# Characterization of high fidelity Raman qubits, possessing the Morris-Shore
symmetry

Stancho G. Stanchev and Nikolay V. Vitanov  Department of Physics, St Kliment
Ohridski University of Sofia, 5 James Bourchier blvd, 1164 Sofia, Bulgaria

# Characterization of high-fidelity Raman qubit gates

Stancho G. Stanchev and Nikolay V. Vitanov  Department of Physics, St Kliment
Ohridski University of Sofia, 5 James Bourchier blvd, 1164 Sofia, Bulgaria

###### Abstract

Raman qubits, represented by two ground or metastable quantum states coupled
via an intermediate state, hold some advantages over directly coupled qubits,
most notably much longer radiative lifetimes, shorter gate duration and lower
radiation intensity due to using electric-dipole allowed optical transitions.
They are also relatively simple to implement and control, making them an
attractive option for building quantum gates for quantum computers. In this
work, we present a simple and fast tomographic method to measure the errors of
Raman qubit gates possessing the Morris-Shore dynamic symmetry. The latter
occurs when the qubit states are on two-photon resonance and the driving
fields have the same time dependence. The method is based on repeating the
same gate multiple times, which amplifies the small coherent errors to
sufficiently large values, which can be measured with high accuracy and
precision. Then the (small) gate errors can be determined from the amplified
errors by using the analytical connections between them.

##  I Introduction

Raman qubits — qubits formed of the long-lived end states $\ket{0}$ and
$\ket{1}$ of a three-state quantum system in a chainwise-coupled Raman
configuration $\ket{0}\leftrightarrow\ket{a}\leftrightarrow\ket{1}$ — are a
popular implementation of qubits for quantum technologies [1, 3, 4, 2]. They
are particularly suitable for trapped ions and ultracold atoms, wherein Raman
linkage patterns are ubiquitous [5, 6, 8, 7, 9]. Compared to directly-coupled
qubits they have the advantage of using the electric-dipole allowed
transitions $\ket{0}\leftrightarrow\ket{a}$ and
$\ket{1}\leftrightarrow\ket{a}$ instead of the electric-dipole forbidden
transition $\ket{0}\leftrightarrow\ket{1}$. This allows one to use convenient
optical transitions with much less laser power resulting in faster gates with
negligible light shifts and unwanted couplings [10, 11, 12]. Moreover, the
availability of two fields brings more control parameters and the possibility
to use more sophisticated methods for quantum control, such as composite [13],
optimal-control and shortcut approaches [14, 15, 16, 17, 18]. However, Raman
qubits are more demanding in regard to their control, as now three, rather
than two, states are involved, with the necessity to avoid population leakage
to the auxiliary intermediate state $\ket{a}$. Moreover, the characterization
of the fidelity also requires dealing with three states and hence SU(3)
dynamics instead of SU(2).

Figure 1:  Reduction of a three-state Raman $\Lambda$ system (left) to an
effective two-state system (right) by the Morris-Shore transformation. The
original system consists of two main states $\ket{\psi_{0}}$ and
$\ket{\psi_{1}}$ and an auxiliary exited state $\ket{\psi_{A}}$. The Rabi
frequencies of the original system share same time dependence $f(t)$ and same
detunings $\Delta(t)$. The reduced system consists of an upper state
$\ket{\psi_{a}}$ (same as the original upper state), a bright state
$\ket{\varphi_{0}}$ and a dark state $\ket{\varphi_{1}}$.

In certain cases, the three-state dynamics can be reduced to two-state one.
Such is the case when the intermediate state $\ket{a}$ is far off resonance
with the driving fields; then it can be eliminated adiabatically [19, 21, 20],
which generates an approximate SU(2) dynamics involving the qubit states only,
with an effective coupling between the qubit states and ac Stark (light)
shifts. Another case of SU(3) $\to$ SU(2) reduction, this time exact, occurs
when the Raman system possesses the Wigner-Majorana angular-momentum symmetry
[23, 24, 22].

A third case, which is the focus of this paper, takes place when the Raman-
coupled system possesses the Morris-Shore symmetry [25, 28, 22, 27, 26, 29];
then the three-state system can be exactly decomposed into a two-state system
and an uncoupled (dark) state. This symmetry requires the two-photon resonance
between the end states $\ket{0}$ and $\ket{1}$, while the middle state
$\ket{a}$ can be off single-photon resonances. Moreover, the two Raman
couplings must have the same time dependence but their magnitudes and phases
can be different; indeed the leeway in the choice of the coupling magnitudes
and phases has allowed the design of accurate quantum control schemes. The
Morris-Shore transformation can be generalized to drop the two-photon
resonance and timing conditions, although then the SU(3) $\to$ SU(2) reduction
is only approximate [30, 31].

The objective of the present paper is to develop a tomographic method for
determination of coherent gate errors in Raman-coupled qubits, obeying the
Morris-Shore (MS) symmetry. The method builds upon the one presented in [32,
34, 33] for two-level systems, where a certain high-fidelity gate is repeated
multiple times with subsequent measurements of the population in the end of
the sequence. The method takes advantage of the constructive interference
created through the repetitions leading to the amplification of the errors to
large enough values. These values can be measured reliably, from which one can
determine the single-gate errors due to the availability of analytic relations
between the single-pass and multi-pass probabilities.

This paper is organized in the following manner. First, in Sec. II we consider
in detail the case when the Raman-coupled qubit is driven by two pulses of
rectangular temporal shape in order to benefit from the simplicity of the
solution. After deriving the basic tomographic principle, based on error
amplification (the NR approximation, see below Secs. III and IV) , we proceed
in Sec. V to smooth pulse shapes and show that the simple solutions based on
the rectangular shapes are applicable for smooth shapes too. Finally, Sec. VI
presents some discussion and conclusions.

##  II Single-pass and multi-pass transitions

###  II.1 Single-pass transition

Consider a three-state Raman $\Lambda$ system under the conditions of the
Morris-Shore (MS) transformation [25], shown in Fig. 1, with the original
system on the left and MS-transformed system on the right. The Hamiltonian of
the system has the form

$\mathbf{H}(t)=\frac{1}{2}\left[\begin{array}[]{ccc}0&0&\Omega_{0}f(t)\\\
0&0&\Omega_{1}f(t)\\\
\Omega^{*}_{0}f(t)&\Omega^{*}_{1}f(t)&2\Delta(t)\end{array}\right],$ (1)

where $\Omega_{1}f(t)$ and $\Omega_{2}f(t)$ are the Rabi frequencies, which
have the same time dependence $f(t)$, and $\Omega_{1}$ and $\Omega_{2}$ are
complex constants. $\Delta(t)$ is the detuning, which is same for both fields.
The MS transformation reduces the original Hamiltonian (1) to an effective
two-state Hamiltonian [26, 19],

$\widetilde{\mathbf{H}}(t)=\mathbf{S}\mathbf{H}(t)\mathbf{S}^{\dagger}=\left[\begin{array}[]{ccc}0&0&0\\\
0&0&\frac{1}{2}\Omega f(t)\\\ 0&\frac{1}{2}\Omega
f(t)&\Delta(t)\end{array}\right],$ (2)

where $\mathbf{S}$ is the transforming complex-valued time-independent matrix

$\mathbf{S}=\left[\begin{array}[]{ccc}\frac{\Omega_{1}^{*}}{\Omega}&\frac{\Omega_{0}}{\Omega}&0\\\
-\frac{\Omega_{0}^{*}}{\Omega}&\frac{\Omega_{1}}{\Omega}&0\\\
0&0&1\end{array}\right],$ (3)

and $\Omega$ is the root-mean-square (RMS) Rabi frequency, which is a real
constant,

$\Omega=\sqrt{|\Omega_{0}|^{2}+|\Omega_{1}|^{2}}.$ (4)

Note that the MS Hamiltonian $\widetilde{\mathbf{H}}(t)$ is real and the
complexity of the original Hamiltonian $\mathbf{H}(t)$ is mapped onto the
transformation matrix $\mathbf{S}$. In the MS basis, the upper state
$\ket{\psi_{a}}$ is the same as in the original system, whereas the two MS
lower states are superpositions of the original lower states,

$\displaystyle\ket{\varphi_{0}}$
$\displaystyle=\frac{\Omega_{1}^{*}\ket{\psi_{0}}-\Omega_{0}^{*}\ket{\psi_{1}}}{\Omega},$
(5a) $\displaystyle\ket{\varphi_{1}}$
$\displaystyle=\frac{\Omega_{0}\ket{\psi_{0}}+\Omega_{1}\ket{\psi_{1}}}{\Omega}.$
(5b)

One of these — the bright state $\ket{\varphi_{1}}$ — is coupled to the upper
state $\ket{\psi_{a}}$ with the RMS coupling $\Omega f(t)$. The other — the
dark state $\ket{\varphi_{0}}$ — is uncoupled and hence, the original three-
state system reduces to a two-state one,
$\ket{\varphi_{1}}\leftrightarrow\ket{\psi_{a}}$. This reduction casts the
original U(3) dynamics to an effective U(2) dynamics, which greatly
facilitates the analysis.

Without loss of generality, consider the initial time to be $t_{i}=0$ and the
final time is denoted by $T$. The propagator in the MS basis can be written as

$\widetilde{\mathbf{U}}(T)=\left[\begin{array}[]{ccc}1&0&0\\\ 0&a&b\\\
0&-b^{*}e^{-i\delta}&a^{*}e^{-i\delta}\end{array}\right],$ (6)

where $a$ and $b$ are complex-valued Cayley-Klein (CK) parameters, restricted
by the relation

$|a|^{2}+|b|^{2}=1,$ (7)

and $\delta$ is a phase defined by

$\delta=\int_{0}^{T}\Delta(t)\,dt\ .$ (8)

By using the inverse of the transformation (6), the original propagator takes
the form

$\displaystyle\mathbf{U}$
$\displaystyle=\mathbf{S}^{\dagger}\widetilde{\mathbf{U}}\mathbf{S}=$
$\displaystyle=\left[\begin{array}[]{ccc}1+(a-1)\frac{|\Omega_{0}|^{2}}{\Omega^{2}}&(a-1)\frac{\Omega_{0}\Omega_{1}^{*}}{\Omega^{2}}&b\frac{\Omega_{0}}{\Omega}\\\
(a-1)\frac{\Omega_{0}^{*}\Omega_{1}}{\Omega^{2}}&1+(a-1)\frac{|\Omega_{1}|^{2}}{\Omega^{2}}&b\frac{\Omega_{1}}{\Omega}\\\
-b^{*}\frac{\Omega_{0}^{*}}{\Omega}e^{-i\delta}&-b^{*}\frac{\Omega_{1}^{*}}{\Omega}e^{-i\delta}&a^{*}e^{-i\delta}\end{array}\right].$
(12)

If the system starts in state $\ket{\psi_{0}}$, Eq. (II.1) dictates the
following populations in the end,

$\displaystyle P_{0}$
$\displaystyle=\left|1+(a-1)\,\frac{|\Omega_{0}|^{2}}{\Omega^{2}}\right|^{2}$
(13a) $\displaystyle P_{1}$
$\displaystyle=\left|(a-1)\,\frac{\Omega_{0}\Omega_{1}}{\Omega^{2}}\right|^{2}$
(13b) $\displaystyle P_{a}$
$\displaystyle=\left|b\,\frac{\Omega_{0}}{\Omega}\right|^{2}.$ (13c)

Hereafter we shall refer to the propagator (II.1) and the probabilities (13)
as _single-pass propagator_ and _single-pass probabilities_.

Let us assume that the system in Fig. 1 is a qubit with qubit states
$\ket{\psi_{0}}=\ket{0}$ and $\ket{\psi_{1}}=\ket{1}$. Then we must have all
population in the qubit subspace, which means that the CK parameter $b$ must
be zero, $b=0$. Then, due to the probability conservation condition (7), the
other CK parameter $a$ will be a phase factor, i.e. $a=e^{i\varphi}$. In fact,
its phase $\varphi$ is an important control parameter. The other control
parameter is the ratio $\Omega_{0}/\Omega_{1}$, which determines which quantum
gate is created.

The condition $b=0$, viewed in the MS basis, implies no transition between the
MS state $\ket{\varphi_{0}}$ and the upper state $\ket{\psi_{a}}$. Obviously,
we are not interested in the trivial case of no interaction because then $a=1$
and the propagator is the identity matrix. The condition $b=0$ in the presence
of interaction can be achieved in two scenarios. The simplest one is by a
resonant pulse of temporal area $2\pi$. Then $a=-1$, $b=0$ and the propagator
(II.1) reduces to

$\mathbf{U}(T)=\left[\begin{array}[]{ccc}1-2\frac{|\Omega_{0}|^{2}}{\Omega^{2}}&-2\frac{\Omega_{0}\Omega_{1}^{*}}{\Omega^{2}}&0\\\
-2\frac{\Omega_{0}^{*}\Omega_{1}}{\Omega^{2}}&1-2\frac{|\Omega_{1}|^{2}}{\Omega^{2}}&0\\\
0&0&-e^{-i\delta}\end{array}\right].$ (14)

The second possibility is far off resonance when $|\Delta|\gg\Omega$ and the
three-state problem can be reduced to a two-state one. In this case, the phase
$\varphi$ can be expressed approximately as

$\varphi\approx\frac{\Omega^{2}}{\Delta}\int_{0}^{T}f^{2}(t)\,dt\ .$ (15)

Equation (15) shows that the far-off resonance case is suitable for
constricting phase gates.

In this paper, we consider only the resonance case. The reason is that in the
far-off resonance case, due to the significant increase in detuning and Rabi
frequencies, the gates require much larger pulse area and hence are much
slower. Moreover, probabilities for transitions to higher energy levels
outside the three-state Raman system become prominent. This would compromise
the quantum gates due to detrimental leakage errors.

###  II.2 Target gate parameters and errors

In the resonance case, we have $\varphi=\pi$, hence $a=-1$. The target gates
have the following general form

$U_{tar}=\left[\begin{array}[]{ccc}\cos\zeta&e^{-i\phi}\sin\zeta&0\\\
e^{i\phi}\sin\zeta&-\cos\zeta&0\\\ 0&0&-1\end{array}\right],$ (16)

where the phase factor $e^{i\phi}$ is coming from the complexity of
$\Omega_{0}$ and $\Omega_{1}$, while $\zeta$ is the mixing angle defined as

$|\Omega_{0}|/\Omega=\sin(\zeta/2),\quad|\Omega_{1}|/\Omega=\cos(\zeta/2).$
(17)

In order to construct X gate, we must have $\zeta=\pi/2$ i.e.,
$|\Omega_{0}|/\Omega=|\Omega_{1}|/\Omega=1/\sqrt{2}$. For the Hadamard gate,
we need $\zeta=\pi/4$, i.e., $|\Omega_{0}|/\Omega=\sin(\pi/8)$ and
$|\Omega_{1}|/\Omega=\cos(\pi/8)$.

In order to quantify the gate errors, stemming from imprecise resonance
(nonzero $\Delta$) and inaccurate pulse area, it is convenient to express the
complex-valued Cayley-Klein parameters $a$ and $b$, restricted by Eq. (7), by
three real parameters as

$\displaystyle a$ $\displaystyle=-e^{-i\alpha}\cos\gamma$ (18a) $\displaystyle
b$ $\displaystyle=-ie^{-i\beta}\sin\gamma,$ (18b)

where $\alpha$, $\beta$ and $\gamma$ have all target values of $0$ in order to
retrieve values of $a$ and $b$ in the ideal case. Therefore they are measures
of _coherent gate errors_. From the resonance requirement $\Delta\to 0$ and
Eq. (8), it follows that the target value of the phase $\delta$ is also zero,
$\delta\to 0$, i.e. it is also an error measure. For high-fidelity quantum
gates these errors are very small and their determination is challenging. The
concept of this paper is to amplify these errors by gate repetitions to
sufficiently large values which can be measured reliably, with high accuracy
and precision.

The parameter $\zeta$ is considered as known. Indeed, it can be determined
from a single-pass measurement of the probabilities. For example, it follows
from Eq. (16) that $P_{1}\approx\sin^{2}(\zeta)$, hence the parameter $\zeta$
can be found from here. Then by substituting of $\zeta$ in Eq. (17) both
$|\Omega_{0}|/\Omega$ and $|\Omega_{1}|/\Omega$ can be found as well.

###  II.3 Multi-pass transition

In our previous work [22], we found the $N$-pass propagator of a three-state
Raman system. In Schrödinger’s representation, the $N$-pass propagator is the
$N$th power of the single propagator $U$ (II.1); it reads

$\mathbf{U}^{N}\\!\\!=\\!\\!\left[\\!\begin{array}[]{ccc}1+(a_{N}-1)\frac{|\Omega_{0}|^{2}}{\Omega^{2}}&(a_{N}-1)\frac{\Omega_{0}\Omega_{1}^{*}}{\Omega^{2}}&b_{N}\frac{\Omega_{0}}{\Omega}\\\
(a_{N}-1)\frac{\Omega_{0}^{*}\Omega_{1}}{\Omega^{2}}&1+(a_{N}-1)\frac{|\Omega_{1}|^{2}}{\Omega^{2}}&b_{N}\frac{\Omega_{1}}{\Omega}\\\
-b_{N}^{*}\frac{\Omega_{0}^{*}}{\Omega}e^{-iN\delta}&-b_{N}^{*}\frac{\Omega_{1}^{*}}{\Omega}e^{-iN\delta}&a_{N}^{*}e^{-iN\delta}\end{array}\\!\right]\\!,$
(19)

where the $N$-pass Cayley-Klein parameters $a_{N}$ and $b_{N}$ are connected
to the single-pass ones $a$ and $b$ by the relations

$\displaystyle a_{N}$
$\displaystyle=\left[\cos(N\vartheta)+i\textrm{Im}(a_{\delta})\dfrac{\sin(N\vartheta)}{\sin(\vartheta)}\right]e^{-iN\delta/2},$
(20a) $\displaystyle b_{N}$
$\displaystyle=b_{\delta}\dfrac{\sin(N\vartheta)}{\sin(\vartheta)}e^{-iN\delta/2},$
(20b)

with

$\displaystyle a_{\delta}$ $\displaystyle=a\,e^{i\delta/2},$ (21a)
$\displaystyle b_{\delta}$ $\displaystyle=b\,e^{i\delta/2},$ (21b)
$\displaystyle\vartheta$ $\displaystyle=\arccos(\textrm{Re}\,a_{\delta}).$
(21c)

The multi-pass probabilities are

$\displaystyle P_{0}^{(N)}$
$\displaystyle=\left|1+(a_{N}-1)\frac{|\Omega_{0}|^{2}}{\Omega^{2}}\right|^{2},$
(22a) $\displaystyle P_{1}^{(N)}$
$\displaystyle=\left|(a_{N}-1)\frac{\Omega_{0}\Omega_{1}}{\Omega^{2}}\right|^{2},$
(22b) $\displaystyle P_{a}^{(N)}$
$\displaystyle=\left|b_{N}\frac{\Omega_{0}}{\Omega}\right|^{2}.$ (22c)

In Eqs. (20), the parameters $\vartheta$ and $\delta$ are multiplied by the
factor $N$ in some terms, therefore we are able to amplify them after the
repetitions. We note that the parameters $|\Omega_{0}|/\Omega$ and
$|\Omega_{1}|/\Omega$ in Eqs. (22) remain the same as in the single pass
propagator (II.1).

For high gate fidelity, we must have

  * •

$P_{a}^{(N)}$ to be very small after every pass, i.e.

$P_{a}^{(N)}\ll 1\quad(N=1,2,\ldots),$ (23)

  * •

$P_{1}^{(N)}$ to be very small after every even pass, i.e

$P_{1}^{(2M)}\ll 1\quad(M=1,2,\ldots),$ (24)

We use these probabilities as indicators by which to determine the errors
$\alpha,\beta$ and $\gamma$ for X and H gates.

##  III Near-resonance (NR) approximation

In this section, our objective is to find out the connection between errors in
the Hamiltonian and those in the propagator. To achieve this, we use the Rabi
model and apply a near-resonance (NR) approximation derived from it. We find
that this approximation is not only suitable for the Rabi model but also
applicable to other models lacking analytical solutions.

###  III.1 Assumptions

Because our objective is to design a protocol for determining the errors of
high-fidelity Raman gates, we assume that their errors are small. Hence we
make three general assumptions.

  * •

We assume that the detuning $\Delta$ is small and constant,

$|\delta|\ll\pi\quad(\text{with }\delta=\Delta T),$ (25a) which we call the
_detuning error_.

  * •

For the pulse shape $f(t)$, we define the _filling ratio_

$r=\frac{1}{T}\int_{0}^{T}f(t)\,dt\quad(0\leq r\leq 1),$ (25b)

the role of which will be revealed below.

  * •

Because at resonance the Cayley-Klein parameter $a$ is $a=\cos(A/2)$, where
$A=\int_{0}^{T}\Omega f(t)\,dt$ is the RMS pulse area, and because the target
value of $a$ is $-1$, the RMS pulse area $A$ must be very close to $2\pi$;
hence we should have

$A=\Omega\int_{0}^{T}f(t)\,dt\,=\Omega rT=2(\pi-\epsilon),$ (25c)

where $|\epsilon|\ll\pi$ is the _pulse area error_.

We will make these assumptions throughout the text hereafter.

###  III.2 Rabi model and NR approximation

Figure 2:  First three pulses of the periodical time dependence $f_{R}(t)$,
related to Rabi model and given in Eq. (26).

The Rabi model is convenient in two aspects: first, it is an exactly solvable
model and second, it allows for any filling ratio $0\leq r\leq 1$. The Rabi
model periodical time dependence is shown on fig. 2 and it has the following
form

$f_{R}(t)=\sum_{n=0}^{N-1}R\Big{[}\frac{t}{rT}-\frac{(2n+1)}{2r}\Big{]},$ (26)

where $R$ denotes the rectangular function. In this case, both CK parameters
are given by the exact expressions

$\displaystyle a$
$\displaystyle=\left[\cos(\sigma/2)+i\frac{\delta}{\sigma}\sin(\sigma/2)\right]e^{-i\delta
r/2},$ (27a) $\displaystyle b$
$\displaystyle=-i\frac{A}{\sigma}\sin(\sigma/2)e^{-i\delta/2},$ (27b)

where $\sigma=\sqrt{\delta^{2}+A^{2}/r^{2}}$. Taking into account conditions
(25), we find the following expressions, which will be referred to as the NR
approximation,

$\displaystyle a$ $\displaystyle\approx-\cos(\epsilon)e^{-i\delta r/2},$ (28a)
$\displaystyle b$ $\displaystyle\approx-i\sin(\epsilon)e^{-i\delta/2},$ (28b)
$\displaystyle a_{\delta}$
$\displaystyle\approx-\cos(\epsilon)e^{i\delta(1-r)/2},$ (28c) $\displaystyle
b_{\delta}$ $\displaystyle\approx-i\sin(\epsilon),$ (28d)
$\displaystyle\vartheta$
$\displaystyle\approx\pi-\sqrt{\epsilon^{2}+{\delta^{2}}(1-r)^{2}/4}.$ (28e)

From here and Eq. (18) we find

$\quad\alpha\approx\delta
r/2,\quad\beta\approx\delta/2,\quad\gamma\approx\epsilon.$ (29)

Note that the parameters $\alpha,\beta$ and $\gamma$ are propagator (gate)
parameters, while $\delta,r$ and $\epsilon$ are Hamiltonian parameters, hence
we have direct connections between them.

###  III.3 Fidelity

For any unitary gate $U$ the fidelity is

$F=\frac{|\textrm{Tr}(U_{0}U^{\dagger})|^{2}}{d^{2}}$ (30)

where $U_{0}$ is the traget gate and $d$ is the Hilbert space dimension. In
our case $d=3$.

  * •

For $r=1$ i.e. $\alpha=\beta$, we find from Eq. (30) for the fidelity

$\displaystyle F$
$\displaystyle=\frac{1}{9}\left[\cos^{2}\zeta^{\prime}+2\cos\alpha\cos\zeta^{\prime}\cos\gamma(1+\cos\zeta^{\prime})\right.$
$\displaystyle+\left.(1+\cos\zeta^{\prime})^{2}\cos^{2}\gamma\right],$ (31)

where $\zeta^{\prime}$ is the error of $\zeta$, i.e. for X gate
$\zeta^{\prime}=\frac{\pi}{2}-\zeta$ and for Hadamard gate
$\zeta^{\prime}=\frac{\pi}{4}-\zeta$. For small error, $|\zeta^{\prime}|\ll
1$, we find from here

$\displaystyle F$
$\displaystyle=\frac{1+4\cos\alpha\cos\gamma+4\cos^{2}\gamma}{9}$
$\displaystyle-\frac{1+3\cos\alpha\cos\gamma+2\cos^{2}\gamma}{9}\zeta^{\prime
2}.$ (32)

For $\zeta^{\prime}=0$, only the first term survives. Obviously, if all errors
vanish, $\alpha=\gamma=\zeta^{\prime}=0$, then $F=1$.

  * •

For $r<1$, the result derived from Eq. (30) for the fidelity is too cumbersome
to be presented here. For $\zeta^{\prime}=0$, the fidelity can also be
expressed using the parameters $\epsilon$, $\delta$, and $r$ that characterize
the Hamiltonian. Thus, we have:

$\displaystyle F$
$\displaystyle=\frac{1}{9}\left[1+2\cos\epsilon\Big{(}\cos(r\delta/2)+\cos(\delta-r\delta/2)\right.$
$\displaystyle+\left.\big{(}1+\cos(\delta-r\delta)\big{)}\cos\epsilon\Big{)}\right].$
(33)

##  IV Determination of the gate errors

Now we shall determine the errors $\alpha$, $\beta$ and $\gamma$ specified in
Eqs. (29) by the multi-pass probabilities in the NR approximation. All figures
use the exact solution of the Rabi model. Nevertheless, for the error range of
$0.05$, which is of interest to us, the plots are practically identical with
those for the NR approximation. In Sec. V, we apply the NR approximation (28)
for other models with various pulse shapes $f(t)$, respectively other filling
ratio $r$, and compare the results with the exact (or numerical) solutions. In
all figures, we choose $|\Omega_{0}|/\Omega=|\Omega_{1}|/\Omega=1/\sqrt{2}$,
which corresponds to the X(NOT) gate.

###  IV.1 Determination of $\gamma$

Figure 3:  Multi-pass probability $P_{a}^{(N)}$ according to the exact Rabi
model solutions (27). The plots are nearly identical for the NR approximation
(34). For small $\epsilon$ and $\delta$, the probability $P_{a}$ depends
slightly on both $\delta$ and $r$ and can be approximated according to Eq.
(35). This allows the error $\gamma=\epsilon$ to be determined by the multi-
pass probability in Eq. (36). The values of $r$ are selected for the sake of
comparison because they naturally emerge for other pulse shapes in Sec. V.

By using of connections (20) and the exact Rabi model solution (27), we can
obtain the multi-pass probabilities $P_{a}^{(N)}$, according to Eq. (22).
According to the NR approximation (28) the probability is

$P_{a}^{(N)}=\frac{|\Omega_{0}|^{2}}{\Omega^{2}}\frac{\sin^{2}\epsilon\sin^{2}N\vartheta}{\sin^{2}\vartheta},$
(34)

where $\vartheta$ is given in Eq. (28e). It is shown in Fig. 3. Equation (34)
gives almost identical results in the range $|\delta|<0.05$ and
$|\epsilon|<0.05$ as the exact one in Fig. 3 and therefore the NR
approximation plot is not shown. From Fig. 3 we see that at small $\delta$,
the multi-pass probability $P_{a}^{(N)}$ depends weakly on $\delta$ and $r$
and at $\delta=0$ we simply have

$P_{a}^{(N)}=\frac{|\Omega_{0}|^{2}}{\Omega^{2}}\sin^{2}(N\epsilon),$ (35)

from which $\gamma=\epsilon$ can be found as

$\gamma=\frac{1}{N}\arcsin[\frac{\Omega}{|\Omega_{0}|}\sqrt{P_{a}^{(N)}}\Big{]}.$
(36)

###  IV.2 Determination of $\alpha$ and $\beta$

Having already the error $\gamma=\epsilon$ determined, we proceed to determine
$\alpha=r\delta/2$. Having found $\alpha$ and knowing the value of $r$ a
priori, we can find the value of $\beta=\delta/2$ as simply $\beta=\alpha/r$.
Hence we focus our attention on the determination of $\alpha$. We will show
that depending of the filling ratio $r$, two approaches are required - one for
$r<0.5$ and another for $r>0.5$.

Figure 4: Multi-pass probability $P_{a}^{(N)}$ according to the exact Rabi
model solutions (27) at $\epsilon=0.025$ and $r=0.25$. At smaller number of
repeated gates $N$ (the dashed line region), all curves almost overlapped at a
given $\epsilon$, that allows to determine $\gamma=\epsilon$. At bigger $N$ ,
the curves diverge that allows to determine $\delta$ Figure 5:  Multi-pass
probability $P_{1}^{(2M)}$. At known $\epsilon$ and $r>0.5$, the detuning
error $\delta$ can be found numerically. For small $\epsilon\leq 0.005$,
$P_{1}$ can be approximated according to (39). For small $\epsilon$ and
$r<0.5$, the amplification is not sufficient and it is necessary to use the
procedures described in IV.2.1.

####  IV.2.1 Determination of $\alpha$ for $r<0.5$

For $r<0.5$, the probability $P_{a}$ depends very strongly on $\delta,r$ and
$N$, which is visible on Fig. 3. For already known $\epsilon$ (measured at
smaller number of $N$), see above, we could also perform another experiment
with larger $N$. In Fig. 4 we see these two regions of the probabilities. The
first region (dashed lines) corresponds to the smaller $N$, where $\epsilon$
is determined for any value of $\delta$. The inflection point is almost the
same for all curves and after it the curves begin to diverge for different
$\delta$. By using the Taylor series up to $\delta^{2}$ Eq. 34 we find

$\displaystyle P_{a}^{(N)}$
$\displaystyle=\frac{|\Omega_{0}|^{2}}{\Omega^{2}}\sin^{2}(N\epsilon)$
$\displaystyle\times\left[1-\frac{\delta^{2}(1-r)^{2}(1-N\epsilon\cot
N\epsilon)}{4\epsilon^{2}}\right],$ (37)

from which $\delta$ and respectively the error $\alpha=r\delta/2$ can be
found.

####  IV.2.2 Determination of $\alpha$ for $r>0.5$

For $r>0.5$ the curves on Fig. 4 come close to each other and at $r=1$, they
overlap for any $N$. In this case we need other approaches in order to
determine $\alpha$. By using a similar approach as in Eq. IV.1, connections
(20) and the exact Rabi model solution (27), we can find the multi-pass
probabilities $P_{1}^{(2M)}$ of Eq. (22), shown in Fig. 5. According to the NR
approximation (28) the probability is

$\displaystyle P_{1}^{(2M)}$
$\displaystyle=\frac{2|\Omega_{0}|^{2}|\Omega_{1}|^{2}}{\Omega^{4}}\left[1-\cos\frac{N\delta}{2}\cos
N\vartheta\right.$ $\displaystyle+\left.\frac{\sin\frac{N\delta}{2}\sin
N\vartheta\sin\frac{\delta(1-r)}{2}}{\sin\vartheta}-\frac{\sin^{2}\epsilon\sin^{2}N\vartheta}{2\sin^{2}\vartheta}\right].$
(38)

In the NR approximation, the probabilities in Eqs. (IV.2.2) give almost
indistinguishable results as the exact ones in Fig. 5 and therefore the NR
approximation plot is not shown. If $r$ is known approximately, then $\delta$
can be found numerically from Eq. (IV.2.2).

For small $\epsilon$ ($\epsilon<0.005$) it can be approximated to

$P_{1}^{(2M)}\approx
4\frac{|\Omega_{0}|^{2}|\Omega_{1}|^{2}}{\Omega^{4}}\sin^{2}(N\delta r/4),$
(39)

from which $\alpha=\delta r/2$ can be found.

##  V Comparisons of NR approximation with other models

In Secs. III.2 and IV, we stated that the NR approximation nearly coincides
with the exact Rabi model for error ranges up to $0.05$. In this section, we
present results for three additional models with various time dependencies
$f(t)$ and filling ratios $r$ and compare the results with those of NR
approximation. We will see that for a Raman qubit, driven by a MS-Hamiltonian,
the NR approximation is a convenient approximation also for other pulse
shapes, which considerably broadens the applicability of the NR approach.

###  V.1 Rosen-Zener (RZ) model

The Rosen-Zener (RZ) model [35], which assumes a hyperbolic-secant pulse shape
sech$(t/T)$ (running from $-\infty$ to $+\infty$) is exactly solvable.
Strictly speaking, even a single pass requires an infinitely long duration,
meaning a filling ratio $r\to 0$. However, for a sech pulse of a finite
duration $[-\tau,\tau]$, truncated sufficiently far from its maximum, such
that $r\leq 0.1$ (meaning $\tau\geq 15.7T$, which in turn corresponds to an
amplitude value of less than $3\times 10^{-7}$ of the maximum value), the RZ
model is essentially exact. According to assumptions in Sec. III.1 the
periodical time dependence is

$f_{RZ}(t)=\sum_{n=0}^{N-1}\sech[\frac{\pi}{r}\Big{(}\frac{t}{T}-\frac{2n+1}{2}\Big{)}\Big{]},$
(40)

shown in Fig. 6. In this example, we choose a filling ratio $r=0.1$

$r=\frac{1}{T}\int_{0}^{T}f_{RZ}(t)\,dt=0.1.$ (41)

Figure 6: Illustration of the pulses of the time dependence $f_{RZ}(t)$ for RZ
model (40) for $r=0.1$ .

Figure 7: Comparison between the multi-pass probabilities $P_{a}^{(N)}$
obtained from the exact RZ-model solutions (42) and the NR approximation (34),
for $r=0.1$.

Considering the conditions in Sec. III.1, the CK parameters have the following
exact solution

$\displaystyle a$
$\displaystyle=\frac{\Gamma^{2}\big{(}\frac{1}{2}+i\frac{\delta
r}{2\pi}\big{)}}{\Gamma\big{(}\frac{1}{2}+\frac{A}{2\pi}+i\frac{\delta
r}{2\pi}\big{)}\Gamma\big{(}\frac{1}{2}-\frac{A}{2\pi}+i\frac{\delta
r}{2\pi}\big{)}},$ (42a) $\displaystyle b$
$\displaystyle=-i\frac{\sin(A/2)}{\cosh(\delta r/2)}e^{-i\delta/2}.$ (42b)

The multi-pass probabilities (22) can be found from Eqs. (42) and (20). In
Fig. 7 we show the comparison between the exact (RZ) probabilities
$P_{a}^{(N)}$ and the NR approximated ones (34). It is visible that both plots
are practically the same for the error range of $0.05$. The plots for the
populations $P_{1}^{(2M)}$ are not shown but they are practically
indistinguishable as the ones shown in Fig. 5 for $r=0.1$. Based on these
findings we conclude that the NR approximation and the method for derivation
presented in the preceding section are perfectly applicable for sech pulses.

###  V.2 $\sin^{2}$ model

Now we present an example where the time dependence is

$f_{S}(t)=\sin^{2}(\pi t/T).$ (43)

The benefit of such a pulse shape is that it has a well-defined finite
duration (contrary to the sech shape), smooth pulse shape (contrary to the
rectangular pulse), but unfortunately, the Schrödinger equation can not be
solved analytically. Yet, it is easily integrated numerically. The filling
ratio for this pulse shape is

$r=\frac{1}{T}\int_{0}^{T}f_{S}(t)\,dt=0.5.$ (44)

The probability map for $P_{a}^{(N)}$ is shown in Fig. 8. The NR approximation
for the same filling ratio $r=0.5$ (top row) is almost identical to the
numerical results (bottom row). The plots for the populations $P_{1}^{(2M)}$
are not shown because they are very similar to the ones shown in Fig. 5 for
$r=0.5$.

Figure 8: The Multi-pass probability $P_{a}^{(N)}$. The upper row shows the NR
approximation (34) for $r=0.5$ and the bottom row shows the numerically
calculated probability for a time dependence according to Eq. (43), which also
corresponds to $r=0.5$.

###  V.3 Second trigonometric model

We now proceed to another numerically solved pulse shape with the time
dependence of

$f_{C}(t)=1-\cos^{10}[\pi t/T],$ (45)

shown in Fig. 9. Compared to the $\sin^{2}$ model, it features a filling ratio
$r=\frac{1}{T}\int_{0}^{T}f_{C}(t)\,dt=0.754$, hence the choice of $r$ in the
corresponding frames for this value of $r$ in Figs. 4 and 5.

Figure 9: The first 3 pulses of the time dependence according to Eq. (45),
with the filling ratio $r=0.754$.

For this model, instead of comparing $P_{a}^{(N)}$, and to provide additional
information, we compare the probability map for $P_{1}^{(2M)}$, as depicted in
Fig. 10. The NR approximation for the same filling ratio, $r=0.754$ (top row),
is nearly identical to the numerical results (bottom row). The plots for the
populations $P_{a}^{(N)}$ are not shown because they are very similar to the
ones shown in Fig. 3 for $r=0.754$.

Figure 10: The Multi-pass probability $P_{1}^{(2M)}$. The upper row refers to
the NR approximation (34) at $r=0.754$. The lower row refers to numerically
solved probability for a time dependence expressed in Eq. (45), which also
corresponds to $r=0.754$.

##  VI Conclusions

We presented a tomographic method designed for the characterization of high-
fidelity Raman qubit gates, which obey the Morris-Shore transformation, the
most important condition for which is the two-photon resonance condition
between the qubit states. The proposed method makes use of coherent
amplification of the gate errors by repeating the same gate numerous times. By
examining the multi-pass probabilities, we establish their dependence on four
key parameters: pulse area error $\epsilon$, detuning error $\delta$, filling
ratio $r$, and the number of pulses (passes) $N$.

From these expressions, it becomes feasible to directly calculate the errors
$\epsilon$ and $\delta$, which determine the gate errors $\alpha,\beta,$ and
$\gamma$. Since the Raman system is reduced to an effective two-state system
in the near-resonance regime, employing the NR approximation with a filling
factor $r$ serves as a convenient and practical approach. Additionally, this
approximation can be extended to other pulse shapes, thereby removing the
restriction of the rectangular shape.

###### Acknowledgements.

This research is supported by the Bulgarian national plan for recovery and
resilience, contract BG-RRP-2.004-0008-C01 (SUMMIT), project number 3.1.4.

## References

  * [1] D.F.V.James et al. (1999). Trapped Ion Quantum Computer Research at Los Alamos. arXiv:quant-ph/9807071v1
  * [2] Hughes R.J. Quantum computation. Technical report LA-UR-98-288, 1998
  * [3] D.F.V. James. Quantum dynamics of cold trapped ions with application to quantum computation. Applied Physics B, 66(2):181–190, Feb 1998.
  * [4] S. E. Economou. Proposal for optical U(1) rotations of electron spin trapped in a quantum dot , Phys. Rev. B 74, 205415 (2006)
  * [5] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Reviews of Modern Physics 75, 281 (2003)
  * [6] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998).
  * [7] 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).
  * [8] M. D. Barrett, B. DeMarco, T. Schaetz, V. Meyer, D. Leibfried, J. Britton, J. Chiaverini, W. M. Itano, B. Jelenkovi´c, J. D. Jost, C. Langer, T. Rosenband, and D. J. Wineland, Phys. Rev. A 68, 042302 (2003).
  * [9] N. V. Vitanov and K.-A. Suominen, Phys. Rev. A 56, R4377 (1997).
  * [10] B. W. Shore, _The Theory of Coherent Atomic Excitation_ (Wiley, 1990).
  * [11] B. W. Shore, _Manipulating Quantum Structures Using Laser Pulses_ (Cambridge University Press, 2011)
  * [12] N. V. Vitanov, Z. Kis, and B. W. Shore, Phys. Rev. A 68, 063414 (2003).
  * [13] B.T. Torosov and N. V. Vitanov, Phys. Rev. Res. 2, 043194 (2020).
  * [14] Oliver Corfield. Quantum Coherence in Trapped Ions, PhD thesis, Imperial College London, 2022.
  * [15] Chungsun Lee, Characterisation and Control of Trapped-ion Qubit, PhD thesis, Imperial College London, 2022.
  * [16] L. N. Egan, Scaling Quantum Computers with Long Chains of Trapped Ions, PhD thesis, Joint Quantum Institute (2021)
  * [17] S. A. Schulz, Scalable Microchip Ion Traps for Quantum Computation, PhD thesis, Ulm University (2009)
  * [18] John McCann, A Novel Approach to Single Ion Addressing by means of a Multicore Photonic Crystal Fibre, Master thesis, ETH Zürich (2018)
  * [19] E. Kyoseva, N. V. Vitanov and B. W. Shore, Journal of Modern Optics 54, (2008).
  * [20] N. V. Vitanov, T. Halfmann, B. W. Shore, and K. Bergmann, Annu. Rev. Phys. Chem. 52, 763 (2001).
  * [21] N. V. Vitanov, A. A. Rangelov, B. W. Shore and K. Bergmann, Rev. Mod. Phys. 89, 015006 (2017).
  * [22] S. G. Stanchev and N. V. Vitanov, J. Phys. B: At. Mol. Opt. Phys. 56 014001 (2023).
  * [23] E. Wigner, _Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren_ (F. Vieweg und sohn Braunschweig, 1931).
  * [24] E. Majorana, Nuovo Cimento 9, 43 (1932).
  * [25] Morris, James R. and Shore, Bruce W., Phys. Rev. A 27, 906 (1983).
  * [26] E. S. Kyoseva, and N. V. Vitanov, Phys. Rev. A 73, 023420 (2006).
  * [27] A. A. Rangelov, N. V. Vitanov, and B. W. Shore, Phys. Rev. A 74, 053402 (2006); Erratum Phys. Rev. A 76, 039901 (E) (2007).
  * [28] B. W. Shore, J. Mod. Optics 61, 787 (2013).
  * [29] H. Kim, Y. Song, H.-G. Lee, and J. Ahn, Phys. Rev. A 91, 053421 (2015).
  * [30] K. N. Zlatanov, G. S. Vasilev and N. V. Vitanov, Phys. Rev. A 102, 063113 (2020).
  * [31] K.N. Zlatanov, A.A. Rangelov and N. V. Vitanov, J. Phys. B: At. Mol. Opt. Phys. 55, 204001 (2022).
  * [32] N. V. Vitanov, New Journal of Physic 22, 023015 (2020).
  * [33] N. V. Vitanov and P. L. Knight, Phys. Rev. A 52, 2245 (1995).
  * [34] N. V. Vitanov, Phys. Rev. A 97, 053409 (2018).
  * [35] N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932).