\downarrow\downarrow}+\ket{\downarrow\downarrow\uparrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\downarrow\uparrow\uparrow}+\ket{\uparrow\downarrow\downarrow}+\ket{\uparrow\downarrow\uparrow}+\ket{\uparrow\uparrow\downarrow}+\ket{\uparrow\uparrow\uparrow})$
(3)
$\displaystyle\ket{\psi_{1}}=\dfrac{1}{2\sqrt{2}}(\ket{\downarrow\downarrow\downarrow}+\omega\ket{\downarrow\downarrow\uparrow}+i\ket{\downarrow\uparrow\downarrow}+i\omega\ket{\downarrow\uparrow\uparrow}-\ket{\uparrow\downarrow\downarrow}-\omega\ket{\uparrow\downarrow\uparrow}-i\ket{\uparrow\uparrow\downarrow}-i\omega\ket{\uparrow\uparrow\uparrow})$
(4)
$\displaystyle\ket{\psi_{2}}=\dfrac{1}{2\sqrt{2}}(\ket{\downarrow\downarrow\downarrow}+i\ket{\downarrow\downarrow\uparrow}-\ket{\downarrow\uparrow\downarrow}-i\ket{\downarrow\uparrow\uparrow}+\ket{\uparrow\downarrow\downarrow}+i\ket{\uparrow\downarrow\uparrow}-\ket{\uparrow\uparrow\downarrow}-i\ket{\uparrow\uparrow\uparrow})$
(5)
$\displaystyle\ket{\psi_{3}}=\dfrac{1}{2\sqrt{2}}(\ket{\downarrow\downarrow\downarrow}+i\omega\ket{\downarrow\downarrow\uparrow}-i\ket{\downarrow\uparrow\downarrow}+\omega\ket{\downarrow\uparrow\uparrow}-\ket{\uparrow\downarrow\downarrow}-i\omega\ket{\uparrow\downarrow\uparrow}+i\ket{\uparrow\uparrow\downarrow}-\omega\ket{\uparrow\uparrow\uparrow})$
(6)
$\displaystyle\ket{\psi_{4}}=\dfrac{1}{2\sqrt{2}}(\ket{\downarrow\downarrow\downarrow}-\ket{\downarrow\downarrow\uparrow}+\ket{\downarrow\uparrow\downarrow}-\ket{\downarrow\uparrow\uparrow}+\ket{\uparrow\downarrow\downarrow}-\ket{\uparrow\downarrow\uparrow}+\ket{\uparrow\uparrow\downarrow}-\ket{\uparrow\uparrow\uparrow})$
(7)
$\displaystyle\ket{\psi_{5}}=\dfrac{1}{2\sqrt{2}}(\ket{\downarrow\downarrow\downarrow}-\omega\ket{\downarrow\downarrow\uparrow}+i\ket{\downarrow\uparrow\downarrow}-i\omega\ket{\downarrow\uparrow\uparrow}-\ket{\uparrow\downarrow\downarrow}+\omega\ket{\uparrow\downarrow\uparrow}-i\ket{\uparrow\uparrow\downarrow}+i\omega\ket{\uparrow\uparrow\uparrow})$
(8)
$\displaystyle\ket{\psi_{6}}=\dfrac{1}{2\sqrt{2}}(\ket{\downarrow\downarrow\downarrow}-i\ket{\downarrow\downarrow\uparrow}-\ket{\downarrow\uparrow\downarrow}+i\ket{\downarrow\uparrow\uparrow}+\ket{\uparrow\downarrow\downarrow}-i\ket{\uparrow\downarrow\uparrow}-\ket{\uparrow\uparrow\downarrow}+i\ket{\uparrow\uparrow\uparrow})$
(9)
$\displaystyle\ket{\psi_{7}}=\dfrac{1}{2\sqrt{2}}\left(\ket{\downarrow\downarrow\downarrow}-i\omega\ket{\downarrow\downarrow\uparrow}-i\ket{\downarrow\uparrow\downarrow}-\omega\ket{\downarrow\uparrow\uparrow}-\ket{\uparrow\downarrow\downarrow}+i\omega\ket{\uparrow\downarrow\uparrow}+i\ket{\uparrow\uparrow\downarrow}+\omega\ket{\uparrow\uparrow\uparrow}\right)$
(10)

and we have set the phase factor $\omega=\exp(i\frac{\pi}{4})$. At this point,
we show the possibilities that lead to the circulant symmetries. We end up
with two scenarios when we require adequate requirements to be met by the
physical parameters. Indeed, the first configuration is

$\displaystyle J_{1}=\Omega_{2}$ (11) $\displaystyle J=J_{2}=J_{3}=\Omega_{3}$
(12) $\displaystyle\Omega_{1}=0$ (13)
$\displaystyle\theta_{2}=\theta_{3}=\phi_{32}=\phi_{3}=\phi_{21}=-\phi_{31}=\varphi$
(14)
$\displaystyle\theta_{1}=\phi_{23}=\phi_{2}=\phi_{1}=\phi_{12}=\phi_{13}=0$
(15)

which can be injected into (2) to get the first circulant Hamiltonian

$\displaystyle{H}_{\sf
cir}^{(1)}=\begin{pmatrix}0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}\\\
Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{-i\varphi}\\\
J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}\\\
Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&0\\\
0&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}\\\
Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}\\\
J_{1}e^{i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}\\\
Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0\end{pmatrix}$
(16)

whereas the second configuration looks like

$\displaystyle J_{1}=\Omega_{2}$ (17) $\displaystyle J=J_{2}=J_{3}=\Omega_{3}$
(18)
$\displaystyle\theta_{2}=\theta_{3}=\phi_{32}=\phi_{3}=\phi_{21}=-\phi_{31}=\varphi$
(19)
$\displaystyle\theta_{1}=\phi_{23}=\phi_{2}=\phi_{1}=\phi_{12}=\phi_{13}=0$
(20)

with $\Omega_{1}\neq 0$ and then, from (2) we obtain the second circulant
Hamiltonian

$\displaystyle{H}_{\sf
cir}^{(2)}=\begin{pmatrix}0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&\Omega_{1}&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}\\\
Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&\Omega_{1}&Je^{i\varphi}&J_{1}e^{-i\varphi}\\\
J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&\Omega_{1}&Je^{i\varphi}\\\
Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&\Omega_{1}\\\
\Omega_{1}&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}\\\
Je^{-i\varphi}&\Omega_{1}&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}\\\
J_{1}e^{i\varphi}&Je^{-i\varphi}&\Omega_{1}&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&Je^{i\varphi}\\\
Je^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&\Omega_{1}&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0\end{pmatrix}.$
(21)

At this level, we underline that the presence of the trilinear interaction is
necessary to realize the QFT gate. This type of interaction is required to
achieve such circulant symmetry. Generalizing to more than three qubits is not
trivial because the Hamiltonian will demand multilinear interactions. As a
result, discussing the adiabatic transition and shortcut to adiabacity will be
difficult mathematically. In the forthcoming analysis, we focus only on one of
the above Hamiltonians, let say ${H}_{\sf cir}^{(1)}$, and investigate its
basic features.

##  3 Adiabatic transition to Fourier modes

Some controls can be used to create an adiabatic transition to Fourier modes.
Two of these will be discussed here: the energy offset and Rabi frequencies.

###  3.1 Controlling by energy offset

To realize an adiabatic evolution to the circulant Hamiltonian states (Fourier
modes), we add an energy offset ${H}_{0}(t)$

${H}_{0}(t)=\Delta_{1}(t){\sigma}_{1}^{z}+\Delta_{2}(t){\sigma}_{2}^{z}+\Delta_{3}(t){\sigma}_{3}^{z}$
(22)

where the time-dependent detuning $\Delta_{j}(t)$ of $j^{th}$ spin are
necessary to control the adiabatic transition from computational spin states
to quantum Fourier states (3-10). Consequently, we have now the Hamiltonian

${H}(t)={H}_{0}(t)+{H}_{\sf cir}^{(1)}(t).$ (23)

Remember that in the adiabatic limit, the system is always in the same
eigenstates of ${H}(t)$ [20]. The eigenstates of $H(t)$ will be those of
$H_{0}(t)$ at $t_{i}$, and the dynamics will drive them to the Fourier modes
at $t_{f}$ if the couplings and detunings have a specific time dependency.
Therefore, adiabatic evolution translates each computational spin state to a
Fourier mode, resulting in a single interaction step that generates the QFT.
The non-degeneracy between the eigen-frequencies of ${H}(t)$ must be bigger at
any time than the non-adiabatic coupling between each pair of $H(t)$
eigenstates $\ket{\lambda_{\pm}}$, $\ket{\delta_{\pm}}$, $\ket{\mu_{\pm}}$,
and $\ket{\gamma_{\pm}}$ of ${H}(t)$, according to adiabatic evolution.
Otherwise, we have the conditions

$\displaystyle|\mu_{\pm}(t)-\lambda_{\pm}(t)|\gg|\bra{\partial_{t}\mu_{\pm}(t)}\ket{\lambda_{\pm}(t)}|$
(24)
$\displaystyle|\lambda_{+}(t)-\lambda_{-}(t)|\gg|\bra{\partial_{t}\lambda_{+}(t)}\ket{\lambda_{-}(t)}|$
(25)
$\displaystyle|\lambda_{\pm}(t)-\delta_{\pm}(t)|\gg|\bra{\partial_{t}\lambda_{\pm}(t)}\ket{\delta_{\pm}(t)}|$
(26)
$\displaystyle|\delta_{+}(t)-\delta_{-}(t)|\gg|\bra{\partial_{t}\delta_{+}(t)}\ket{\delta_{-}(t)}|$
(27)
$\displaystyle|\delta_{\pm}(t)-\gamma_{\pm}(t)|\gg|\bra{\partial_{t}\delta_{\pm}(t)}\ket{\gamma_{\pm}(t)}|$
(28)
$\displaystyle|\gamma_{+}(t)-\gamma_{-}(t)|\gg|\bra{\partial_{t}\gamma_{+}(t)}\ket{\gamma_{-}(t)}|.$
(29)

Let us choose $\varphi=\frac{\pi}{2}$ to simplify our problem, and then we
prove in Appendix A that the eigenvalues
$\lambda_{\pm},\delta_{\pm},\mu_{\pm},\gamma_{\pm}$ of the Hamiltonian
${H}(t)$ (23) are provided by (A.1-A.4). Now, it is worthwhile to mention that
the circulant symmetry is broken. We suppose that the system is initially
prepared in the computational product states
$\ket{\psi_{s_{1}s_{2}s_{3}}}=\ket{s_{1}s_{2}s_{3}}$
$(s_{j}=\downarrow_{j},\uparrow_{j})$, which are eigenstates of the
Hamiltonian ${H}_{0}(t)$. As a result, the initial parameters should verify
the conditions

$\displaystyle\Delta_{1,2,3}(t_{i})\gg J(t_{i}),\>J_{1}(t_{i})$ (30)

and then ${H}(t)$ goes to ${H}_{0}(t)$. Consequently the eigenvalues become

$\displaystyle\lambda_{\pm}(t_{i})=\pm\left[\Delta_{1}(t_{i})+\Delta_{2}(t_{i})+\Delta_{3}(t_{i})\right]$
(31)
$\displaystyle\delta_{\pm}(t_{i})=\pm\left[\Delta_{1}(t_{i})+\Delta_{2}(t_{i})-\Delta_{3}(t_{i})\right]$
(32)
$\displaystyle\mu_{\pm}(t_{i})=\pm\left[\Delta_{1}(t_{i})-\Delta_{2}(t_{i})+\Delta_{3}(t_{i})\right]$
(33)
$\displaystyle\gamma_{\pm}(t_{i})=\pm\left[\Delta_{1}(t_{i})-\Delta_{2}(t_{i})-\Delta_{3}(t_{i})\right]$
(34)

and the ${H}(t)$ eigenvectors are exactly the computational spin states, i.e.
$\ket{\psi(t_{i})}=\ket{s_{1}s_{2}s_{3}}$,

$\displaystyle\ket{\lambda_{+}}=\ket{\downarrow\downarrow\downarrow},\quad\ket{\lambda_{-}}=\ket{\uparrow\uparrow\uparrow}$
(35)
$\displaystyle\ket{\delta_{+}}=\ket{\downarrow\downarrow\uparrow},\qquad\ket{\delta_{-}}=\ket{\uparrow\uparrow\downarrow}$
(36)
$\displaystyle\ket{\mu_{+}}=\ket{\downarrow\uparrow\downarrow},\qquad\ket{\mu_{-}}=\ket{\uparrow\downarrow\uparrow}$
(37)
$\displaystyle\ket{\gamma_{+}}=\ket{\downarrow\uparrow\uparrow},\qquad\ket{\gamma_{-}}=\ket{\uparrow\downarrow\downarrow}.$
(38)

To avoid degeneracy, the condition $\Delta_{i}(t_{i})\neq\Delta_{j}(t_{i})$
with $i,j=1,2,3$, and we can have equidistant eigen-frequencies by requiring
$\Delta_{1}(t_{i})=2\Delta_{2}(t_{i})=4\Delta_{3}(t_{i})$. Furthermore, to
obtain the Fourier modes at the final time $t_{f}$ of transition, the coupling
parameters together with detunings should verify the conditions

$\displaystyle\Delta_{1,2,3}(t_{f})\ll J(t_{f}),\>J_{1}(t_{f}).$ (39)

As a result, the total Hamiltonian evolves to a circulant one, i.e.
${H}(t)\rightarrow{H}_{\sf cir}^{(1)}(t)$, and its eigenspectrum becomes that
of ${H}_{\sf cir}^{(1)}(t)$, as shown below

$\displaystyle\ket{\lambda_{+}}=\ket{\psi_{0}},\qquad\ket{\lambda_{-}}=\ket{\psi_{7}}$
(40)
$\displaystyle\ket{\delta_{+}}=\ket{\psi_{1}},\qquad\ket{\delta_{-}}=\ket{\psi_{6}}$
(41)
$\displaystyle\ket{\mu_{+}}=\ket{\psi_{2}},\qquad\ket{\mu_{-}}=\ket{\psi_{5}}$
(42)
$\displaystyle\ket{\gamma_{+}}=\ket{\psi_{3}},\qquad\ket{\gamma_{-}}=\ket{\psi_{4}}.$
(43)

Additionally, the realization of the QFT relies on the adiabatic following of
each of the instantaneous eigenvectors

$\displaystyle\ket{\downarrow\downarrow\downarrow}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{i\alpha_{1}}\>\ket{\psi_{0}}$ (44)
$\displaystyle\ket{\downarrow\downarrow\uparrow}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{i(\alpha_{2}-\frac{\pi}{2})}\>\ket{\psi_{1}}$ (45)
$\displaystyle\ket{\downarrow\uparrow\downarrow}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{i\alpha_{3}}\>\ket{\psi_{2}}$ (46)
$\displaystyle\ket{\downarrow\uparrow\uparrow}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{i\alpha_{4}}\>\ket{\psi_{3}}$ (47)
$\displaystyle\ket{\uparrow\downarrow\downarrow}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\alpha_{4}}\>\ket{\psi_{4}}$ (48)
$\displaystyle\ket{\uparrow\downarrow\uparrow}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\alpha_{3}}\>\ket{\psi_{5}}$ (49)
$\displaystyle\ket{\uparrow\uparrow\downarrow}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\alpha_{2}}\>\ket{\psi_{6}}$ (50)
$\displaystyle\ket{\uparrow\uparrow\uparrow}\>$ $\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\alpha_{1}}\>\ket{\psi_{7}}$ (51)

and the global adiabatic phases $\alpha_{j}$ appear due to the adiabatic
evolution [20, 27, 28, 22]

$\displaystyle\alpha_{1}=\int_{t_{i}}^{t_{f}}\lambda_{+}(t)\,\mathrm{d}t,\qquad\alpha_{2}=\int_{t_{i}}^{t_{f}}\delta_{+}(t)\,\mathrm{d}t,\qquad\alpha_{3}=\int_{t_{i}}^{t_{f}}\mu_{+}(t)\,\mathrm{d}t,\qquad\alpha_{4}=\int_{t_{i}}^{t_{f}}\gamma_{+}(t)\,\mathrm{d}t$
(52)

with the relations

$\displaystyle\lambda_{-}(t)=-\lambda_{+}(t),\qquad\delta_{-}(t)=-\delta_{+}(t),\qquad\mu_{-}(t)=-\mu_{+}(t),\qquad\gamma_{-}(t)=-\gamma_{+}(t).$
(53)

After a specific tuning of the detuning $\Delta_{j}(t)$, $\alpha_{j}$ reduce
to

$\alpha_{1}=2p\pi,\qquad\alpha_{2}=2m\pi,\qquad\alpha_{3}=2n\pi,\qquad\alpha_{4}=2k\pi$
(54)

with $p,m,n,k$ are four integer numbers. This choice leads to the realization
of the following unitary quantum gate

$\displaystyle\mathcal{G}=\dfrac{1}{2\sqrt{2}}\begin{pmatrix}1&-i&1&1&1&1&1&1\\\
1&-i\omega&i&i\omega&-1&-\omega&-i&-i\omega\\\ 1&1&-1&-i&1&i&-1&-i\\\
1&\omega&-i&\omega&-1&-i\omega&i&-\omega\\\ 1&i&1&-1&1&-1&1&-1\\\
1&i\omega&i&-i\omega&-1&\omega&-i&i\omega\\\ 1&-1&-1&i&1&-i&-1&i\\\
1&-\omega&-i&-\omega&-1&i\omega&i&\omega\end{pmatrix}.$ (55)

As a result, one can show that up to an additional phase $-\frac{\pi}{2}$ the
determinant of $\mathcal{G}$ is equal to one, i.e. $\det(\mathcal{G})=1$,
which is necessary for adiabatic evolution. Therefore, it is worthwhile to
note that the gate $\mathcal{G}$ is a QFT one.

###  3.2 Controlling by Rabi frequencies

Now we will discuss how to control the Rabi frequencies $\Omega_{2}(t)$ and
$\Omega_{3}(t)$ by considering the Hamiltonian ${H}^{(1)}(t)$ (B.2) in
Appendix B without using the energy offset ${H}_{0}(t)$ (22). To achieve this
goal, we drive $\Omega_{2}(t)$ and $\Omega_{3}(t)$ in a way that they become
equal to the couplings $J_{1}$ and $J$, respectively. In what follows, we
summarize the process of control in three steps.

  * •

Initially: Let us assume that $\Omega_{2}(t_{i})\gg J_{1}$ and
$\Omega_{3}(t_{i})\gg J$, then the eigenvectors of ${H}^{(1)}(t)$ (B.2) are
the rotating computational spin states
$\ket{\psi({t_{i}})}=\ket{s^{\prime}_{1}s^{\prime}_{2}s^{\prime}_{3}}$
($s^{\prime}_{j}=\pm_{j})$, such as

$\displaystyle\ket{\pm_{1}}=\dfrac{1}{\sqrt{2}}\left(\ket{\downarrow_{1}}\pm\ket{\uparrow_{1}}\right)$
$\displaystyle\ket{\pm_{2}}=\dfrac{1}{\sqrt{2}}\left(e^{i\varphi}\ket{\downarrow_{2}}\pm\>\ket{\uparrow_{2}}\right)$
(56)
$\displaystyle\ket{\pm_{3}}=\dfrac{1}{\sqrt{2}}\left(\ket{\downarrow_{3}}\pm\ket{\uparrow_{3}}\right).$

  * •

Transition: The adiabatic transition from the initial eigenvectors to Fourier
modes is given by the mappings

$\displaystyle\ket{---}\>$ $\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\beta_{0}}\>\ket{\psi_{0}}$ $\displaystyle\ket{--+}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\beta_{1}}\>\ket{\psi_{1}}$ $\displaystyle\ket{-+-}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\beta_{2}}\>\ket{\psi_{2}}$ $\displaystyle\ket{-++}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\beta_{3}}\>\ket{\psi_{3}}$ (57)
$\displaystyle\ket{+--}\>$ $\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\beta_{4}}\>\ket{\psi_{4}}$ $\displaystyle\ket{+-+}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\beta_{5}}\>\ket{\psi_{5}}$ $\displaystyle\ket{++-}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\beta_{6}}\>\ket{\psi_{6}}$ $\displaystyle\ket{+++}\>$
$\displaystyle\longrightarrow$
$\displaystyle\>e^{-i\beta_{7}}\>\ket{\psi_{7}}$

where the adiabatic phases $\beta_{i}$ read as

$\displaystyle\beta_{i}=\int_{t_{i}}^{t_{f}}\Lambda_{i}(t)\ dt,\qquad
i=1,\cdots,7$ (58)

and Appendix B shows the eigenfrequencies $\Lambda_{i}(t)$ (B.3-B.10) of
${H}^{(1)}(t)$ (B.2).

  * •

Finally: We adiabatically decrease $\Omega_{2}(t)$ together with
$\Omega_{3}(t)$ to end up with

$\Omega_{2}(t_{f})=J_{1},\qquad\Omega_{3}(t_{f})=J.$ (59)

As a result, the circulant symmetry will be established, and the Fourier modes
will be derived as well.

At this stage, we mention that in the case of the gate realization based on
the energy offset, the exact eigenvectors of the Hamiltonian (23) can not be
obtained exactly. However, under the consideration made here, the derivation
of eigenvectors can be achieved, see Appendix B. Thus, to accelerate the gate,
we combine the gate scheme with the short-cut to adiabaticity.

##  4 Gate implementation

To give a physical implementation of our system, we generalize the process
proposed by Ivanov and Vitanov [20] to the three qubit case with trilinear
coupling. Indeed, to realize the circulant Hamiltonian, we proceed with
trapped ions [29, 30, 31]. A crystal with $3N$ ions with mass $M$ is
considered, the trap axis are $z$ and $x$ with the frequencies $\Omega_{z}$
and $\Omega_{x}$, respectively. Each ion of the crystal is a qubit described
by two typical levels: $\ket{\uparrow},\ket{\downarrow}$ with an energy gap
$\omega_{0}$. Moreover, the virtual excitations generated between two coupled
ions undergo small radial vibrations around the equilibrium positions of the
ions. Then, as an illustration, we present our physical implementation,
depicted in Figure 2.

Figure 2: (color online) A proposal for the physical implementation of the
interaction between three coupled trapped ions with laser frequencies
$\Omega_{j,L_{r}}=\omega_{0}-\nu-\Delta_{j}(t)$ and
$\Omega_{j,L_{b}}=\omega_{0}+\nu-\Delta_{j}(t)$ that generates a spin-
dependent force at the frequency $\nu$.

As for our Hamiltonian (2), the implementation of kinetic terms together with
bilinear couplings is perfectly discussed in [20]. Regarding the
implementation of the trilinear coupling, we suggest that it can be seen as a
coupling between two coupled ions with an extra third ion [25]. To be clear,
the small radial vibrations around the equilibrium positions between two
coupled ions are displayed by a set of collective vibrational modes with the
Hamiltonians [23, 32]

${H}_{\sf
ph1}=\sum_{n}\Omega_{n}\widehat{a}_{n}^{\dagger}\widehat{a}_{n},\qquad{H}_{\sf
ph2}=\sum_{n}\Omega_{n}\widehat{b}_{n}^{\dagger}\widehat{b}_{n},\qquad{H}_{\sf
ph3}=\sum_{n}\Omega_{n}\widehat{c}_{n}^{\dagger}\widehat{c}_{n}$ (60)

whereas the internal energy is

${H}_{\sf q}=\frac{1}{2}\sum_{j}\omega_{0}\sigma_{j}^{z}.$ (61)

and the free Hamiltonian then becomes

${H}_{0}={H}_{\sf q}+\sum_{j}{H}_{\sf phj}.$ (62)

As claimed above, generating the trilinear coupling will be dependent on the
interaction between two coupled ions and a third one. For instance, this can
be achieved by using two pairs of noncopropagating laser beams along the
radial directions of frequencies

$\displaystyle\Omega_{j,L_{r}}=\omega_{0}-\nu-\Delta_{j}(t)$ (63)
$\displaystyle\Omega_{j,L_{b}}=\omega_{0}+\nu-\Delta_{j}(t)$ (64)

that generate a spin-dependent force at frequency $\nu$.

Besides, using the weak coupling assumption, one can legitimately apply the
optical rotating-wave approximation and, as a result, the interacting part of
our system reads now as

$\displaystyle{H}_{\sf I}=$
$\displaystyle\sum_{j}\Delta_{j}\sigma_{j}^{z}+\Omega_{x}e^{ik\widehat{x}_{1}}\cos(\nu
t)\left(\sigma_{1}^{+}e^{i\phi_{1}}+\sigma_{1}^{-}e^{-i\phi_{1}}\right)+\Omega_{x}e^{ik\widehat{x}_{2}}\cos(\nu
t)\left(\sigma_{2}^{+}e^{i\phi_{2}}+\sigma_{2}^{-}e^{-i\phi_{2}}\right)$
$\displaystyle+\Omega_{z}e^{ik\widehat{z}_{3}}\cos(\nu
t)\left(\sigma_{3}^{+}e^{i\phi_{3}}+\sigma_{3}^{-}e^{-i\phi_{3}}\right)+\sum_{j}\Omega_{j}\left(\sigma_{j}^{+}e^{i\theta_{j}}+\sigma_{j}^{-}e^{-i\theta_{j}}\right)$
(65) $\displaystyle+\Omega_{\alpha}e^{ik\widehat{z}_{3}}\sin(\nu
t)\left(\sigma_{1}^{+}e^{i{\varphi_{1}}}+\sigma_{1}^{-}e^{-i{\varphi_{1}}}\right)\left(\sigma_{2}^{+}e^{i{\varphi_{2}}}+\sigma_{2}^{-}e^{-i{\varphi_{2}}}\right)$

where $\Omega_{x},\Omega_{z},\Omega_{j},\Omega_{\alpha}$ are the Rabi
frequencies, $\phi_{j},\theta_{j}$ are the laser phases, $\varphi_{1,2}$ are
the phases resulted from trilinear interaction, and the spatial arguments

$\displaystyle k\widehat{x}_{1}$ $\displaystyle=$
$\displaystyle\sum_{n}\eta_{1,n}(\widehat{a}_{n}^{\dagger}e^{i\Omega_{n}t}+\widehat{a}_{n}e^{-i\Omega_{n}t})$
(66) $\displaystyle k\widehat{x}_{2}$ $\displaystyle=$
$\displaystyle\sum_{n}\eta_{2,n}(\widehat{b}_{n}^{\dagger}e^{i\Omega_{n}t}+\widehat{b}_{n}e^{-i\Omega_{n}t})$
(67) $\displaystyle k\widehat{z}_{3}$ $\displaystyle=$
$\displaystyle\sum_{n}\eta_{3,n}(\widehat{c}_{n}^{\dagger}e^{i\Omega_{n}t}+\widehat{c}_{n}e^{-i\Omega_{n}t})$
(68)

are involving the Lamb-Dicke parameters

$\displaystyle\eta_{j,n}=b_{j,n}k\sqrt{\hbar/2M\Omega_{n}}$ (69)

with $b_{j,n}$ are the normal mode transformation matrix for the $j^{th}$ ion.
Hence, since the dimensionless parameter $\eta_{j,n}$ is small, we can make
the Lamb-Dicke approximation, $\Delta k\langle\widehat{x}_{1}\rangle\ll 1$,
$\Delta k\langle\widehat{x}_{2}\rangle\ll 1$, $\Delta
k\langle\widehat{z}_{3}\rangle\ll 1$, to end up with

$\displaystyle{H}_{\sf I}=$
$\displaystyle\sum_{j}\Delta_{j}\sigma_{j}^{z}+\sum_{n}J_{1,n}\cos(\nu
t)\left(\sigma_{1}^{+}e^{i\phi_{1}}+\sigma_{1}^{-}e^{-i\phi_{1}}\right)\left(\widehat{a}_{n}^{\dagger}e^{i\Omega_{n}t}+\widehat{a}_{n}e^{-i\Omega_{n}t}\right)$
$\displaystyle+\sum_{j}\Omega_{j}\left(\sigma_{j}^{+}e^{i\theta_{j}}+\sigma_{j}^{-}e^{-i\theta_{j}}\right)+\sum_{n}J_{2,n}\cos(\nu
t)\left(\sigma_{2}^{+}e^{i\phi_{2}}+\sigma_{2}^{-}e^{-i\phi_{2}}\right)\left(\widehat{b}_{n}^{\dagger}e^{i\Omega_{n}t}+\widehat{b}_{n}e^{-i\Omega_{n}t}\right)$
$\displaystyle+\sum_{n}J_{3,n}\cos(\nu
t)\left(\sigma_{3}^{+}e^{i\phi_{3}}+\sigma_{3}^{-}e^{-i\phi_{3}}\right)\left(\widehat{c}_{n}^{\dagger}e^{i\Omega_{n}t}+\widehat{c}_{n}e^{-i\Omega_{n}t}\right)$
(70) $\displaystyle+\sum_{n}h_{n}\sin(\nu
t)\left(\sigma_{1}^{+}e^{i{\varphi_{1}}}+\sigma_{1}^{-}e^{-i{\varphi_{1}}}\right)\left(\sigma_{2}^{+}e^{i{\varphi_{2}}}+\sigma_{2}^{-}e^{-i{\varphi_{2}}}\right)\left(\widehat{c}_{n}^{\dagger}e^{i\Omega_{n}t}+\widehat{c}_{n}e^{-i\Omega_{n}t}\right)$

where $J_{1,n}=\eta_{1,n}\Omega_{x}$, $J_{2,n}=\eta_{2,n}\Omega_{x}$ and
$J_{3,n}=\eta_{3,n}\Omega_{z}$ are the spin-phonon coupling and
$h_{n}=\eta_{3,n}\Omega_{\alpha}$ is the trilinear coupling. Moreover, during
a slow dynamics, the beat-note frequency $\nu$ isn’t resonant with any radial
vibration mode, i.e. $|\Omega_{n}-\nu|\gg J_{j,n},h_{n}$. Additionally, if the
phonons are virtually excited, then they should be eliminated from the
dynamics, and as a consequence, the spin states in different sites become
coupled to each other. For the different three sites $j^{th}$, $p^{th}$ and
$q^{th}$, we have

$\displaystyle{H}_{\sf I}=$
$\displaystyle\Delta_{j}\sigma_{j}^{z}+\Delta_{p}\sigma_{p}^{z}+\Delta_{q}\sigma_{q}^{z}+\Omega_{j}\left(\sigma_{j}^{+}e^{i\theta_{j}}+\sigma_{j}^{-}e^{-i\theta_{j}}\right)+\Omega_{p}\left(\sigma_{p}^{+}e^{i\theta_{p}}+\sigma_{p}^{-}e^{-i\theta_{p}}\right)$
$\displaystyle+\Omega_{q}\left(\sigma_{q}^{+}e^{i\theta_{q}}+\sigma_{q}^{-}e^{-i\theta_{q}}\right)+J_{1}\left(\sigma_{j}^{+}e^{i\phi_{j}}+\sigma_{j}^{-}e^{-i\phi_{j}}\right)\left(\sigma_{p}^{+}e^{i\phi_{p}}+\sigma_{p}^{-}e^{-i\phi_{p}}\right)$
(71)
$\displaystyle+J_{2}\left(\sigma_{p}^{+}e^{i\phi_{p}}+\sigma_{p}^{-}e^{-i\phi_{p}}\right)\left(\sigma_{q}^{+}e^{i\phi_{q}}+\sigma_{q}^{-}e^{-i\phi_{q}}\right)+J_{3}\left(\sigma_{j}^{+}e^{i\phi_{j}}+\sigma_{j}^{-}e^{-i\phi_{j}}\right)\left(\sigma_{q}^{+}e^{i\phi_{q}}+\sigma_{q}^{-}e^{-i\phi_{q}}\right)$
$\displaystyle+J\left(\sigma_{j}^{+}e^{i{\varphi_{j}}}+\sigma_{j}^{-}e^{-i{\varphi_{j}}}\right)\left(\sigma_{p}^{+}e^{i{\varphi_{p}}}+\sigma_{p}^{-}e^{-i{\varphi_{p}}}\right)\left(\sigma_{q}^{+}e^{i{\varphi_{q}}}+\sigma_{q}^{-}e^{-i{\varphi_{q}}}\right)$

where the couplings between two ions are given by

$\displaystyle
J_{1}=\sum_{n}J_{j,n}J_{p,n}\frac{1}{\nu^{2}-\Omega_{n}^{2}},\qquad
J_{2}=\sum_{n}J_{p,n}J_{q,n}\frac{1}{\nu^{2}-\Omega_{n}^{2}},\qquad
J_{3}=\sum_{n}J_{j,n}J_{q,n}\frac{1}{\nu^{2}-\Omega_{n}^{2}}$ (72)

and that of the trilinear coupling between three ions

$\displaystyle J=\sum_{n}J_{j,n}J_{p,n}h_{n}\frac{1}{\nu^{2}-\Omega_{n}^{2}}.$
(73)

At this level, it is clearly seen that one can realize the circulant
Hamiltonian $H_{\sf cir}^{(1)}(t)$ (16) by adjusting the coupling parameters.

##  5 Numerical analysis

In what follows, we choose the following time modulations of the couplings
$J_{1}(t)$, $J(t)$ and the detunings $\Delta_{j}(t)$ for the gate
implementation:

$\displaystyle J_{1}(t)=J_{01}\sin^{2}(\omega^{\prime}t)$ (74) $\displaystyle
J(t)=J_{0}\sin^{2}(\omega^{\prime}t)$ (75)
$\displaystyle\Delta_{j}(t)=\Delta_{j}\cos^{2}(\omega^{\prime}t)$ (76)

where the characteristic parameter $\omega^{\prime}$ controls the adiabaticity
of the transition and the interaction time $t$ varies as $t\in[0,t_{max}]$
with $t_{max}=\frac{\pi}{2\omega{{}^{\prime}}}$. This time dependence
guarantees the following conditions

$\displaystyle\Delta_{1,2,3}(0)\gg
J(0),J_{1}(0),\qquad\Delta_{1,2,3}(t_{max})\ll J(t_{max}),J_{1}(t_{max}).$
(77)

The adiabatic transition to Fourier modes can be carried out without using the
detuning $\Delta_{j}$. In fact, we can simply vary the Rabi frequencies to
finally get the Fourier modes, such as

$\displaystyle J(t)=J_{0}\sin^{2}(\omega^{\prime}t)$ (78) $\displaystyle
J_{1}(t)=J_{01}\sin^{2}(\omega^{\prime}t)$ (79)
$\displaystyle\Omega_{2}(t)=J_{01}+\Upsilon_{0}\cos^{2}(\omega^{\prime}t)$
(80)
$\displaystyle\Omega_{3}(t)=J_{0}+\Upsilon^{\prime}_{0}\cos^{2}(\omega^{\prime}t)$
(81)

with $\Upsilon_{0}$ and $\Upsilon^{\prime}_{0}$ are the adding amplitudes for
the control of the adiabaticity of transition in the second and third qubits,
respectively.

###  5.1 Eigenfrequencies

We numerically show in Figure 3 the eigenfrequencies
$\lambda_{\pm}(t),\delta_{\pm}(t),\mu_{\pm}(t),\gamma_{\pm}(t)$ of the
Hamiltonian ${H}(t)$ (23) versus time under suitable choices of the coupling
parameters and detunings. As expected, all eigenfrequencies are separated from
each other, which entails, in its turn the suppression of any transition to a
superposition of eigenstates. The degeneracy of the energies should be avoided
during the simulated time, and this is due to the fact that degeneracy entails
the prevention of the gate implementation at hand. The gap between the
eigenvalues decreases during the simulation time. To avoid their degeneracy
during the evolution, we have introduced the detuning frequencies
$\Delta_{j}$. Additionally, we mention that the amplitude couplings $J_{0}$
and $J_{01}$ are important to prevent the degeneracy at final time $t_{max}$.

Figure 3: (color online) Eigenfrequencies of the Hamiltonian ${H}(t)$ (23) as
a function of the time. The parameters are chosen as $J_{0}/2\pi=1$ kHz,
$J_{01}/2\pi=2$ kHz, $\Delta_{1}/2\pi=120$ kHz, $\Delta_{2}/2\pi=60$ kHz,
$\Delta_{3}/2\pi=30$ kHz, $\varphi=\dfrac{\pi}{2}$,
$\omega^{\prime}/2\pi=0.15$ kHz.

###  5.2 Gate fidelity

Gate fidelity is a tool to compare how close two gates, or more generally
operations, are to each other [33]. In other words, it expresses the
probability that one state will pass a test to identify itself as the other
one. We recall that fidelities higher than 99.99$\%$ for a single-qubit gate
and 99.9 $\%$ for an entangling gate in a two-ion crystal have been developed
in [34, 35, 36]. Generally, for the theoretical density matrix $\rho_{0}$ and
reconstructed density matrix $\rho$, it is defined by

$\displaystyle
F(\rho_{0},\rho)=\left(\Tr\sqrt{\sqrt{\rho}_{0}\rho\sqrt{\rho}_{0}}\right)^{2}.$
(82)

By applying the Uhlmann theorem [37], (82) can take a simple form

$\displaystyle F(\rho_{0},\rho)=\left|\langle\psi_{0}|\psi\rangle\right|^{2}.$
(83)

with $\psi_{0}$ and $\psi$ are theoretical and reconstructed purified state
vectors, respectively. As for our system, we have [20]

$\displaystyle F_{\sf
Gate}(t)=\dfrac{1}{16}\left|\sum_{s_{1},s_{2},s_{3}}\bra{s_{1}s_{2}s_{3}}\mathcal{G}^{+}\mathcal{G}^{\prime}(t)\ket{s_{1}s_{2}s_{3}}\right|^{2}$
(84)

where $s_{j}=\uparrow_{j},\downarrow_{j}$, $\mathcal{G}$ is the three-qubit
QFT (55) and $\mathcal{G}^{\prime}(t)$ is the real transform. In Figure 4, we
present the gate fidelity versus the evolution time by choosing the detunings
$\Delta_{1,2,3}$ such that the adiabatic phases are given in (54). The unitary
propagator $\mathcal{G}^{\prime}(t)$ converges to $\mathcal{G}$ as time
progresses. We notice that for a nonlinear coupling $J_{0}=1$ kHz and
$t=0.4875$ ms, the gate reaches a high fidelity ($96$⁒).

Figure 4: (color online) The gate fidelity is calculated using the Hamiltonian
${H}(t)$ (23) in a numerical simulation. The parameters are chosen as
$J_{0}/2\pi=J_{01}/2\pi=1$ kHz, $\Delta_{1}/2\pi=20$ kHz, $\Delta_{2}/2\pi=10$
kHz, $\Delta_{3}/2\pi=6$ kHz, $\varphi=\dfrac{\pi}{2}$ and
$\omega^{\prime}=0.505$ kHz.

By using the Hamiltonian (B.2) together with the quantum Fourier states
(3-10), one can end up with the fidelity of the adiabatic transitions between
the rotating computational spin states
$\ket{s^{\prime}_{1}s^{\prime}_{2}s^{\prime}_{3}}(s^{\prime}_{j}=\pm_{j})$

$\displaystyle F_{\sf
ad}(t)=\dfrac{1}{16}\left|\sum_{i=0}^{i=7}\bra{\psi_{i}}\ket{\Lambda_{i}}\right|^{2}$
(85)

and more explicitly, we have

$\displaystyle F_{\sf ad}(t)=$
$\displaystyle\dfrac{1}{16}\left|\bra{\psi_{0}}\ket{\Lambda_{0}}+\bra{\psi_{1}}\ket{\Lambda_{1}}+\bra{\psi_{2}}\ket{\Lambda_{2}}+\bra{\psi_{3}}\ket{\Lambda_{3}}+\bra{\psi_{4}}\ket{\Lambda_{4}}+\bra{\psi_{5}}\ket{\Lambda_{5}}+\bra{\psi_{6}}\ket{\Lambda_{6}}+\bra{\psi_{7}}\ket{\Lambda_{7}}\right|^{2}.$
(86)

Figure 5: (color online) Fidelity of adiabatic transition with
$J_{0}/2\pi=2.1$ kHz, $\Upsilon_{0}/2\pi=1.9$ kHz, $J_{01}/2\pi=2.4$ kHz,
$\Upsilon^{\prime}_{0}/2\pi=2$ kHz, $\varphi=\dfrac{\pi}{4}$ and
$\omega^{\prime}/2\pi=0.3$ kHz.

Figure 5 presents the good fidelity of the adiabatic transition within a
shorter interaction time, $t_{max}=0.835$ ms. It is clearly seen that our
results show the possibility for obtaining high fidelity ($71$⁒).

###  5.3 Creation of entangled states

To create entangled states, one has to suitably prepare the initial state in a
superposition of spin states, which is due to the fact that the action of the
QFT on the computational basis creates a superposition, but they are not
entangled. For concreteness, we assume that the system is initially prepared
in the following state:

$\displaystyle\ket{\psi(0)}=\frac{1}{2}\ket{\downarrow_{1}}(e^{i\alpha_{1}}\ket{\downarrow_{2}\downarrow_{3}}+e^{i\alpha_{2}}\ket{\downarrow_{2}\uparrow_{3}}+e^{i\alpha_{3}}\ket{\uparrow_{2}\downarrow_{3}}+e^{i\alpha_{4}}\ket{\uparrow_{2}\uparrow_{3}}).$
(87)

Performing our three qubit gates, we obtain the entangled state

$\displaystyle\ket{\psi(0)}\longrightarrow\ket{\psi(t_{f})}=\frac{1}{2}(\ket{\psi_{0}}+\ket{\psi_{1}}+\ket{\psi_{2}}+\ket{\psi_{3}}).$
(88)

Let us emphasize that by rotating the initially prepared state such as

$\displaystyle\ket{\psi(0)}=\frac{1}{2}\ket{-_{1}}(e^{-i\beta_{0}}\ket{-_{2}-_{3}}+e^{-i\beta_{1}}\ket{-_{2}+_{3}}+e^{-i\beta_{2}}\ket{+_{2}-_{3}}+e^{-i\beta_{3}}\ket{+_{2}+_{3}})$
(89)

we end up with the same transformed entangled state (88). Therefore, the
fidelity of the creation of the entangled state is defined by

$\displaystyle
F(t)=\dfrac{1}{2}|\bra{\psi(t_{f})}(e^{-i\beta_{0}}\ket{\Lambda_{0}(t)}+e^{-i\beta_{1}}\ket{\Lambda_{1}(t)}+e^{-i\beta_{2}}\ket{\Lambda_{2}(t)}+e^{-i\beta_{3}}\ket{\Lambda_{3}(t)})|^{2}.$
(90)

By adjusting the parameters $\omega^{\prime}$, $J_{01}$ and $J_{0}$ one can
achieve high fidelity in the creation of entangled states as presented in
Figure 6-A, Figure 6-B and Figure 6-C.

Figure 6: (color online) The fidelity of the entangled state is calculated
from the numerical simulation of the Hamiltonian (B.2). (A):
$\Upsilon_{0}=1.8$ kHz,$\Upsilon^{\prime}_{0}=1.7$ kHz, $J_{01}=2.1$ kHz,
$J_{0}=2.3$ kHz and the gate time $t=0.31$ ms. (B): The same values with
$\omega^{\prime}/2\pi=0.5$ kHz and vary the coupling strength $J_{01}$. (C):
The same values in (A) with $\omega^{\prime}/2\pi=0.605$ kHz and vary the
coupling strength $J_{0}$.

##  6 Short-cut to adiabaticity

Now we add an auxiliary interaction ${H}_{\sf CD}(t)$ (counter-driving-field)
[38, 22] to the reference Hamiltonian ${H}^{(1)}(t)$ (B.2) in order to
suppress the non-adiabatic transitions and reduce the gate time. As a result,
the Hamiltonian will take the following form:

${H}_{\sf T}(t)={H}^{(1)}(t)+{H}_{\sf CD}(t)$ (91)

such that the interaction is

$\displaystyle{H}_{\sf
CD}(t)=i\hbar\sum_{i=0}^{7}\ket{\partial_{t}\Lambda_{i}(t)}\bra{\Lambda_{i}(t)}$
(92)

and (B.11-B.18) show the time-dependent eigenvectors $\ket{\Lambda_{i}(t)}$ of
${H}^{(1)}(t)$. After some algebra, we obtain

$\displaystyle{H}_{\sf
CD}(t)=-\partial_{t}\kappa(t)\begin{pmatrix}0&0&0&0&1&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\end{pmatrix}.$ (93)

Using the time-dependent coupling parameters (78-81) and the eigenvectors
(B.11-B.18), we explicitly determine the counter-driving-field

$\displaystyle{\partial_{t}\kappa(t)=\dfrac{1}{2}\left(\dfrac{\omega^{\prime}J_{01}(J_{01}+\Upsilon_{0})\sin(2\omega^{\prime}t)}{J_{01}^{2}\sin^{4}(\omega^{\prime}t)+[\Upsilon_{0}\sin^{2}(\omega^{\prime}t)-(J_{01}+\Upsilon_{0})]^{2}}+\dfrac{\omega^{\prime}J_{0}(J_{0}+\Upsilon_{0}^{\prime})\sin(2\omega^{\prime}t)}{J_{0}^{2}\sin^{4}(\omega^{\prime}t)+[\Upsilon_{0}^{\prime}\sin^{2}(\omega^{\prime}t)-(J_{0}+\Upsilon_{0}^{\prime})]^{2}}\right).}$
(94)

In  Figure 7, we show the shape of the counter-driving field (94) as a
function of the time and by varying $\omega^{\prime}$, $J_{01}$ and $J_{0}$.
The counter-driving term should be zero at $t=0$, because the system starts in
the rotational computational spin states. We mention also that at $t_{max}$,
the system ends up with the Fourier modes.

Figure 7: (color online) Behavior of a counter-driving field with
$\Upsilon_{0}/2\pi=0.5$ kHz and $\Upsilon^{\prime}_{0}/2\pi=2$ kHz, while the
remaining parameters are chosen as $J_{01}=1.5$ kHz, $J_{0}=1$ kHz (blue
line), $J_{01}=1.9$ kHz, $J_{0}=1.3$ kHz (cyan line),$J_{01}=2$ kHz,
$J_{0}=1.7$ kHz (red line).

##  7 Conclusion

We generalized the work done by Ivanov and Vitanov [20] dealing with two-qubit
quantum gates and entanglement protected by circulant symmetry to a system of
three-qubit quantum gates. In fact, we have constructed a discrete system
based on three qubits emerging in a magnetic field. A special symmetry called
circulant can be obtained only by adjusting the Rabi frequencies and the
coupling parameters characterizing our system. We have shown that our
eigenvectors do not depend on the magnitude of the physical parameters, which
entails the protection of entanglement. These eigenvectors lead to obtaining
the quantum Fourier transform (QFT) modes, which imply the realization of a
QFT gate. To discuss the implementation of the gate, the eigenfrequencies
should be non-degenerate. For this aim, we have added an Hamiltonian
$H_{0}(t)$ that breaks the circulant symmetry and favors the adiabatic
transition process.

Subsequently, instead of adding an energy offset, we have shown that it is
possible to control the transition by using only the Rabi frequencies. As a
result, the gate scheme with the short-cut to adiabaticity has been discussed,
and therefore we have found the suppression of the non-adiabatic transition
and the acceleration of the gate. Additionally, in the framework of the
trapped ions, we have suggested a possible physical implementation of the
constructed Hamiltonian. By assuming a particular sinusoidal modulation,
several fidelities are discussed, and the results show the possibility to
achieve high fidelities only by adjusting the physical parameters. The
physical realization of a three-qubit QFT is the key subroutine in several
quantum algorithms. Then, it turns out that our present three-qubit gates can
significantly reduce the number of gates in a quantum algorithm.

In this work, we have added the trilinear coupling, which is necessary to
retain the circulant symmetry in a three-qubit system. Indeed, a three-qubit
system with only bilinear interaction terms, as done in [18], does not lead to
the wanted circulant symmetry. Additionally, constructing a circulant
Hamiltonian based on $N$-qubits is not obvious. Indeed, the Hamiltonian
demands multilinear interaction terms that are not easily realizable.
Moreover, discussing the adiabatic transition and accelerating the gate using
a shortcut to adiabacity will be hard.

##  Appendix A Eigenfrequencies of ${H}(t)$ with energy offset

As for ${H}(t)$ (23) including the circulant Hamiltonian (16) and the energy
offset (22), we show that the corresponding eigenfrequencies are given by

$\displaystyle\lambda_{\pm}=\pm\sqrt{\left|-\frac{A}{4}-S+\frac{1}{2}\sqrt{\left|-4S^{2}-2p+\frac{q}{S}\right|}\right|}$
(A.1)
$\displaystyle\delta_{\pm}=\pm\sqrt{\left|-\frac{A}{4}-S-\frac{1}{2}\sqrt{\left|-4S^{2}-2p+\frac{q}{S}\right|}\right|}$
(A.2)
$\displaystyle\mu_{\pm}=\pm\sqrt{\left|-\frac{A}{4}+S+\frac{1}{2}\sqrt{\left|-4S^{2}-2p-\frac{q}{S}\right|}\right|}$
(A.3)
$\displaystyle\gamma_{\pm}=\pm\sqrt{\left|-\frac{A}{4}+S-\frac{1}{2}\sqrt{\left|-4S^{2}-2p-\frac{q}{S}\right|}\right|}$
(A.4)

where we have set

$\displaystyle p=\frac{1}{8}\left(8B-3\right)$ (A.5) $\displaystyle
q=\frac{1}{8}\left(-1+4B+8C\right)$ (A.6) $\displaystyle
S=\frac{1}{2}\sqrt{\frac{1}{3}\left|-2p+\left(Q+\frac{\Delta_{0}}{Q}\right)\right|}$
(A.7) $\displaystyle\Delta_{0}=B^{2}+3C+12D$ (A.8) $\displaystyle
Q=\sqrt[3]{\frac{1}{2}\left|\Delta+\sqrt{\left|\Delta^{2}-4\Delta_{0}^{3}\right|}\right|}$
(A.9) $\displaystyle\Delta=2B^{3}+9BC+27D+27C^{2}-72BD$ (A.10)

and the associated quantities are

$\displaystyle A=$
$\displaystyle-16J^{2}-4\Delta_{1}^{2}-4\Delta_{2}^{2}-4\Delta_{3}^{2}-8J_{1}^{2}$
$\displaystyle B=$ $\displaystyle
32{J}^{2}{\Delta_{{1}}}^{2}+32\,{J}^{2}{\Delta_{{2}}}^{2}+48{J}^{2}{\Delta_{{3}}}^{2}+128{J}^{2}{J_{{1}}}^{2}+6{\Delta_{{1}}}^{4}+4\,{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{2}+4{\Delta_{{1}}}^{2}{\Delta_{{3}}}^{2}$
$\displaystyle+16{\Delta_{{1}}}^{2}{J_{{1}}}^{2}+6\,{\Delta_{{2}}}^{4}+4{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{2}+24{\Delta_{{2}}}^{2}{J_{{1}}}^{2}+6{\Delta_{{3}}}^{4}+8{\Delta_{{3}}}^{2}{J_{{1}}}^{2}+16{J_{{1}}}^{4}$
$\displaystyle C=$
$\displaystyle-16{J}^{2}{\Delta_{{1}}}^{4}-32{J}^{2}{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{2}-128{J}^{2}{\Delta_{{1}}}^{2}{J_{{1}}}^{2}-16{J}^{2}{\Delta_{{2}}}^{4}-128{J}^{2}{\Delta_{{2}}}^{2}{J_{{1}}}^{2}$
$\displaystyle-48{J}^{2}{\Delta_{{3}}}^{4}-256\,{J}^{2}{J_{{1}}}^{4}-4{\Delta_{{1}}}^{6}+4{\Delta_{{1}}}^{4}{\Delta_{{2}}}^{2}+4{\Delta_{{1}}}^{4}{\Delta_{{3}}}^{2}-8{\Delta_{{1}}}^{4}{J_{{1}}}^{2}+4{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{4}$
$\displaystyle-40{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{2}+4{\Delta_{{1}}}^{2}{\Delta_{{3}}}^{4}-32{\Delta_{{1}}}^{2}{\Delta_{{3}}}^{2}{J_{{1}}}^{2}-4\,{\Delta_{{2}}}^{6}+4{\Delta_{{2}}}^{4}{\Delta_{{3}}}^{2}-24\,{\Delta_{{2}}}^{4}{J_{{1}}}^{2}$
$\displaystyle+4{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{4}+16\,{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{2}{J_{{1}}}^{2}-32\,{\Delta_{{2}}}^{2}{J_{{1}}}^{4}-4\,{\Delta_{{3}}}^{6}+8\,{\Delta_{{3}}}^{4}{J_{{1}}}^{2}-32\,{\Delta_{{3}}}^{2}{J_{{1}}}^{4}$
$\displaystyle D=$ $\displaystyle
4\,{\Delta_{{1}}}^{4}{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{2}+4\,{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{4}{\Delta_{{3}}}^{2}+4\,{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{4}-4\,{\Delta_{{1}}}^{6}{\Delta_{{2}}}^{2}-4\,{\Delta_{{1}}}^{6}{\Delta_{{3}}}^{2}+6\,{\Delta_{{1}}}^{4}{\Delta_{{2}}}^{4}$
$\displaystyle+6\,{\Delta_{{1}}}^{4}{\Delta_{{3}}}^{4}-4\,{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{6}-4\,{\Delta_{{1}}}^{2}{\Delta_{{3}}}^{6}-4\,{\Delta_{{2}}}^{6}{\Delta_{{3}}}^{2}+6\,{\Delta_{{2}}}^{4}{\Delta_{{3}}}^{4}-4\,{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{6}+16\,{\Delta_{{2}}}^{4}{J_{{1}}}^{4}$
$\displaystyle-8{\Delta_{{3}}}^{6}{J_{{1}}}^{2}+16\,{\Delta_{{3}}}^{4}{J_{{1}}}^{4}+16\,{J}^{2}{\Delta_{{3}}}^{6}+8\,{\Delta_{{2}}}^{6}{J_{{1}}}^{2}+16\,{\Delta_{{1}}}^{2}{\Delta_{{3}}}^{4}{J_{{1}}}^{2}-24\,{\Delta_{{2}}}^{4}{\Delta_{{3}}}^{2}{J_{{1}}}^{2}$
$\displaystyle+24\,{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{4}{J_{{1}}}^{2}-32\,{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{2}{J_{{1}}}^{4}+8{\Delta_{{1}}}^{4}{\Delta_{{2}}}^{2}{J_{{1}}}^{2}-8\,{\Delta_{{1}}}^{4}{\Delta_{{3}}}^{2}{J_{{1}}}^{2}-16\,{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{4}{J_{{1}}}^{2}$
$\displaystyle-128\,{J}^{2}{\Delta_{{3}}}^{4}{J_{{1}}}^{2}+256\,{J}^{2}{\Delta_{{3}}}^{2}{J_{{1}}}^{4}+16\,{J}^{2}{\Delta_{{1}}}^{4}{\Delta_{{3}}}^{2}-32\,{J}^{2}{\Delta_{{1}}}^{2}{\Delta_{{3}}}^{4}+16\,{J}^{2}{\Delta_{{2}}}^{4}{\Delta_{{3}}}^{2}$
$\displaystyle-32\,{J}^{2}{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{4}+32\,{J}^{2}{\Delta_{{1}}}^{2}{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{2}+128\,{J}^{2}{\Delta_{{1}}}^{2}{\Delta_{{3}}}^{2}{J_{{1}}}^{2}+128\,{J}^{2}{\Delta_{{2}}}^{2}{\Delta_{{3}}}^{2}{J_{{1}}}^{2}+{\Delta_{{1}}}^{8}+{\Delta_{{2}}}^{8}+{\Delta_{{3}}}^{8}.$

##  Appendix B Energy spectrum of $H^{(1)}(t)$

The Hamiltonian $H^{(1)}(t)$ with $\Omega_{2}(t)$ (i.e. $J_{1}=\Omega_{2}(t)$
and $J=\Omega_{3}(t)$ are not always respected) has the following form:

$\displaystyle{H}^{(1)}(t)=$ $\displaystyle
J_{1}(\sigma_{1}^{+}+\sigma_{1}^{-})(\sigma_{2}^{+}e^{-i\varphi}+\sigma_{2}^{-}e^{i\varphi})+\Omega_{3}(t)(\sigma_{2}^{+}+\sigma_{2}^{-})(\sigma_{3}^{+}e^{-i\varphi}+\sigma_{3}^{-}e^{i\varphi})$
$\displaystyle+\Omega_{3}(t)(\sigma_{1}^{+}+\sigma_{1}^{-})(\sigma_{3}^{+}e^{i\varphi}+\sigma_{3}^{-}e^{-i\varphi})+\Omega_{2}(t)(\sigma_{2}^{+}e^{i\varphi}+\sigma_{2}^{-}e^{-i\varphi})$
(B.1)
$\displaystyle+\Omega_{3}(t)(\sigma_{3}^{+}e^{i\varphi}+\sigma_{3}^{-}e^{-i\varphi})+J(\sigma_{1}^{+}+\sigma_{1}^{-})(\sigma_{2}^{+}+\sigma_{2}^{-})(\sigma_{3}^{+}e^{-i\varphi}+\sigma_{3}^{-}e^{i\varphi})$

and we have in the matrix notation

$\displaystyle{H}^{(1)}(t)=$ (B.2)
$\displaystyle\begin{pmatrix}0&\Omega_{3}(t)e^{i\varphi}&\Omega_{2}(t)e^{i\varphi}&\Omega_{3}(t)e^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}\\\
\Omega_{3}(t)e^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}&\Omega_{2}(t)e^{i\varphi}&\Omega_{3}(t)e^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{-i\varphi}\\\
\Omega_{2}(t)e^{-i\varphi}&\Omega_{3}(t)e^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}&J_{1}e^{i\varphi}&Je^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}\\\
\Omega_{3}(t)e^{i\varphi}&\Omega_{2}(t)e^{-i\varphi}&\Omega_{3}(t)e^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{i\varphi}&\Omega_{3}(t)e^{-i\varphi}&0\\\
0&\Omega_{3}(t)e^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}&\Omega_{2}(t)e^{i\varphi}&\Omega_{3}(t)e^{-i\varphi}\\\
\Omega_{3}(t)e^{-i\varphi}&0&Je^{i\varphi}&J_{1}e^{-i\varphi}&Je^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}&\Omega_{2}(t)e^{i\varphi}\\\
J_{1}e^{i\varphi}&Je^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}&\Omega_{2}(t)e^{-i\varphi}&\Omega_{3}(t)e^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}\\\
Je^{i\varphi}&J_{1}e^{i\varphi}&\Omega_{3}(t)e^{-i\varphi}&0&\Omega_{3}(t)e^{i\varphi}&\Omega_{2}(t)e^{-i\varphi}&\Omega_{3}(t)e^{-i\varphi}&0\end{pmatrix}.$

We choose $\varphi=\dfrac{\pi}{4}$ for simplicity and show that the time-
dependent eigenfrequencies ${H}^{(1)}(t)$ (B.2) are given by

$\displaystyle\Lambda_{0}(t)=\sqrt{(\Omega_{3}-J)^{2}+J_{1}^{2}+\Omega_{2}^{2}+\sqrt{2(\Omega_{3}-J)^{2}(J_{1}-\Omega_{2})^{2}}}$
(B.3)
$\displaystyle\Lambda_{1}(t)=-\sqrt{(\Omega_{3}-J)^{2}+J_{1}^{2}+\Omega_{2}^{2}+\sqrt{2(\Omega_{3}-J)^{2}(J_{1}-\Omega_{2})^{2}}}$
(B.4)
$\displaystyle\Lambda_{2}(t)=\sqrt{(\Omega_{3}-J)^{2}+J_{1}^{2}+\Omega_{2}^{2}-\sqrt{2(\Omega_{3}-J)^{2}(J_{1}-\Omega_{2})^{2}}}$
(B.5)
$\displaystyle\Lambda_{3}(t)=-\sqrt{(\Omega_{3}-J)^{2}+J_{1}^{2}+\Omega_{2}^{2}-\sqrt{2(\Omega_{3}-J)^{2}(J_{1}-\Omega_{2})^{2}}}$
(B.6)
$\displaystyle\Lambda_{4}(t)=\sqrt{5\Omega_{3}^{2}+2J\Omega_{3}+J^{2}+J_{1}^{2}+\Omega_{2}^{2}+\sqrt{2\Omega_{3}^{2}(9\Omega_{2}^{2}+2J_{1}\Omega_{2}+9J_{1}^{2})+2J(J_{1}+\Omega_{2})^{2}(2\Omega_{3}+J)}}$
(B.7)
$\displaystyle\Lambda_{5}(t)=-\sqrt{5\Omega_{3}^{2}+2J\Omega_{3}+J^{2}+J_{1}^{2}+\Omega_{2}^{2}+\sqrt{2\Omega_{3}^{2}(9\Omega_{2}^{2}+2J_{1}\Omega_{2}+9J_{1}^{2})+2J(J_{1}+\Omega_{2})^{2}(2\Omega_{3}+J)}}$
(B.8)
$\displaystyle\Lambda_{6}(t)=\sqrt{5\Omega_{3}^{2}+2J\Omega_{3}+J^{2}+J_{1}^{2}+\Omega_{2}^{2}-\sqrt{2\Omega_{3}^{2}(9\Omega_{2}^{2}+2J_{1}\Omega_{2}+9J_{1}^{2})+2J(J_{1}+\Omega_{2})^{2}(2\Omega_{3}+J)}}$
(B.9)
$\displaystyle\Lambda_{7}(t)=-\sqrt{5\Omega_{3}^{2}+2J\Omega_{3}+J^{2}+J_{1}^{2}+\Omega_{2}^{2}-\sqrt{2\Omega_{3}^{2}(9\Omega_{2}^{2}+2J_{1}\Omega_{2}+9J_{1}^{2})+2J(J_{1}+\Omega_{2})^{2}(2\Omega_{3}+J)}}$
(B.10)

and the corresponding eigenvectors can be expressed as

$\displaystyle\ket{\Lambda_{0}(t)}=\dfrac{1}{2\sqrt{2}}$ (B.11)
$\displaystyle\left(e^{-i\alpha(t)}\ket{\downarrow\downarrow\downarrow}+e^{-i\alpha(t)}\ket{\downarrow\downarrow\uparrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\downarrow\uparrow\uparrow}+e^{-i\alpha(t)}\ket{\uparrow\downarrow\downarrow}+e^{-i\alpha(t)}\ket{\uparrow\downarrow\uparrow}+\ket{\uparrow\uparrow\downarrow}+\ket{\uparrow\uparrow\uparrow}\right)$
$\displaystyle\ket{\Lambda_{1}(t)}=\dfrac{1}{2\sqrt{2}}$ (B.12)
$\displaystyle\left(e^{i\alpha(t)}\ket{\downarrow\downarrow\downarrow}+\omega
e^{i\alpha(t)}\ket{\downarrow\downarrow\uparrow}+i\ket{\downarrow\uparrow\downarrow}+i\omega\ket{\downarrow\uparrow\uparrow}-e^{i\alpha(t)}\ket{\uparrow\downarrow\downarrow}-\omega
e^{i\alpha(t)}\ket{\uparrow\downarrow\uparrow}-i\ket{\uparrow\uparrow\downarrow}-i\omega\ket{\uparrow\uparrow\uparrow}\right)$
$\displaystyle\ket{\Lambda_{2}(t)}=\dfrac{1}{2\sqrt{2}}$ (B.13)
$\displaystyle\left(e^{-i\alpha(t)}\ket{\downarrow\downarrow\downarrow}+ie^{-i\alpha(t)}\ket{\downarrow\downarrow\uparrow}-\ket{\downarrow\uparrow\downarrow}-i\ket{\downarrow\uparrow\uparrow}+e^{-i\alpha(t)}\ket{\uparrow\downarrow\downarrow}+ie^{-i\alpha(t)}\ket{\uparrow\downarrow\uparrow}-\ket{\uparrow\uparrow\downarrow}-i\ket{\uparrow\uparrow\uparrow}\right)$
$\displaystyle\ket{\Lambda_{3}(t)}=\dfrac{1}{2\sqrt{2}}$ (B.14)
$\displaystyle\left(e^{i\alpha(t)}\ket{\downarrow\downarrow\downarrow}+i\omega
e^{i\alpha(t)}\ket{\downarrow\downarrow\uparrow}-i\ket{\downarrow\uparrow\downarrow}+\omega\ket{\downarrow\uparrow\uparrow}-e^{i\alpha(t)}\ket{\uparrow\downarrow\downarrow}-i\omega
e^{i\alpha(t)}\ket{\uparrow\downarrow\uparrow}+i\ket{\uparrow\uparrow\downarrow}-\omega\ket{\uparrow\uparrow\uparrow}\right)$
$\displaystyle\ket{\Lambda_{4}(t)}=\dfrac{1}{2\sqrt{2}}$ (B.15)
$\displaystyle\left(e^{-i\alpha(t)}\ket{\downarrow\downarrow\downarrow}-e^{-i\alpha(t)}\ket{\downarrow\downarrow\uparrow}+\ket{\downarrow\uparrow\downarrow}-\ket{\downarrow\uparrow\uparrow}+e^{-i\alpha(t)}\ket{\uparrow\downarrow\downarrow}-e^{-i\alpha(t)}\ket{\uparrow\downarrow\uparrow}+\ket{\uparrow\uparrow\downarrow}-\ket{\uparrow\uparrow\uparrow}\right)$
$\displaystyle\ket{\Lambda_{5}(t)}=\dfrac{1}{2\sqrt{2}}$ (B.16)
$\displaystyle\left(e^{i\alpha(t)}\ket{\downarrow\downarrow\downarrow}-\omega
e^{i\alpha(t)}\ket{\downarrow\downarrow\uparrow}+i\ket{\downarrow\uparrow\downarrow}-i\omega\ket{\downarrow\uparrow\uparrow}-e^{i\alpha(t)}\ket{\uparrow\downarrow\downarrow}+\omega
e^{i\alpha(t)}\ket{\uparrow\downarrow\uparrow}-i\ket{\uparrow\uparrow\downarrow}+i\omega\ket{\uparrow\uparrow\uparrow}\right)$
$\displaystyle\ket{\Lambda_{6}(t)}=\dfrac{1}{2\sqrt{2}}$ (B.17)
$\displaystyle\left(e^{-i\alpha(t)}\ket{\downarrow\downarrow\downarrow}-ie^{-i\alpha(t)}\ket{\downarrow\downarrow\uparrow}-\ket{\downarrow\uparrow\downarrow}+i\ket{\downarrow\uparrow\uparrow}+e^{-i\alpha(t)}\ket{\uparrow\downarrow\downarrow}-ie^{-i\alpha(t)}\ket{\uparrow\downarrow\uparrow}-\ket{\uparrow\uparrow\downarrow}+i\ket{\uparrow\uparrow\uparrow}\right)$
$\displaystyle\ket{\Lambda_{7}(t)}=\dfrac{1}{2\sqrt{2}}$ (B.18)
$\displaystyle\left(e^{i\alpha(t)}\ket{\downarrow\downarrow\downarrow}-i\omega
e^{i\alpha(t)}\ket{\downarrow\downarrow\uparrow}-i\ket{\downarrow\uparrow\downarrow}-\omega\ket{\downarrow\uparrow\uparrow}-e^{i\alpha(t)}\ket{\uparrow\downarrow\downarrow}+i\omega
e^{i\alpha(t)}\ket{\uparrow\downarrow\uparrow}+i\ket{\uparrow\uparrow\downarrow}+\omega\ket{\uparrow\uparrow\uparrow}\right)$

where we have defined

$\displaystyle\alpha(t)=\frac{\pi}{4}-\kappa(t)$ (B.19)
$\displaystyle\tan[\kappa(t)]=\frac{\Omega_{2}(t)}{2J_{1}}+\frac{\Omega_{3}(t)}{2J}.$
(B.20)

## References

  * [1] Paul Benioff, J. Stat. Phys. 22, 563 (1980).
  * [2] R. Feynman, Int. J. Theo. Phys. 21, 467 (1982).
  * [3] S. Lloyd, Science 261, 1569 (1993).
  * [4] Y. S. Weinstein, S. Lloyd, and D. G. Cory, Phys. Rev. Lett. 86, 1889 (2001).
  * [5] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
  * [6] P. W. Shor, SIAM J. Sci. Statist. Comput. 26, 1484 (1997).
  * [7] L. Ruiz-Perez and J. C. Garcia-Escartin, Qua. Inf. Process. 16, 152 (2017).
  * [8] S. A. Daniel and S. Lloyd, Phys. Rev. Lett. 83, 5162 (1999).
  * [9] G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Contemp. Math. 305, 53 (2002).
  * [10] P. J. Davis, Circulant Matrices (Wiley, New York, 1970).
  * [11] Robert M. Gray, Found. Trends Commun. Inf. Theory 2, 155 (2006).
  * [12] G. Dzhelepov, I. Dokuzova, and D. Razpopov, Plovdiv Univ. Paisiǐ Khilendarski Nauchn. Trud. Mat. 38, 17 (2011).
  * [13] X. Y. Jiang and K. Hong, Explicit Determinants of the k-Fibonacci and k-Lucas RSFPLR Circulant Matrix in Codes. In: Y. Yang, M. Ma, and B. Liu, (eds) Information Computing and Applications (ICICA 2013). Communications in Computer and Information Science (Springer, Berlin, Heidelberg, 2013, vol 391), https://doi.org/10.1007/978-3-642-53932-9$\\_$61\.
  * [14] M. Muzychuk, Proc. London Math. Soc. 88, 1 (2004).
  * [15] B. Olson, S. Shaw, C. Shi, C. Pierre, and R. G. Parker, Appl. Mech. Rev. 66, 040803 (2014).
  * [16] D. Razpopov, Four-dimensional Riemannian manifolds with two circulant structures. In: Mathematics Education Mathematics, Proceedings of 44-th Spring Conference of UBM, SOK Kamchia, Bulgaria, vol. 44, pp. 179–185 (2015).
  * [17] R. M. Roth and A. Lempel, IEEE Trans. Inform. Theory 36 , 1157 (1990).
  * [18] S. S. Zhou, T. Loke, J. A. Izaac, and J. B. Wang, Qua. Inf. Process. 16, 82 (2017).
  * [19] F. R. Gantmacher, Matrix Theory (Springer, Berlin, 1986).
  * [20] P. A. Ivanov and N. V. Vitanov, Sci. Rep. 10, 5030 (2020).
  * [21] H. Wu, X. Huang, C. Hu, Z. Yang, and S. Zheng, Phys. Rev. A 96, 022321 (2017).
  * [22] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J.-G. Muga, Rev. Mod. Phys. 91, 045001 (2019).
  * [23] A. Rueda, W. Hease, S. Barzanjeh, and J. M. Fink, npj Quantum Information 5, 108 (2019).
  * [24] R. Hablützel, Nonlinear Quantum Optics and Thermodynamics with Three Trapped Ions (Thesis, National University of Singapore 2018).
  * [25] J. K. Pachos, Int. J. Quan. Inf. 4, 541 (2006).
  * [26] T. Hatomura, J. Phys. Soc. Jpn. 86, 094002 (2017).
  * [27] M. Born and V. A. Fock, Zeitschrift für Physik A. 51, 165 (1928).
  * [28] M. V Berry, Proc. R. Soc. Lond. A 392, 45 (1984).
  * [29] Michael R. Hush, Weibin Li, Sam Genway, Igor Lesanovsky, and Andrew D. Armour, Phys. Rev. A 91, 061401(R) (2015).
  * [30] K. Kim, M.-S. Chang, R. Islam, S. Korenblit, L.-M. Duan, and C. Monroe, Phys. Rev. Lett. 103, 120502 (2009).
  * [31] S. X. Wang, Quantum Gates, Sensors, and Systems with Trapped Ions (Thesis, Massachusetts Institute of Technology 2012).
  * [32] Shiqian Ding, Gleb Maslennikov, Roland Hablützel, and Dzmitry Matsukevich, Phys. Rev. Lett. 121, 130502 (2018).
  * [33] Easwar Magesan, Robin Blume-Kohout, and Joseph Emerson, Phys. Rev. A 84, 012309 (2011).
  * [34] C. J. Ballance, T. P. Harty, N. M. Linke, M. A. Sepiol, and D. M. Lucas, Phys. Rev. Lett. 117, 060504 (2016).
  * [35] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. J. Wineland, Phys. Rev. Lett. 117, 060505 (2016).
  * [36] Yukai Wu, Sheng-Tao Wang, and L.-M. Duan, Phys. Rev. A 97, 062325 (2018).
  * [37] Armin Uhlmann, Phys. Rev. A 62, 032307 (2000).
  * [38] Xi Chen, I. Lizuain, A. Ruschhaupt, D. Guéry-Odelin, and J. G. Muga, Phys. Rev. Lett. 105, 123003 (2010).