# Optical control protocols for high-fidelity spin rotations of single SiV-
and SnV- centers in diamond

Evangelia Takou  Department of Physics, Virginia Polytechnic Institute and
State University, 24061 Blacksburg, VA, USA  Sophia E. Economou  Department of
Physics, Virginia Polytechnic Institute and State University, 24061
Blacksburg, VA, USA

###### Abstract

Silicon-vacancy and tin-vacancy defects in diamond are of interest as
alternative qubits to the NV center due to their superior optical properties.
While the availability of optical transitions in these defects is one of their
assets, high-fidelity optical coherent control has not been demonstrated.
Here, we design novel optical control schemes tailored to these defects. We
evaluate the performance of arbitrary single-qubit rotations of the electron
spin qubit both in the absence and presence of an external magnetic field, by
taking into account both coherent and incoherent errors. We find that
rotations in excess of $98.0\%$ ($T=4$ K) and $99.71\%$ ($T=6$ K) can be
achieved for Si-V and Sn-V respectively in the presence of realistic
relaxation and leakage errors.

††preprint: APS/123-QED

##  I Introduction

Over the past decades, color centers in diamond have been investigated for
their potential use as the hardware for solid-state quantum processing
applications. The most well-known defect in a diamond host is the NV center.
Its most prominent features are the long coherence times of the NV electron
spin [1] and the room temperature operation [2, 3]. Practical experimental
demonstrations regarding the NV center involve readout [4, 5, 6, 7],
initialization [8, 9], entanglement generation schemes, as well as control of
the surrounding nuclear bath [10, 11]. However, the optical control usually
relies on the application of external perturbations such as strain/electric or
magnetic fields, to lift the ground state degeneracy, or allow for spin-
flipping transitions [12, 13]. In addition, NV centers have poor optical
properties, with large phonon sideband and low probability of emission in the
zero phonon line (ZPL). While this can be boosted using coupling to cavities
and waveguides [14], the low ZPL emission still limits the performance of
entanglement generation schemes. Moreover, the sensitivity of NV centers to
charge noise introduces spectral diffusion to optical transitions [15].

As a result, alternative defects are explored in diamond for quantum
information applications. Two that stand out are the negatively charged
silicon vacancy (SiV-) [16, 17, 18, 19, 20, 21, 22, 23, 24] in diamond and the
newly emerging tin vacancy (SnV-) color centers [25, 26, 27, 28, 29, 30].
These spin $S=1/2$ systems, have excellent optical properties, such as narrow
linewidths of the ZPL transition which comprises $70-80\%$ of the emitted
light [31], small phonon sidebands, and spectral stability [32, 33]. They
belong to the $D_{3\text{d}}$ point group [16] and display inversion symmetry,
which renders them robust to charge fluctuations, since, unlike the NV, they
lack permanent electric dipole moment to first order [34]. Consequently, they
are excellent indistinguishable single photon sources [35, 36], and they are
immune to noise emerging from integration into photonic devices [37].
Furthermore, due to the large ground state splitting of the SnV- defect [38,
25], no spin mixing is observed in the presence of external magnetic fields
[25], which can lead to improved optical control. In addition, nuclear spin
control has been achieved for SiV- in nano-waveguides [39, 40]. Initialization
and readout of the SiV- have also been demonstrated in [41, 42], while
coherent control and single qubit rotations have been shown in [43, 44, 45,
46, 47]. In [46], the control of the electronic spin is achieved using MW
pulses, resulting in slower rotations. This approach also requires microwave
frequency generators and amplifiers, thus increasing the experimental demands.
In [45], full SU(2) spin rotations were demonstrated via Ramsey interference
generated by two temporally separated pulses. Each of these pulses originated
from a single broadband laser that addressed both transitions of a
$\Lambda$-system. To avoid driving unwanted transitions, a far off-resonant
Raman pulse was used, which restricted the achievable rotation angles due to
limitation of the laser power and did not mitigate the decoherence entirely
due to the excitation of an unwanted excited state. Moreover, the fidelity of
the rotations was not quantified nor were the error mechanisms investigated in
detail.

So far, the theoretical models of optical control usually involve
approximations under which the off-resonant transitions are ignored. This is a
good approximation for longer gate durations, i.e., narrowband pulses. For
faster pulses, which are needed in these systems to ensure the control takes
place well within the optical coherence and relaxation times, such off-
resonant transitions cannot be ignored. Another issue typically present in
defect systems is orbital mixing of the states caused by the Jahn-Teller
effect or crystal strain. In the case of $\Lambda$-system schemes, each
transition dipole couples to both driving fields, leading to additional errors
during the optical control. Thus, these error mechanisms present in many
defects give rise to the need for high-fidelity control techniques tailored to
these systems.

In this paper, we address the aforementioned challenges by developing all-
optical control protocols for the SiV- and SnV- color centers in diamond. We
start by demonstrating that existing approaches, in particular coherent
population trapping, do not suffice for high-fidelity gates. We show how to
eliminate cross-talk errors and reduce the number of unwanted leakage
transitions by appropriately selecting the polarization of the lasers. We
further optimize the gates by analyzing the full dynamics of the systems, and
we identify the coherent errors as well as incoherent errors that arise due to
unwanted excitations of the multi-level electronic structures. We resolve the
leakage through two corrective methods, one available in the literature and
one developed here, that allow for faster implementation of the rotations
without compromising the gate fidelity.

This paper is organized as follows. In Sec. II we describe the two main
sources of errors of the optical control; the cross-talk and leakage. In Sec.
III, we discuss the cross-talk issue and provide an effective and simple
solution based on polarization schemes. In Sec. IV we analyze two approaches
for leakage mitigation. Finally, in Secs. V and VI we apply our protocols to
the SiV- and SnV- defects, respectively, and quantify the fidelity for various
rotations angles and pulse durations.

##  II Overview of the problem and error mechanisms

A well-known technique that provides fast all-optical control of
$\Lambda$-systems is coherent population trapping (CPT). In CPT, the two
transitions of a $\Lambda$-system are driven with two laser fields $E_{1}$ and
$E_{2}$, each acting on a distinct transition, as shown in Fig. 1(a), which
satisfy the two-photon resonance condition. CPT is based on the destructive
interference of the quantum processes driven by the different fields, which
leads to trapping of the population into a dark state. In this so-called dark-
bright frame, the dark state is completely decoupled from the dynamics of the
other two levels in the system [Fig. 1(b)]. The transformation to the dark-
bright frame defines the rotation axis of the qubit; by combining CPT with
hyperbolic secant pulses, we can design arbitrary single-qubit rotations, as
explained in Appendix A and in Refs. [48, 49].

In an ideal CPT scheme, the distinct couplings can be satisfied by either
energy separation of the ground states, or polarization selection rules that
ensure each transition is accessed by a single laser. However, energy
separation alone does not guarantee negligible cross-talk errors for all gate
durations, and the approximation of distinguishable couplings breaks down for
broadband pulses. Unfortunately, the two transition dipoles are not orthogonal
for the SiV- and the SnV- systems, leading to the cross talk (dashed arrows)
shown in Fig. 1(c). The source of the cross talk and our solution to this
problem will be explained in Sec. III. For now, we stress that this setting is
unavoidable if each laser field is chosen according to the polarization
selection rules, i.e., such that its coupling to one of the two
$\Lambda$-transitions is maximized. Henceforth, we refer to this approach as
“naive”.

In addition to the cross-talk, each laser field removes population from the
$\Lambda$-subspace inducing thus leakage errors to the control. As an example,
we show the leakage transitions of the SiV- system in Fig. 1(e), which occur
with an off-resonant energy cost $\delta_{\text{es}}$. In the dark-bright
frame, these errors translate into couplings between the dark/bright states
and the unwanted excited level, $|\text{C}\rangle$ [Fig. 1(f)].

In the following sections, we propose schemes to resolve the cross-talk by
polarization tuning of the lasers, as well as to counteract the leakage errors
via pulse modulation. These protocols are analyzed in Sec. III and Sec. IV
respectively. The readers who are more interested in the numerical results
could directly proceed with Sec. V (for the SiV-) and Sec. VI (for the SnV-).

Figure 1: Summary of the error mechanisms for the SiV- system. (a) Ideal CPT
scheme performed with two fields $E_{1}$ and $E_{2}$ acting on distinct
transitions. The ground state splitting is denoted as $\delta_{\text{gs}}$.
(b) Transformation to the dark-bright basis for the case of (a) successfully
decouples the dark state $|\text{d}\rangle$ and transitions are driven between
the bright $|\text{b}\rangle$ and excited state $|\text{A}\rangle$. The
effective Rabi frequency is expressed in terms of the Rabi frequencies in the
lab frame, i.e.
$\Omega_{\text{eff}}=\sqrt{|\Omega_{1}|^{2}+|\Omega_{2}|^{2}}$. (c) Cross-talk
within the $\Lambda$-system leads to off-resonant errors (dashed green and
blue arrows), that oscillate with an additional energy shift,
$\delta_{\text{gs}}$. (d) In the presence of cross-talk couplings as shown in
(c), the dark state is not completely decoupled. (e) Both laser fields drive
the leakage transitions C1 and C4, introducing errors to the optical control.
$\delta_{\text{es}}$ is the excited states splitting. (f) In addition to the
cross-talk shown in (d), each laser drives the
$|\text{b}\rangle\leftrightarrow|\text{C}\rangle$ and
$|\text{d}\rangle\leftrightarrow|\text{C}\rangle$ transitions in the db-frame.

##  III Addressing Cross-talk errors

One advantage of SiV- and SnV- defects is that $\Lambda$-schemes can be
realized even at zero-magnetic fields, which simplifies the dynamics and
facilitates experimental implementations of optical control. In Fig. 2 we show
the electronic structure for the SiV- and SnV- at zero magnetic fields; each
ground- and excited-state manifold is pairwise degenerate. We follow the
literature convention of labeling the ground states as $|1\rangle-|4\rangle$,
and the excited states as $|\text{A}\rangle-|\text{D}\rangle$ (for more
precise labeling we use the eigenstates of the spin-orbit coupling,
$|e_{\pm}\rangle$, for the orbital part of the states).

Based on group theory, the allowed optical transitions can be accessed by
either linear $z$-polarization, which drives transitions between orbital
states with the same symmetry, or by circular polarization, which drives
transitions between the states
$|e_{\text{g},\pm}\rangle\leftrightarrow|e_{\text{u},\mp}\rangle$ [16].
However, a small orbital mixing of the states caused by the Jahn-Teller effect
[16] (or by crystal strain) introduces non-zero $z$-dipoles ($x,y$ dipoles) to
the transitions mainly accessed by $\sigma^{\pm}$ ($z$) polarization.
Consequently, the choice of polarizations as dictated by selection rules would
give rise to a cross-talk, i.e. coupling of each laser field to both
$\Lambda$-transitions. In such a setting, the dark state is not completely
decoupled from the dynamics, as shown in Fig. 1(d).

Figure 2: Electronic structure for the negatively silicon vacancy (SiV-) (a),
and for the negatively charged tin vacancy (SnV-) in diamond (b), for zero
magnetic fields. We label the states using the eigenstates of the spin-orbit
coupling (i.e. $|e_{\pm}\rangle$) which is the largest perturbation term, but
in the text, we use the notation $|1\rangle-|4\rangle$ for the ground states
and $|\text{A}\rangle-|\text{D}\rangle$ for the excited states.

The contribution of the off-resonant cross-talk is quantified by the ground
state splitting of the qubit states. For the SnV- system, the errors average
out more efficiently due to its large ground-state splitting
($\delta_{\text{gs}}^{\text{SnV${}^{-}$}}=825$ GHz). These errors however are
not negligible for SiV- (for which
$\delta_{\text{gs}}^{\text{SiV${}^{-}$}}=50$ GHz). Nevertheless, when
broadband pulses are considered, the cross-talk becomes the central source of
infidelity for both defects since off-resonant transitions are more strongly
coupled.

We propose a simple cross-talk elimination scheme that is achieved by tuning
the laser field polarizations. We consider as an example the SiV- system and
express the direction and polarization of the laser fields in the defect
coordinate frame. In an experimental setup, most diamond samples are cut with
$[0~{}0~{}1]$ as the surface normal. Thus, the polarization directions that we
express in the internal coordinate frame would require a non-zero angle of
incidence on the sample.

We assume two lasers of $xz$-polarization and $y$-propagation, where the
former drives the A1 transition and the latter the A4 transition. To fix the
orthogonality of each laser with one of the $\Lambda$-transitions, we require
that its polarization is orthogonal to the corresponding dipole,
$\textbf{d}_{ij}$. In this particular case of $\Lambda$-system, we require
$\textbf{d}_{\text{A1}}\cdot\textbf{E}_{2}=0$ and
$\textbf{d}_{\text{A4}}\cdot\textbf{E}_{1}=0$, from which we find that the
electric fields need to be defined as:

$\textbf{E}_{1}=E_{01}\left(\hat{\textbf{x}}-\frac{\langle
p_{x}\rangle_{\text{A4}}}{\langle
p_{z}\rangle_{\text{A4}}}\hat{\textbf{z}}\right)e^{i(k_{1}y-\omega_{1}t)}+\text{cc.},$
(1) $\textbf{E}_{2}=E_{02}\left(\hat{\textbf{x}}-\frac{\langle
p_{x}\rangle_{\text{A1}}}{\langle
p_{z}\rangle_{\text{A1}}}\hat{\textbf{z}}\right)e^{i(k_{2}y-\omega_{2}t)}+\text{cc.}.$
(2)

Here $\langle p_{k}\rangle_{ij}=\langle\psi_{i}|p_{k}|\psi_{j}\rangle$ (with
$k\in\\{x,y,z\\}$) is the transition dipole overlap that can be calculated
according to group theory. The polarizations of the lasers that satisfy the
orthogonality conditions are not unique; we choose to restrict the
polarization vectors in the $xz$ plane, in which case the polarizations are
uniquely determined. The definitions of Eq. (1) and Eq. (2) can be generalized
easily to other choices of $\Lambda$-systems or other polarization directions.
For the SnV- system, we chose $yz$-polarization instead, and the reasons for
this choice are explained in Sec. VI.1 and in Appendix F.

Throughout the paper, we combine the sech-based CPT scheme with E-field
polarizations that satisfy the orthogonality conditions to design arbitrary
gates free from cross-talk errors.

##  IV Corrective methods for leakage suppression

###  IV.1 General strategy of Magnus expansion

As we mentioned in Sec. III, we resolve the cross-talk issue of the
$\Lambda$-system by redefining the polarization of the laser fields. However,
leakage errors reduce the gate fidelity of fast optical control schemes. To
counteract this problem, we use a Magnus-based expansion approach developed in
Ref. [50]. Here we outline the basic steps of the method, and we provide
further details about the procedure we follow in Appendix F.

Let us consider a generic Hamiltonian $H(t)$ given by:

$H(t)=H_{0}(t)+\epsilon V(t),$ (3)

where $H_{0}(t)$ implements our analytically solvable target gate, and $V(t)$
introduces an error to the dynamics generated by $H_{0}(t)$. The error term is
assumed to be perturbative, as it contains off-resonant terms, oscillating
faster than $H_{0}$. To mitigate these errors, we additionally consider a
control Hamiltonian $W(t)$, such that the total Hamiltonian is modified into

$\bar{H}(t)=H_{0}(t)+\epsilon V(t)+W(t).$ (4)

Further, the control Hamiltonian is expanded in a power series according to:

$W(t)=\sum_{k=0}^{\infty}\epsilon^{k}W^{(k)}(t).$ (5)

By going into the interaction picture of $H_{0}(t)$, the Hamiltonian
transforms into $\bar{H}_{\text{I}}(t)=\epsilon
V_{\text{I}}(t)+W_{\text{I}}(t)$, and the total evolution operator becomes

$U(t)=U_{0}(t)U_{\text{I}}(t),$ (6)

where $U_{0}$ is the ideal gate, and $U_{\text{I}}(t)$ is generated by the
error and control Hamiltonian. The implementation of the ideal gate is
achieved if $U_{\text{I}}(T)=\textbf{1}$ (where $T$ is the gate time), such
that $U(T)=U_{0}(T)$, based on Eq. (6). To this end, the evolution operator
$U_{\text{I}}(t)$ is expanded in a Magnus series, and as was shown in Ref.
[50], the solutions for the control are obtained iteratively via the equation:

$\epsilon^{n}\int_{0}^{T}dt^{\prime}W_{\text{I}}^{(n)}(t^{\prime})=-i\sum_{k=1}^{n}\Omega_{k}^{(n-1)}(T),$
(7)

where $\Omega_{k}$ is the $k$-th Magnus expansion order. In this work, we
focus on first order corrections, i.e. we truncate the Magnus series to the
first order, which leads to the equation:

$\epsilon\int_{0}^{T}dtV_{\text{I}}(t)=-\int_{0}^{T}dtW_{\text{I}}^{(1)}(t).$
(8)

Equation (8) can be reformulated into a linear system of equations via the
decomposition of the error and control part into an operator basis, which
enables to rewrite it as [50]:

$B\textbf{x}^{(1)}=\textbf{y}^{(1)},$ (9)

where $B$ is a matrix that encodes the dynamics of $H_{0}$, $\textbf{y}^{(1)}$
are the error terms, and $\textbf{x}^{(1)}$, is a vector that contains the
solutions to the first order of control expansion.

An essential requirement of the Magnus scheme is that the control Hamiltonian
is decomposed to at least the same operators as the errors, in the final
interaction frame of $H_{0}$. However, it is not a strict requirement that the
control pulse has access to all error transitions in the initial frame. For
both defect systems, the leakage transitions that remove the population
outside of the $\Lambda$-subspace correspond to the C transitions. To cancel
out the leakage in both cases, we need to modify only one of the original sech
pulses that drive the $\Lambda$-transitions.

As we already mentioned, the control Hamiltonian is expanded in a power
series. In our case, we consider the total envelope:

$g^{(n)}(t)=g_{1}^{(n)}(t)\cos(\omega_{\text{d}}t)+g_{2}^{(n)}(t)\sin(\omega_{\text{d}}t),$
(10)

composed of two $\pi/2$-shifted envelopes $g_{l}^{(n)}(t)$, which are expanded
in Fourier series. In particular, we use only the cosine terms with
$g_{l}^{(n)}(t)$ given by:

$g_{l}^{(n)}(t)=\sum_{k}c_{l,k}^{(n)}\left(1-\cos\left(\frac{2\pi
tk}{T}\right)\right),$ (11)

where $n$ denotes the Magnus expansion order and $k$ denotes the Fourier
expansion order. We have also fixed $g_{l}^{(n)}(0)=g_{l}^{(n)}(T)=0$ such
that the corrective pulse is zero at the beginning and end of the evolution.
Throughout this paper, we always truncate the Magnus expansion to the first
order, i.e. we set $n=1$.

The driving frequency of the control $\omega_{\text{d}}$ is another free
parameter that can be tuned to lead to the most effective leakage
cancellation. Nonetheless, introducing and modifying a new laser field is more
challenging experimentally. To that end, we restrict the control pulse to have
the same frequency with the original pulse that we modulate.

###  IV.2 DRAG framework

An alternative route to leakage suppression is based on the adiabatic removal
of errors, which we analyze in this subsection. The DRAG technique is a widely
known method, extensively used for correcting leakage errors in
superconducting qubits [51, 52, 53]. Based on the DRAG formalism, analytically
derived controls are obtained via a time-dependent Schrieffer-Wolff
transformation. The generator of the transformation is $A(t)=e^{-iS(t)}$,
where $S(t)$ is a Hermitian operator, and leads to the effective DRAG
Hamiltonian

$H_{\text{D}}=A^{\dagger}H_{\text{db}}A+i\dot{A}^{\dagger}A.$ (12)

The dark-bright frame Hamiltonian is given by:

$H_{\text{db}}=(\Omega_{\text{eff}}f(t)\sigma_{\text{be}}+\text{H.c.})-\Delta\sigma_{\text{ee}},$
(13)

where $\Delta$ is the two-photon detuning and
$f(t)=\text{sech}(\sigma(t-t_{0}))$. Also, $|\text{b}\rangle$ is the bright
state and $|\text{e}\rangle$ the excited state. By requiring that the frame
transformation vanishes at the boundaries (i.e. $A(0)=A(T)=\textbf{1}$), the
target evolution in the initial (in this case, dark-bright) and DRAG frames
remains the same at the end of the pulse. To reduce the leakage errors, one
needs to find an appropriate adiabatic transformation, $S(t)$ that respects
the boundary conditions. Besides this restriction, $S(t)$ can be an arbitrary
Hermitian operator, which allows for the suppression of leakage errors.

The original DRAG scheme is designed to cancel out leakage errors of a ladder-
type system (e.g. transmon), which are caused by transitions between
consecutive levels. In our work, we extend this formalism to a
$\Lambda$-system. This is qualitatively different, since in our case the
population is removed from the system via transitions that link the ground
(qubit) states to an unwanted excited level. Moreover, the complexity is
increased, since each leakage transition is driven by both laser fields used
for the CPT control.

In the DRAG framework, $H_{\text{D}}$ has to be constrained in a way that it
implements an ideal evolution dictated by a target Hamiltonian. In our case,
the target Hamiltonian as defined in the CPT frame has the form:

$H_{\text{target}}=\frac{h_{x}^{(0)}(t)}{2}\sigma_{x,\text{be}}+h_{z}^{(0)}(\sigma_{\text{bb}}-\sigma_{\text{ee}}),$
(14)

where $|\text{d}\rangle$ ($|\text{b}\rangle$) is the dark (bright) state,
$|\text{e}\rangle$ is the excited state of the $\Lambda$-system
($|\text{e}\rangle=|\text{A}\rangle$), and $h_{z}^{(0)}$ is the two-photon
detuning. Also, we have defined $\sigma_{x,ij}=|i\rangle\langle
j|+|j\rangle\langle i|$ and $\sigma_{y,ij}=-i(|i\rangle\langle
j|-|j\rangle\langle i|)$. At this point, we should emphasize that our
treatment is different from Ref. [51], where the leakage-robust gates are
designed according to a target qubit-Hamiltonian in the rotating frame.
Instead, to reduce the leakage errors from the qubit subspace
($|\text{d}\rangle$, $|\text{b}\rangle$), we formulate an indirect treatment
which involves the bright-excited subspace.

Based on the target Hamiltonian of Eq. (14), our target constraints are:

$\displaystyle h_{x}^{(n)}$ $\displaystyle=$
$\displaystyle\text{Tr}[H_{\text{D}}^{(n)}\sigma_{x,\text{bA}}],$ (15)
$\displaystyle h_{y}^{(n)}$ $\displaystyle=$
$\displaystyle\text{Tr}[H_{\text{D}}^{(n)}\sigma_{y,\text{bA}}]=0,$ (16)
$\displaystyle h_{z}^{(n)}$ $\displaystyle=$
$\displaystyle\text{Tr}[H_{\text{D}}^{(n)}(\sigma_{\text{bb}}-\sigma_{\text{AA}})].$
(17)

The zero-th order target constraints ensure that
$H_{\text{D}}^{(0)}=H_{\text{target}}$. To satisfy the decoupling of the
$\Lambda$-system from the $|\text{C}\rangle$ leakage subspace we require the
following decoupling constraints, with $k\in\\{x,y\\}$:

$\displaystyle\text{Tr}[H_{\text{D}}^{(n)}\sigma_{k,\text{dC}}]$
$\displaystyle=$ $\displaystyle 0,$ (18)
$\displaystyle\text{Tr}[H_{\text{D}}^{(n)}\sigma_{k,\text{bC}}]$
$\displaystyle=$ $\displaystyle 0,$ (19)
$\displaystyle\text{Tr}[H_{\text{D}}^{(n)}\sigma_{k,\text{AC}}]$
$\displaystyle=$ $\displaystyle 0,$ (20)

as well as:

$\text{Tr}[H_{\text{D}}^{(n)}\sigma_{k,\text{dA}}]=0,$ (21)

which ensures that in the DRAG frame there is no transition between the dark
and excited states. Intuitively, for any order of $H_{\text{D}}^{(n)}$ with
$n\geq 0$, the elements of the DRAG Hamiltonian that do not correspond to the
target subspace should be zero. To obtain the $n$-th order DRAG Hamiltonian,
we expand both $S(t)$ and $H_{\text{db}}$ in power series. The appropriate
pulse modifications to the initial fields are obtained by satisfying the
constraints consistently. In the particular case of $R_{x}(\pi)$ rotations
(where the two-photon detuning is zero), the modulation of one laser vanishes,
which in terms of experimental requirements matches the Magnus scheme. More
details about the analytic derivation of the corrective envelopes are provided
in Appendix G.

##  V Control of SiV- system

###  V.1 Zero magnetic fields

We begin by testing our protocols for the SiV- system at $B=0$ T. The bright
transitions at zero magnetic fields are the A1, C1, B2, D2, B3, D3, A4 and C4
transitions. We consider the $\Lambda$-system formed by the states
$|1\rangle$, $|4\rangle$ and $|\text{A}\rangle$ shown in Fig. 3(b). By
choosing the $|\text{A}\rangle$ state to be the excited state of our
$\Lambda$-system, we avoid downward orbital relaxations (which are more likely
than the upward ones) that are present in the higher excited manifold.

Figure 3: The selection rules-based E-field polarizations of (a) lead to
cross-talk and four leakage transitions shown in (b). (c,d) The redefined
polarizations in the $xz$-plane can eliminate the cross-talk of (b). Figure 4:
Optical control of SiV- at $B=0$: Infidelity of $R_{x}(\phi)$ (a), and optimal
gate time (b), corresponding to four different protocols. The pulse envelopes
for $R_{x}(-\pi)$ rotations are shown in (c), (d), (e), (f) and the pulse
envelopes for $R_{x}(-\pi/2)$ are depicted in (g), (h), (i), (g). Gray: naive,
red: orthogonal, purple: Magnus and blue: DRAG.

We assume no initialization errors and a temperature of $T=4$ K. At this
temperature according to Ref. [41], the spin relaxation time is
$T_{1,\text{spin}}=2.4$ ms, the orbital relaxation is $T_{1,\text{orbit}}=38$
ns, and the dephasing time $T_{2}^{*}=35$ ns. We also set the optical lifetime
to be $\tau=4.5$ ns [45].

We start from the simplest approach of controlling the electronic spin, which
we refer to as naive, which is simply based on the CPT control of an ideal
Lambda system. It utilizes two laser fields according to the polarization
selection rules; in this setup the A1 transition is driven by $z$-polarized
light, and the A4 transition by circularly polarized light [see Figs. 3(a),
(b)]. Notice that we define the polarizations in the defect coordinate frame.
This choice of polarizations is far from optimal, since it introduces many
off-resonant errors. In particular, in Fig. 3(b) we show all the possible
transitions driven by the naive laser fields. The main error is the cross-talk
within the $\Lambda$-system, as this involves a transition that is detuned by
only $\delta_{\text{gs}}=50$ GHz, which is the ground state splitting. Each
laser field additionally drives transitions C1 and C4 (off-resonant by
$\delta_{\text{es}}=260$ GHz), introducing leakage outside of the
$\Lambda$-subspace.

In the naive approach, the leakage issue can be partially resolved by
considering more narrowband pulses. Nevertheless, the relaxation mechanisms
for longer pulses are detrimental to the optical control of the SiV-, leading
to enhanced gate errors. In Fig. 4(a), we show (gray curve) the infidelity of
arbitrary rotations, $R_{x}(\phi)$, and in Fig. 4(b) the optimal gate time.
The naive approach is the slowest of all our proposed protocols, since it
balances the trade-off between leakage and cross-talk errors with relaxation-
induced effects. The sech pulses for the optimal implementation of
$R_{x}(-\pi)$ and $R_{x}(-\pi/2)$ by the naive approach, are shown in the
panels of Fig. 4(c) and Fig. 4(g) respectively.

As mentioned in Sec. III, the cross-talk can be completely eliminated by
redefining the polarization of the two driving fields. In particular, we
choose the polarization vectors to be in the $xz$-plane as shown in Fig. 3(c).
We refer to this approach as orthogonal, shown with the red curve in Fig. 4.
This protocol allows us to use more broadband pulses (still well-protected
from leakage errors), while simultaneously reducing the effect of relaxations.
Consequently, the pulses in the orthogonal scheme can be up to four times
faster compared to the naive approach [see Fig. 4(b)], and the rotations
display lower infidelity [see Fig. 4(a)]. In Fig. 4(d) and Fig. 4(h) we show
the pulse envelopes of the orthogonal method for the optimal implementation of
$R_{x}(-\pi)$ and $R_{x}(-\pi/2)$ rotations, respectively.

The orthogonal approach sets the upper bound to the gate fidelities in the
absence of additional pulse shaping. However, the optimal gate time of this
method still lies within the regime where relaxation errors have non-zero
contribution. It is also apparent that by reducing the gate time, the leakage
errors gradually become more important. To mitigate the unwanted couplings to
the upper excited manifold we use the corrective techniques we described in
Sec. IV, and combine them with the orthogonal scheme to avoid the cross-talk
within the $\Lambda$-system. To minimize the experimental overhead, we do not
introduce new pulse envelopes, but instead modify the initial laser fields.

The first corrective method that we employ is the Magnus-based scheme [50]. In
our protocol, we modify only one of the initial laser fields, which in this
case is the laser field $E_{1}$ that drives the A1 transition, while the laser
field $E_{2}$ that drives the A4 transition remains intact. Both initial
fields introduce leakage outside of the $\Lambda$-system, via the excitation
of the C1 and C4 transitions [each driven by both lasers as shown in Fig.
3(d)]. The modulated pulse has additional cosine envelopes, and the Fourier
coefficients are obtained by solving a linear system of equations that we
describe in Appendix F. To optimize the performance of the Magnus scheme, we
search solutions that reduce the gate error for different Fourier series
truncation and gate time intervals, while keeping the Magnus truncation to the
first order.

We show the infidelity of the Magnus protocol in Fig. 4(a), and the optimal
gate time in Fig. 4(b) (purple curve). The Magnus scheme allows us to reach an
even faster regime while simultaneously restricting the leakage errors
contributions. Therefore, with a simple modulation of one of the initial
pulses we can retain the same fidelity as in the orthogonal scheme. The
optimal pulse envelopes for $R_{x}(\pi)$ and $R_{x}(-\pi/2)$ are shown in Fig.
4(e) and Fig. 4(i) respectively. In both cases, the top panel corresponds to
the modified pulse.

The alternative corrective method for leakage suppression is the DRAG
technique. This scheme requires pulse modulation of both initial laser fields,
but in the particular case of $R_{x}(\pi)$ rotations, the correction to the
field driving the A1 transition goes to zero. This is a consequence of the
condition for achieving $R_{x}(\pi)$ gates, which requires zero two-photon
detuning, leading to vanishing correction for one pulse. We notice that for
rotation angles $\phi>-\pi/2$ the DRAG method (blue curve) has a longer
optimal gate time compared to the Magnus method. For rotation angles
$\phi<-\pi/2$, however, the gate time is further reduced [Fig. 4(b)] and the
fidelity enhancement becomes more apparent [see blue curve of Fig. 4(a)]. We
should also mention that the DRAG pulse modulations are obtained analytically
[see Appendix G], but we also perform a simple optimization by redefining the
amplitude of the corrections. The optimal pulses for $R_{x}(\pi)$ and
$R_{x}(-\pi/2)$ are displayed in Fig. 4(f) and Fig. 4(g) respectively.

###  V.2 Non-zero magnetic fields

In the presence of an external non-axial B-field, the pairwise degeneracy of
each manifold is lifted, and all transitions become allowed and are no longer
spin-conserving. This phenomenon is caused by the off-axial Zeeman interaction
that gives rise to $S_{x}B_{x}$ and $S_{y}B_{y}$ terms, which cause spin-
mixing of the states.

Arbitrary magnetic field directions are more difficult to implement
experimentally since a vector magnet is required. For this reason, we assume a
fixed magnetic field orientation where the $B_{j}$ magnetic field components
in the SiV- frame are expressed in terms of the $B_{\parallel}$ and
$B_{\perp}$ magnetic field strengths in the lab frame. The lab frame magnetic
fields in an experimental setting would be applied parallel and perpendicular
to the cryostat axis, where the sample is placed. The parallel magnetic field
strengths reach up to $|B_{\parallel}|=9$ T and the perpendicular up to
$|B_{\perp}|=3$ T. In the coordinate frame of the SiV- defect, we define the
magnetic fields as:

$B_{x}=B_{\parallel}\cos(54.7^{o})+B_{\perp}\sin(54.7^{o}),$ (22) $B_{y}=0,$
(23)

and

$B_{z}=B_{\parallel}\sin(54.7^{o})-B_{\perp}\cos(54.7^{o}),$ (24)

where $\gamma=54.7~{}^{o}$ is the angle between the symmetry axis [1 1 1] and
the $(1~{}0~{}0)$ sample surface.

The spin coherence of the $S=1/2$ systems shows an angular dependence on the
direction of the external magnetic field. Larger deviation from the symmetry
axis results in enhanced spin-mixing, which consequently reduces the spin
coherence. In particular, according to Ref. [41], the reported
$T_{1,\text{spin}}$ for the SiV- reduces to $3.6~{}\mu$s at $20^{o}$
misaligned field and to 60 ns at $70^{o}$ misalignment. In our simulations, we
assume a spin relaxation time of $T_{1,\text{spin}}=300$ ns for the SiV-.

We consider zero two-photon detuning corresponding to $R_{x}(\pi)$ gates, and
we examine the $\Lambda$-system formed by the states $|1\rangle$, $|2\rangle$
and $|\text{A}\rangle$. We study only the performance of the orthogonal method
and using the results of the analysis of Sec. V.1, we assume a fixed laser
power, that balances the contribution of relaxations and leakage errors.

In Fig. 5, we show the fidelity [Fig. 5(a)] and gate time [Fig. 5(b)] of
$R_{x}(\pi)$ gates for the SiV- for a fixed laser field intensity. The maximum
fidelity for the SiV- corresponds to
$F_{\text{max}}^{\text{SiV${}^{-}$}}=0.975$ for $B_{\perp}=-2.8$ T and
$B_{\parallel}=-2$ T. The corresponding gate time is $T=0.235$ ns (versus
$T=0.3$ ns at $B=0$ T, see Appendix D). We observe a small increase in the
fidelity for the SiV- ($\sim 1\%$), and reduction of the gate time, compared
to orthogonal scheme at zero magnetic fields. Note that even though the laser
intensity is fixed, the gate time varies as the transition dipole overlaps
$\langle\psi_{i}|p_{k}|\psi_{j}\rangle$ (which change the effective Rabi
frequency and hence the bandwidth) are different for each magnetic field
strength.

Figure 5: Optical control of SiV- at $B\neq 0$: Dependence of the fidelity (a)
and duration (b) of $R_{x}(\pi)$ gates on the parallel and perpendicular (with
respect to the cryostat axis) magnetic field strengths. For $B\neq 0$ we
consider the A1-A2 $\Lambda$-system. The regions of low fidelity correspond to
weakly excited $\Lambda$-system.

In the low fidelity range, one or both $\Lambda$-transitions become weakly
allowed, while other transitions are driven more strongly. As an example, in
Fig. 5(b), the regions of longest gate time correspond to weakly allowed
$\Lambda$-transitions, which consequently lowers the fidelity in Fig. 5(a),
for the same magnetic field values. The long gate time of these weakly excited
transitions is associated with the transitionless-pulse condition that we
explain in Appendix. A, so for weak effective Rabi frequency, the pulses are
narrow-band. In the remaining low fidelity range, one of the
$\Lambda$-transitions is weakly driven, thus requiring an increase of the
laser power driving the particular transition to match the Rabi frequency of
the second $\Lambda$-transition. This is a requirement that we impose to the
CPT transformation to achieve $R_{x}$ gates. Consequently, with higher laser-
power, other bright transitions are driven more strongly, which results in low
overall fidelity. Nevertheless, our choice of $\Lambda$-system is not
restricted, and for the magnetic field values of low fidelity shown in Fig. 5,
we could instead select a different $\Lambda$-system.

##  VI Control of SnV- system

###  VI.1 Zero magnetic fields

The main advantage of the SnV- defect is its large ground and excited states
splittings. This suppresses both incoherent and coherent errors. Due to the
large energy separation, orbital relaxations are further suppressed compared
to the SiV-, meaning that high fidelity control is possible without
millikelvin cooling. The larger splitting also reduces the cross-talk of the
lasers driving the transitions. For zero magnetic fields, we find that the
bright transitions are A2, C2, B1, D1, B3, D3, A4, and C4. We form the
$\Lambda$-system by selecting the $|2\rangle$, $|4\rangle$ and
$|\text{A}\rangle$ states, as shown in Fig. 6(b). Again, we assume no
initialization errors, and in this case, a temperature of $T=6$ K. The spin
relaxation time is set to $T_{1,\text{spin}}=1.26$ ms, the orbital relaxation
time to $T_{1,\text{orbit}}=38$ ns, the dephasing time to $T_{2}^{*}=59$ ns
and the optical lifetime to $\tau=4.5$ ns [25].

Figure 6: The selection rules-based E-field polarizations of (a) lead to
cross-talk and four leakage transitions shown in (b). (c,d) The redefined
polarizations in the $yz$-plane can eliminate the cross-talk and additionally
two leakage transitions of (b). Figure 7: Optical control of SnV- at $B=0$:
Infidelity of $R_{x}(\phi)$ (a), and optimal gate time (b), corresponding to
four different protocols. The pulse envelopes for $R_{x}(-\pi)$ rotations are
shown in (c), (d), (e), (f) and the pulse envelopes for $R_{x}(-\pi/2)$ are
depicted in (g), (h), (i), (g). Gray: naive, red: orthogonal, purple: Magnus
and blue: DRAG.

In the naive scheme, which is based on selection rules, the A2 transition is
strongly driven by $z$-polarized light and the A4 by circularly polarized
light [Fig. 6(a)]. As a result, both cross-talk and leakage errors are present
in the optical control [Fig. 6(b)]. However, due to the large ground and
excited states splittings of the SnV-, these errors average out more
effectively than the SiV- system, when the pulses are not extremely broadband.

Again, we test the performance of our four protocols; the naive, the
orthogonal, the Magnus and the DRAG approaches. All of these schemes can
achieve faster and higher fidelity gates compared to the SiV- system. The
large excited state splitting of almost $3$ THz allows to implement the
rotations at less than hundreds of picoseconds, while preserving selectivity
of the A transitions. Consequently, in contrast to the SiV-, the gates can be
performed in a completely relaxation-free duration range, even for the naive
approach. Thus, there is no trade-off between frequency selectivity and
relaxation errors.

Starting with the naive approach (gray curve) we show the infidelity of
$R_{x}(\phi)$ rotations in Fig. 7(a) and the optimal gate time in Fig. 7(b).
We notice that the gates are well-protected and considerably faster than the
SiV-, as the error transitions average out efficiently. The selectivity of the
A transitions and the contribution of the cross-talk set a lower bound for the
optimal gate time, which on average is greater than 50 ps [see Fig. 7(b)].

In the orthogonal scheme, we can remove the cross-talk within the
$\Lambda$-system by redefining the polarization of the laser fields. One
example would be to select $xz$-polarization for the driving fields (such that
$\textbf{E}_{1}\cdot\textbf{d}_{\text{A4}}=0=\textbf{E}_{2}\cdot\textbf{d}_{\text{A2}}$),
similar to the SiV- system. However, we found that a different choice of
polarizations can additionally eliminate two out of the four leakage
transitions.

We model the Jahn-Teller (JT) contribution according to [25], which gives rise
to purely real $\langle p_{z}\rangle$ and $\langle p_{x}\rangle$ transition
dipoles and purely imaginary $\langle p_{y}\rangle$ transition dipoles. Under
this assumption, and considering $yz$-polarization for $\textbf{E}_{1}$ and
$\textbf{E}_{2}$, we find that by setting the polarizations to be:

$\textbf{E}_{1}=E_{01}\left(\textbf{y}-\frac{\langle
p_{y}\rangle_{\text{A4}}}{\langle
p_{z}\rangle_{\text{A4}}}\textbf{z}\right)e^{i(k_{1}x-\omega_{1}t)}+\text{c.c.}$
(25)

$\textbf{E}_{2}=E_{02}\left(\textbf{y}-\frac{\langle
p_{y}\rangle_{\text{A2}}}{\langle
p_{z}\rangle_{\text{A2}}}\textbf{z}\right)e^{i(k_{2}x-\omega_{2}t)}+\text{c.c.},$
(26)

we not only resolve the cross-talk, but we also fulfil the relations
$\textbf{E}_{1}\cdot\textbf{d}_{\text{C2}}=0$ and
$\textbf{E}_{2}\cdot\textbf{d}_{\text{C4}}=0$. Thus, the remaining leakage
transitions correspond to the driving of C2 by the $\textbf{E}_{2}$ field and
to the driving of C4 by the $\textbf{E}_{1}$ field [see Figs. 6(c), (d)]. In
Appendix C, we derive the polarizations we need to define to eliminate the
cross-talk and two out of the four leakage transitions for arbitrary JT
parameters.

With this simple redefinition of the E-field polarizations, the orthogonal
approach (red curve) achieves enhanced gate fidelities compared to the naive
scheme, as shown by the red curve in Fig. 7(a). By removing the cross-talk and
reducing the leakage, we manage to decrease the optimal gate time below 50 ps
[Fig. 7(b)]. The optimal pulse envelopes for the $R_{x}(-\pi)$ and
$R_{x}(-\pi/2)$ are shown in Fig. 7(d) and Fig. 7(h) respectively, which still
correspond to simple sech pulses.

The orthogonal scheme sets the lower bound of gate infidelities and gate
durations for unmodulated pulses. To go beyond this limit, we allow for pulse
modifications by using the Magnus- and DRAG-based protocols. First, we test
the performance of the Magnus protocol. In this method, only the pulse
envelope of the $\textbf{E}_{1}$-field is modified. Even though in the initial
frame the control pulse has access only to the C4 error transition and not the
C2 (which is driven by the $\textbf{E}_{2}$-laser field), we find that the
linear system we solve to specify the control is still well-defined. More
details for the Magnus scheme are given in Appendix F. The Magnus scheme has
the shortest optimal gate-time duration of all methods [see purple curve Fig.
7(b)]. Although it seems to underperform compared to the orthogonal scheme for
larger rotation angles, the pulses are much more broadband than in the former
case. The pulse envelopes are displayed in Fig. 7(e) [$R_{x}(-\pi)$], and Fig.
7(i) [$R_{x}(-\pi/2)$] , where the top panels involve pulse modulation of the
laser driving the A2 transition.

Finally, we evaluate the performance of the DRAG protocol. In Fig. 7(a), we
show with blue curve the infidelity of arbitrary rotations for the DRAG
scheme. For almost all rotation angles the infidelity owed to leakage is
suppressed, and the gate time is reduced compared to the orthogonal method.
The optimal pulse envelopes that implement $R_{x}(-\pi)$ and $R_{x}(-\pi/2)$
rotations are shown in Fig. 7(f) and Fig. 7(g) respectively. In both cases,
the modified pulse shown in the bottom panels corresponds to the laser driving
the A4 transition. In general, for rotations other than $R_{x}(-\pi)$, both
envelopes require modulation. However, we have performed a simple optimization
search on the DRAG corrections, which allows for a redefinition of their
amplitude strength. Thus, in this particular case, the optimal solution for
the $R_{x}(-\pi/2)$ gate involves modification of one of the initial driving
fields.

###  VI.2 Non-zero magnetic fields

Similar to the SiV-, we consider zero two-photon detuning which corresponds to
$R_{x}(\pi)$ rotations, and we select the $\Lambda$-system formed by the
states $|1\rangle$, $|3\rangle$ and $|\text{A}\rangle$. For the spin
relaxation time, we assume $T_{1,\text{spin}}=150$ ns. We consider the
orthogonal approach and in this case we select the $xz$-polarization
definition, similar to Sec. III. For non-zero magnetic field strengths, all
transitions become bright, and selecting the $yz$-polarizations for the lasers
does not offer any advantage (since the transition dipoles are modified due to
the Zeeman Hamiltonian). For each magnetic field strength, we would have to
select the optimal $\Lambda$-system, and define laser field polarizations that
eliminate the cross-talk and also reduce or remove the contribution of the
dominant leakage transitions. Although this analysis would be more complete,
we instead prefer to showcase the performance of a particular polarization of
the E-fields of the orthogonal scheme (that mitigates only the cross-talk),
and optimize in terms of the magnetic field strengths.

Figure 8: Optical control of SnV- at $B\neq 0$: Dependence of the fidelity (a)
and duration (b) of $R_{x}(\pi)$ gates on the parallel and perpendicular (with
respect to the cryostat axis) magnetic field strengths. For $B\neq 0$ we
consider the A1-A3 $\Lambda$-system. The regions of low fidelity correspond to
weakly excited $\Lambda$-system.

We assume the same magnetic field definitions in the defect coordinate frame
as in Sec. V.2, and we vary the parallel and perpendicular magnetic field
strenghts, with respect to the cryostat axis. In Fig. 8(a) we show the
fidelity and in Fig. 8(b) the gate time of $R_{x}(\pi)$ rotations for the
optimal laser field intensity. The maximum fidelity corresponds to
$F_{\text{max}}^{\text{SnV${}^{-}$}}=0.996$ for $B_{\perp}=0.3$ T and
$B_{\parallel}=0.2$ T and the gate time is $T=80$ ps. This is also the maximum
achievable fidelity for the $xz$-polarization in the absence of magnetic field
strengths, and the gate time is also close to the zero magnetic field case
[see Appendix D].

The low fidelity range arises for the same reasons as in the case of the SiV-
system. Specifically, the $\Lambda$-transitions are weakly excited and our
choice of $\Lambda$-system is not optimal for the particular magnetic fields.
As we mentioned for the SiV- system, even though the laser field intensity is
fixed in this case, the bandwidth of the pulse varies, since the transition
dipoles depend on the Zeeman Hamiltonian term. For the magnetic field
strengths where the fidelity is low, the choice of a different
$\Lambda$-system could still maintain high fidelity control.

##  VII Conclusions

In conclusion, we have designed optical control protocols for high-fidelity
rotations of two defect systems: the SiV- and SnV- in diamond. We use coherent
population trapping techniques combined with judicious choice of laser
polarizations to mitigate the cross-talk issue of the $\Lambda$-transitions
caused by the Jahn-Teller effect. Importantly, strain induced due to
integration of the defects in photonic structures can result in enhanced
orbital mixing and modification of the selection rules, and hence, more
intensified cross-talk. Thus, our cross-talk elimination approach could also
be beneficial in such a context. We implement simulations of arbitrary
rotations both in the absence and presence of external magnetic fields and
thoroughly test the maximum fidelity that we can reach without any additional
corrections. To an extended generalization, the choice of polarizations can
ensure both vanishing cross-talk and reduced number of leakage transitions.

For the SiV-, there is a trade-off between faster gates protected from
relaxation and slower gates protected from leakage errors. On the contrary,
for the SnV-, we can safely reach the gate time range where the dissipation
mechanisms are negligible, without causing enhanced leakage. We show that with
our orthogonal pulse scheme we achieve fast and high fidelity control for the
SnV- system, due to its larger ground and excited state splittings.

Further, we use a Magnus expansion technique, as well as a newly developed
version of the DRAG technique, to mitigate leakage errors when considering
broadband pulses. The corrective modifications in the Magnus and DRAG schemes
involve simple cosine envelopes that can be generated using arbitrary waveform
generators and electro-optical modulators, which create modulated pulses from
a CW laser. In general, pulses carved out of a CW laser have limited power and
speed, but a power enhancement could be achieved with a fast response optical
amplifier (e.g. semiconductor amplifiers with up to tens of GHZ repetition
rate [54]). Depending on experimental constraints (e.g. laser power,
duration), one could select the least demanding and most practical approach to
counteract leakage errors.

###### Acknowledgements.

The authors would like to thank Shahriar Aghaei, Jonas Becker, Alison Emiko
Rugar, Jelena Vuckovic, as well as Arian Vevzaee for valuable discussions. The
authors were supported by the United States National Science Foundation under
the grant 1838976.

##  Appendix A CPT control with sech pulses

As we mentioned in the main text, the destructive interference in the CPT
scheme leads to a dark state that is completely decoupled from the dynamics of
the three-level system. This is achieved by tuning the laser parameters
(relative amplitudes and phases), and satisfying the two-photon resonance
condition $(\Delta_{1}=\Delta_{2}\equiv\Delta)$, where $\Delta_{\text{j}}$ is
the detuning of the transition labeled $j$. The mapping from the initial qubit
states in the lab frame to the dark-bright basis can be performed via the
transformation:

$R_{\text{db}}=\begin{pmatrix}\cos\frac{\theta}{2}&-e^{-i\alpha}\sin\frac{\theta}{2}\\\
e^{i\alpha}\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{pmatrix}.$ (27)

Effectively, this transformation defines the rotation axis of the qubit, which
is $\textbf{n}=(\sin\theta\cos\alpha,\sin\theta\sin\alpha,\cos\theta)$, while
it also enables the reduction of the initial problem into a two-level system.
In particular, the Hamiltonian in the dark-bright basis reads:

$H_{\text{db}}=\Omega_{\text{eff}}f(t)e^{i\Delta
t}\sigma_{\text{be}}+\text{H.c.},$ (28)

where $\sigma_{\text{be}}=|\text{b}\rangle\langle\text{e}|$, with
$|\text{b}\rangle$ being the bright state and $|\text{e}\rangle$ the excited
state. Also, the effective Rabi frequency in this frame is expressed in terms
of the original Rabi frequencies as
$\Omega_{\text{eff}}=\sqrt{|\Omega_{1}|^{2}+|\Omega_{2}|^{2}}$. For a general
pulse envelope, the two-level problem is not analytically solvable. Here we
consider hyperbolic secant pulses (i.e. $f(t)=\text{sech}(\sigma t)$), that
have been proven to be analytically solvable [55], and lead to a rotation in
the qubit subspace given by [48, 49]:

$U_{0}=\begin{pmatrix}1&0\\\ 0&e^{-i\phi}\end{pmatrix}.$ (29)

As shown by Eq. (29), the dark state does not evolve, whereas the bright state
picks up a phase given by $\phi=2\tan^{-1}(\sigma/\Delta)$, where $\Delta$ is
the detuning and $\sigma$ the bandwidth. Control of both rotation axis and
angle is achieved by combining CPT with hyperbolic secant pulses, which allows
us to design arbitrary single-qubit gates. The only additional requirement is
that the bandwidth is equal to the effective Rabi frequency in the CPT frame
($\sigma=\Omega_{\text{eff}}$), such that the pulse is transitionless, i.e.
the population returns to the ground states at the end of the pulse.

##  Appendix B Details of the simulations

In this section, we provide further details regarding our simulations. First,
based on Ref. [44], we use the laser power applied on the SiV- system for
$\pi$-rotations to calculate the electric field amplitude and estimate the
Rabi frequencies. For a numerical aperture NA=0.7, the spot-size (radius) of
the laser is given by $w_{0}=\lambda_{0}/(\pi\text{NA})$, where $\lambda_{0}$
is the wavelength of a specific transition for the SiV- or the SnV-, which can
be assumed to be close to the central transition. For the SiV- the central
wavelength is $\lambda_{0}\approx 736$ nm, while for the SnV- it is
$\lambda_{0}\approx 620$ nm. Assuming an emitter focused at the center of the
beam, the intensity is related with the power and spot-size by the expression:

$I=\frac{P_{0}}{\pi w_{0}^{2}},$ (30)

while it can also be expressed as:

$I=\frac{cn\epsilon_{0}}{2}|E_{0}|^{2},$ (31)

where $c$ is the speed of light, $n=2.4$ is the refractive of diamond, $E_{0}$
is the electric field amplitude and $\epsilon_{0}$ is the vacuum permittivity.
The factor of 1/2 comes from averaging the intensity. By combining Eq. (30)
with Eq. (31) we can express the electric field in terms of the laser power
as:

$|E_{0}|=\sqrt{\frac{2P_{0}}{\pi w_{0}^{2}cn\epsilon_{0}}}.$ (32)

From Eq. (32), we calculate the electric field amplitude based on the laser
powers of Ref. [44], shown in Fig. B.1. For the SiV-, the maximum electric
field amplitude we have considered is $E_{0}=8.5\times 10^{4}$ (V/m), while
for the SnV-, we have considered up to $E_{0}\approx 1\times 10^{6}$ (V/m).
(These values exclude the numerically optimized DRAG pulses, whose amplitude
corrections have a multiplicative factor $|c|<4$). Further, for the
$z$-transition dipoles, we have taken into account the experimental
enhancement factor of 2 of the $z$-dipole. In general, the optimal ranges of
operation we found for both defects are smaller than these maximum $E_{0}$
amplitudes, so the laser power should correspond to the experimentally safe
and achievable ranges.

Figure B.1: Electric field amplitude versus the square root of the laser
power, for the SiV- central wavelength (blue) and the SnV- central wavelength
(red).

Further, we calculate the Rabi frequencies for each transition as:

$\Omega^{ij}=\alpha\frac{er_{0}|E_{0}|}{\hbar}\langle\psi_{i}|p_{k}|\psi_{j}\rangle,$
(33)

where $e$ is the electronic charge, and for $r_{0}$ we assume $r_{0}=0.53$ Å.
We estimate the multiplicative factor $\alpha\approx 6.663$ for the SiV- and
$\alpha\approx 3.3$ for the SnV-. In particular, for the SiV-, that would give
rise to a dipole moment of approximately $\mu\approx 16.6$ Debye, which is
close to $\mu=14.3$ Debye reported in [32]. Also,
$\langle\psi_{i}|p_{k}|\psi_{j}\rangle$, is the dipole overlap of the
transition and the matrices $p_{x}$, $p_{y}$, $p_{z}$ are given by group
theory [16]. We should mention that in our simulations we start by defining
the driving Hamiltonian without the factor of $1/2$ in front of the Rabi-
frequencies, which would be a result of the RWA. This means that the $E_{0}$
value should be twice as much as the approximate values we report above. To
calculate the eigensystem of all eight levels of each defect, we consider
three main interaction terms: the spin-orbit coupling, the Jahn-Teller and the
Zeeman effects.

Regarding the Lindblad relaxation operators, we follow a similar convention as
in Ref. [46]. First, for the dephasing mechanism, we assume an equal dephasing
of all states:

$G_{\text{deph}}=\frac{1}{\sqrt{T_{2}^{*}}}|i\rangle\langle i|.$ (34)

Spin-relaxation mechanisms lead to a change of the spin-state while preserving
the same orbital part, which occur within the ground or excited state
manifold. We define the Lindblad spin-relaxation operators as:

$G_{\text{spin}}=\frac{1}{\sqrt{2T_{1,\text{spin}}}}|i\rangle\langle j|.$ (35)

Orbital dissipation mechanisms occur between different orbital states of the
same spin projection. We define the associated Lindblad operator as:

$G_{\text{orbit}}=\sqrt{F_{\text{orbit}}}|i\rangle\langle j|,$ (36)

where the decay rate $F_{\text{orbit}}$ is different for an upward or downward
relaxation. For a downward relaxation we define:

$F_{\text{orbit, down}}=\frac{1}{T_{1,\text{orbit}}(1+e^{-|\Delta
E|/k_{b}T})}$ (37)

and for an upward:

$F_{\text{orbit,up}}=F_{\text{orbit,down}}e^{-|\Delta E|/k_{b}T},$ (38)

i.e. the orbital relaxations are scaled by Boltzman factors, with the upward
orbital relaxations being less probable. $\Delta E$ is the energy difference
between the levels that participate in the orbital relaxation mechanism.
Finally, for the lifetime relaxations, we define the Lindblad operators as:

$G_{\text{lifetime}}=\frac{1}{\sqrt{\tau}}|i\rangle\langle j|,$ (39)

where $\sigma_{ij}=|i\rangle\langle j|$ corresponds to a bright transition,
and $\tau$ is the lifetime.

##  Appendix C General method for removing the cross-talk and one leakage
transition

As we mentioned in the main text, we can always redefine the polarization of
the laser fields to remove completely the cross-talk within the
$\Lambda$-system. We also mentioned that for the SnV- and by using the Jahn-
Teller (JT) parameters of [25], the polarization of the E-fields was found to
be additionally orthogonal to one leakage transition each.

However, this extra property depends on the modeling of the JT interaction. To
resolve this subtlety, we derive analytically the transition dipoles for
arbitrary JT parameters. Assuming no crystal strain, and working at $B=0$ T,
the only non-zero interaction terms are the spin-orbit coupling and the JT
effect.

By expressing the interaction Hamiltonian in the $|e_{\pm}\rangle$ orbital
basis and in the $\\{|\uparrow\rangle,|\downarrow\rangle\\}$ spin basis, the
two interaction terms read:

$H_{\text{g/e}}=\begin{pmatrix}Q_{x,\text{g/e}}&Q_{y,\text{g/e}}\\\
Q_{y,\text{g/e}}&-Q_{x,\text{g/e}}\end{pmatrix}\otimes\textbf{1}-\frac{\lambda_{\text{SO},\text{g/e}}}{2}L_{z}\otimes
S_{z},$ (40)

where $Q_{x,\text{g/e}}=Q_{\text{g/e}}\cos\phi_{\text{g/e}}$,
$Q_{y,\text{g/e}}=Q_{\text{g/e}}\sin\phi_{\text{g/e}}$,
$\lambda_{\text{SO}_{\text{g/e}}}=\Delta
E_{\text{g/e}}\cos\theta_{\text{g/e}}$ and $Q_{\text{g/e}}=\Delta
E_{\text{g/e}}/2\sin\theta_{\text{g/e}}$. The subscript $g$ and $e$ denote
ground and excited state respectively. The parameter $\theta_{\text{g/e}}$ can
be tuned so as to give the relative strength of the SO and JT contributions
that are experimentally observed. The unnormalized eigenvectors are given by:

$\displaystyle v_{1,\text{g/e}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&e^{i\phi_{\text{g/e}}}\tan\frac{\theta_{\text{g/e}}}{2}&0&1\end{pmatrix}^{T}$
(41) $\displaystyle v_{2,\text{g/e}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}e^{i\phi_{\text{g/e}}}\cot\frac{\theta_{\text{g/e}}}{2}&0&1&0\end{pmatrix}^{T}$
(42) $\displaystyle v_{3,\text{g/e}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&-e^{i\phi_{\text{g/e}}}\cot\frac{\theta_{\text{g/e}}}{2}&0&1\end{pmatrix}^{T}$
(43) $\displaystyle v_{4,\text{g/e}}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}-e^{i\phi_{\text{g/e}}}\cot\frac{\theta_{\text{g/e}}}{2}&0&1&0\end{pmatrix}^{T},$
(44)

where $v_{1,\text{g/e}}$ and $v_{2,\text{g/e}}$ correspond to the eigenenergy
$-\Delta E_{\text{g/e}}/2$ and the eigenvectors $v_{3,\text{g/e}}$ and
$v_{4,\text{g/e}}$ to the eigenenergy $\Delta E_{\text{g/e}}/2$.

As an example let’s assume that we use the $\Lambda$-transitions A2 and A4,
and that we want to make the $\textbf{E}_{1}$ field orthogonal to
$\textbf{d}_{\text{A4}}$ and $\textbf{d}_{\text{C2}}$. Under this notation,
the non-zero transitions would be between $v_{1,\text{g}}\leftrightarrow
v_{1,\text{e}}$ and $v_{1,\text{g}}\leftrightarrow v_{3,\text{e}}$. Thus, the
transition dipole $\textbf{d}_{1}=\textbf{d}_{v_{1,\text{g}}v_{1,\text{e}}}$
(of A2) is:

$\textbf{d}_{1}=\frac{1}{|\sec\frac{\theta_{\text{e}}}{2}\sec\frac{\theta_{\text{g}}}{2}|}\begin{pmatrix}-(e^{-i\phi_{\text{e}}}\tan\frac{\theta_{\text{e}}}{2}+e^{i\phi_{\text{g}}}\tan\frac{\theta_{\text{g}}}{2})\\\
i(e^{i\phi_{\text{e}}}\tan\frac{\theta_{\text{e}}}{2}-e^{i\phi_{\text{g}}}\tan\frac{\theta_{\text{g}}}{2})\\\
2(1+e^{i(\phi_{\text{e}}+\phi_{\text{g}})}\tan\frac{\theta_{\text{e}}}{2}\tan\frac{\theta_{\text{g}}}{2})\end{pmatrix}.$
(45)

Similarly the dipole
$\textbf{d}_{2}=\textbf{d}_{v_{1,\text{g}}v_{3,\text{e}}}$ (of C2) is:

$\textbf{d}_{2}=\frac{1}{|\sec\frac{\theta_{\text{e}}}{2}\sec\frac{\theta_{\text{g}}}{2}|}\begin{pmatrix}(e^{i\phi_{\text{e}}}\cot\frac{\theta_{\text{e}}}{2}-e^{i\phi_{\text{g}}}\tan\frac{\theta_{\text{g}}}{2})\\\
-i(e^{i\phi_{\text{e}}}\cot\frac{\theta_{\text{e}}}{2}+e^{i\phi_{\text{g}}}\tan\frac{\theta_{\text{g}}}{2})\\\
2(-1+e^{i(\phi_{\text{e}}+\phi_{\text{g}})}\cot\frac{\theta_{\text{e}}}{2}\tan\frac{\theta_{\text{g}}}{2})\end{pmatrix}.$
(46)

Finally, the dipole $\textbf{d}_{3}=\textbf{d}_{v_{3,\text{g}}v_{1,\text{e}}}$
(of A4) is:

$\textbf{d}_{3}=\frac{1}{|\sec\frac{\theta_{\text{e}}}{2}\sec\frac{\theta_{\text{g}}}{2}|}\begin{pmatrix}e^{i\phi_{\text{g}}}(\cot\theta_{\text{g}}+\csc\theta_{\text{g}})-e^{i\phi_{\text{e}}}\tan\frac{\theta_{\text{e}}}{2}\\\
-i(e^{i\phi_{\text{g}}}(\cot\theta_{\text{g}}+\csc\theta_{\text{g}})+e^{i\phi_{\text{e}}}\tan\frac{\theta_{\text{e}}}{2})\\\
2(1-e^{i(\phi_{\text{e}}+\phi_{\text{g}})}\cot\frac{\theta_{\text{g}}}{2}\tan\frac{\theta_{\text{e}}}{2})\end{pmatrix}.$
(47)

The goal is to make $\textbf{E}_{1}\cdot\textbf{d}_{2}=0$ and
$\textbf{E}_{1}\cdot\textbf{d}_{3}=0$. Thus, considering the general
expression:

$\textbf{E}_{1}=E_{01}(c_{1}\textbf{x}+c_{2}\textbf{y}+c_{3}\textbf{z})e^{i(\textbf{k}\cdot\textbf{r}-\omega
t)}+\text{c.c.},$ (48)

we specify the $c_{2}$ and $c_{3}$ that satisfy the orthogonality relations:

$c_{3}=\frac{-ic_{1}(\cos\theta_{\text{e}}+\cos\theta_{\text{g}})}{2(\sin\theta_{\text{e}}\sin\phi_{\text{e}}+\sin\theta_{\text{g}}\sin\phi_{\text{g}})}$
(49)
$c_{2}=\frac{c_{1}(-\cos\phi_{\text{e}}\sin\theta_{\text{e}}+\cos\phi_{\text{g}}\sin\theta_{\text{g}})}{\sin\theta_{\text{e}}\sin\phi_{\text{e}}+\sin\theta_{\text{g}}\sin\phi_{\text{g}}}.$
(50)

Similarly, we could follow the same procedure to satisfy
$\textbf{E}_{2}\cdot\textbf{d}_{\text{A2}}=0=\textbf{E}_{2}\cdot\textbf{d}_{\text{C4}}$.

Alternatively, we could choose the polarization of $\textbf{E}_{1}$ such that
we satisfy the orthogonality relation to the A4 $\Lambda$-transition while
also minimizing both leakage transitions C2 and C4 that are driven by each
laser field (and similarly for the $\textbf{E}_{2}$ field).

##  Appendix D Gate time dependence of the fidelity

Here we show the time dependence of the fidelity of the gates for the SiV- and
the SnV- defects. For both systems we consider the orthogonal scheme and use a
combination of $xz$-polarizations for the E-fields (such that we cancel the
cross-talk errors). In Fig. D.1 (a) and Fig. D.1(b) we show the fidelity
versus the gate time for $R_{x}(\pi)$ and $R_{x}(-\pi/2)$ rotations for the
SiV-.

Figure D.1: Gate time dependence of the fidelity for $R_{x}(-\pi)$ (a) and
$R_{x}(-\pi/2)$ gates for the SiV-. Fidelity of $R_{x}(\phi)$ rotations (c)
and rotation angles (d) versus the gate time and two-photon detuning.

Narrowband pulses suffer from relaxation errors, while significantly broadband
pulses suffer from enhanced leakage errors. The optimal gate time for both
rotations is $T=0.3$ ns, with a fidelity close to $F=0.97-0.98$. In Fig.
D.1(c) we show the fidelity of arbitrary rotations versus the gate time and
two-photon detuning. For $\Delta\gtrsim 200$ GHz, which corresponds to the
excited state splitting, the upper-excited manifold is driven more strongly
leading to significant leakage errors. Nevertheless, the same rotation angles
can be implemented with negative detuning at high fidelities. The
corresponding rotation angles are shown in Fig. D.1(d).

Similarly, we show the gate time dependence of $R_{x}(\pi)$ and
$R_{x}(-\pi/2)$ for the SnV- system in Fig. D.2(a) and Fig. D.2(b).

Figure D.2: Gate time dependence of the fidelity for $R_{x}(-\pi)$ (a) and
$R_{x}(-\pi/2)$ gates for the SnV-. Fidelity of $R_{x}(\phi)$ rotations (c)
and rotation angles (d) versus the gate time and two-photon detuning.

The infidelity of arbitrary rotations in logarithmic scale is shown in Fig.
D.2(c) and the rotation angles are shown in Fig. D.2(d). In this case, due to
the large excited state splitting of the defect, the positive rotation angles
exhibit low infidelity.

##  Appendix E Effect of relaxations on the fidelity

In Fig. E.1(a), we test the fidelity of $R_{x}(\pi)$ rotations (orthogonal
scheme) for the case of the SnV-, considering two different temperatures. For
$T=3$ K, we assume $T_{2}^{*}=540$ ns and $T_{1,\text{spin}}=10.2$ ms, while
for $T=6$ K we assume $T_{1,\text{spin}}=1.26$ ms and $T_{2}^{*}=59$ ns. We
observe that the two curves are almost identical for gate times $T<0.1$ ns,
while for longer gates the deviation starts to increase further.

In general, for all rotation angles, the optimal fidelity range should lie
below $\sim 0.1$ ns for the SnV-, such that the contribution of the
relaxations is negligible and the fidelity is almost independent of the
temperature. On the other hand, for the SiV-, the optimal range is shifted to
longer times, as the leakage errors tend to increase substantially for
broadband pulses [Fig. E.1(b)]. (Again we show the performance of the
orthogonal scheme.) Upon cooling to mK temperatures, the phonon induced
relaxations can be suppressed substantially [42], since the qubit states
become decoupled from the phonon bath.

Figure E.1: (a) Fidelity of $R_{x}(\pi)$ rotations at zero magnetic fields for
the SnV-, for temperature $T=3$ K (blue) and $T=6$ K (red). For a gate time
$T<$0.1 ns there is no temperature dependence of the fidelity, as below this
gate time the relaxations seize to contribute. (b) Fidelity of $R_{x}(\pi)$
rotations for the SiV- in the presence of relaxations (red) and without
dissipation mechanisms (blue).

##  Appendix F Corrections to leakage errors with the Magnus expansion
approach

In this section, we provide the linear system of equations of the Magnus
methods, for both defect systems. According to the Magnus expansion approach
of [50], we specify the control fields by reducing the problem to a linear set
of equations. We consider zero external magnetic fields since, in this case,
we have a smaller number of unwanted transitions that we want to cancel out.
We further distinguish two cases: i) resonant driving ($R_{x}(\pi)$ rotations)
and ii) off-resonant driving ($R_{x}(\phi)$ rotations). As we will show in
subsequence, we can generalize from the resonant to the off-resonant case by a
slight modification of our linear system of equations.

Starting from the SiV- system, our goal is to find a corrective Hamiltonian
$W(t)$ to suppress the leakage errors. In the main text, we chose the
$\Lambda$-system formed by the states $|1\rangle,|4\rangle$ and
$|\text{A}\rangle$. Assuming perfect initialization, the main error of our
orthogonal scheme is the driving of the C1 and C4 transitions, which leads to
leakage outside of our $\Lambda$-system.

We first decompose the error terms of our Hamiltonian into the Gell-Mann
basis. Starting from the lab frame, we fix two orthogonal polarizations for
the $\Lambda$-transitions, as described in the main text. Thus, in the
interaction frame, our initial Hamiltonian including only the A1, A4 and C1
and C4 transitions reads:

$\begin{split}H&=(|\Omega_{1}^{\text{A1}}|e^{i\phi_{1}}e^{i\Delta
t}\sigma_{1\text{A}}+\Omega_{1}^{\text{C1}}e^{i(\Delta-\delta_{\text{es}})t}\sigma_{1\text{C}}+|\Omega_{1}^{\text{C4}}|e^{i\phi_{\text{C}4}}e^{i(\Delta-(\delta_{\text{es}}-\delta_{\text{gs}}))t}\sigma_{4\text{C}}+\text{H.c.})\\\
&+(|\Omega_{2}^{\text{A4}}|e^{i\phi_{1}}e^{i\Delta
t}\sigma_{4\text{A}}+|\Omega_{2}^{\text{C1}}|e^{-i\phi_{\text{C}4}}e^{i(\Delta-(\delta_{\text{es}}+\delta_{\text{gs}}))t}\sigma_{1\text{C}}-|\Omega_{2}^{\text{C4}}|e^{i(\Delta-\delta_{\text{es}})t}\sigma_{4\text{C}}+\text{H.c.})f(t),\end{split}$
(51)

where $\sigma_{ij}=|i\rangle\langle j|$, $\delta_{\text{es}}=260$ GHz,
$\delta_{\text{gs}}=50$ GHz are the excited and ground state splittings
respectively, $f(t)=\text{sech}(\sigma(t-t_{0}))$ and $\Delta$ is the two-
photon detuning. Note that in order to define the error and ideal Hamiltonians
we need only these four transitions. For the SiV- at zero magnetic fields, we
also find that $|\Omega_{2}^{\text{C1}}|=|\Omega_{1}^{\text{C4}}|$, and
$\phi_{\text{C2}}=-\phi_{\text{C}4}$. Since we are further interested for the
$R_{x}$ rotations, we fix $|\Omega_{1}^{\text{A1}}|=|\Omega_{2}^{\text{A4}}|$.
The transformation matrix to the dark-bright frame is given by:

$\begin{split}R&=\frac{\sigma_{11}-\sigma_{14}+\sigma_{41}+\sigma_{44}}{\sqrt{2}}+\sigma_{22}+\sigma_{33}\\\
&+\sigma_{\text{AA}}+\sigma_{\text{BB}}+\sigma_{\text{CC}}+\sigma_{\text{DD}},\end{split}$
(52)

which transforms our initial ground states into the dark-bright states (i.e.
$|1\rangle\rightarrow|\text{d}\rangle$ and
$|4\rangle\rightarrow|\text{b}\rangle$), and the initial Hamiltonian $H$ into
$H_{\text{db}}=RHR^{\dagger}$. Our target Hamiltonian in the db-frame is given
by:

$H_{0,\text{db}}=(\sigma e^{i\Delta t}\sigma_{\text{bA}}+\text{H.c.})f(t),$
(53)

where we have substituted $\sigma=\sqrt{2}|\Omega_{1}^{\text{A1}}|$, and we
have defined $|\text{b}\rangle=1/\sqrt{2}(|1\rangle+|4\rangle)$ to be the
bright state. Next, we apply one more transformation by going to the
interaction picture generated by the ideal Hamiltonian, $H_{0,\text{db}}$,
which is given by $U_{0}$:

$\begin{split}U_{0}&=\sigma_{11}+\sigma_{22}+\sigma_{33}+\sigma_{\text{BB}}+\sigma_{\text{CC}}+\sigma_{\text{DD}}\\\
&+\cos\theta(\sigma_{44}+\sigma_{\text{AA}})+i\sin\theta(\sigma_{4\text{A}}+\sigma_{\text{A}4}).\end{split}$
(54)

Note that $U_{0}$ of Eq. (54) should not be confused with the target gate
$U_{0}$ of Sec. A. Here, $\theta(t)$ is the integral of the pulse envelope,
which for the resonant case reads:

$\begin{split}\theta(t)&=\int_{0}^{t}\sigma\text{sech}(\sigma(t^{\prime}-t_{0}))dt^{\prime}\\\
&=2(\tan^{-1}(e^{\sigma(t-t_{0})})-\tan^{-1}(e^{-\sigma t_{0}})).\end{split}$
(55)

For the off-resonant case, we need to evaluate:

$\begin{split}\theta_{\pm}(t)&=\int_{0}^{t}\sigma\text{sech}(\sigma(t^{\prime}-t_{0}))e^{\pm
i\Delta t^{\prime}}dt^{\prime}\\\ &=\sigma e^{\pm i\Delta
t_{0}}\int_{-t_{0}}^{t-t_{0}}e^{\pm i\Delta u}\text{sech}(\sigma
u)du.\end{split}$ (56)

The solution of these indefinite integrals is:

$g_{\pm}(u)={{}_{2}}F_{1}(1,\frac{\sigma\pm i\Delta}{2\sigma},\frac{3\sigma\pm
i\Delta}{2\sigma},-e^{2\sigma u})\frac{e^{i[(\sigma\pm\Delta)u\pm\Delta
t_{0}]}}{\frac{\sigma\pm i\Delta}{2\sigma}},$ (57)

where ${}_{2}F_{1}(a,b,c,z)$ is the Gauss hypergeometric function. By
evaluating this expression in the limits of the integration, we obtain
$\theta_{\pm}(t)$. We notice that the two functions $\theta_{\pm}(t)$ are
complex conjugates, which simplifies the equations for the off-resonant case.
To obtain the set of equation for off-resonant driving, we replace $\theta(t)$
of the resonant case by $\tilde{\theta}(t)=|\theta_{+}(t)|=|\theta_{-}(t)|$.

The error terms in the interaction picture of $H_{0,\text{db}}$, are given by
$H_{\text{error}}=U_{0}(H_{\text{db}}-H_{0,\text{db}})U_{0}^{\dagger}$. Using
the Gell-Mann basis we now decompose the error terms (where the RWA has been
applied) into the operators

$\displaystyle H_{\text{er},1}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{(|\Omega_{1}^{\text{C1}}|+|\Omega_{2}^{\text{C4}}|)\cos(t\delta_{\text{es}})-2|\Omega_{1}^{\text{C4}}|\sin(t\delta_{\text{es}})\sin(\phi_{\text{C}4}+t\delta_{\text{gs}})}{\sqrt{2}}f(t)L_{s,17}$
(58) $\displaystyle H_{\text{er},2}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta(t)\cos(\phi_{1}/2+\delta_{\text{es}}t)(|\Omega_{1}^{\text{C1}}|-|\Omega_{2}^{\text{C4}}|+2|\Omega_{1}^{\text{C4}}|\cos(\phi_{\text{C}4}+\delta_{\text{gs}}t))}{\sqrt{2}}f(t)L_{s,47}$
(59) $\displaystyle H_{\text{er},3}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\sin\theta(t)\sin(\phi_{1}/2+\delta_{\text{es}}t)(|\Omega_{1}^{\text{C1}}|-|\Omega_{2}^{\text{C4}}|+2|\Omega_{1}^{\text{C4}}|\cos(\phi_{\text{C}4}+\delta_{\text{gs}}t))}{\sqrt{2}}f(t)L_{s,57}$
(60) $\displaystyle H_{\text{er},4}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{(|\Omega_{1}^{\text{C1}}|+|\Omega_{2}^{\text{C4}}|)\sin(\delta_{\text{es}}t)+2|\Omega_{1}^{\text{C4}}|\cos(\delta_{\text{es}}t)\sin(\phi_{\text{C}4}+\delta_{\text{gs}}t)}{\sqrt{2}}f(t)L_{a,17}$
(61) $\displaystyle H_{\text{er},5}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta(t)\sin(\phi_{1}/2+\delta_{\text{es}}t)(|\Omega_{1}^{\text{C1}}|-|\Omega_{2}^{\text{C4}}|+2|\Omega_{1}^{\text{C4}}|\cos(\phi_{\text{C}4}+\delta_{\text{gs}}t))}{\sqrt{2}}f(t)L_{a,47}$
(62) $\displaystyle H_{\text{er},6}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\sin\theta(t)\cos(\phi_{1}/2+\delta_{\text{es}}t)(-|\Omega_{1}^{\text{C1}}|+|\Omega_{2}^{\text{C4}}|-2|\Omega_{1}^{\text{C4}}|\cos(\phi_{\text{C}4}+\delta_{\text{gs}}t))}{\sqrt{2}}f(t)L_{a,57},$
(63)

where $L_{s}$ are the symmetric and $L_{a}$ the anti-symmetric Gell-Mann
operators given by [56]:

$L_{s,jk}=|j\rangle\langle k|+|k\rangle\langle j|$ (64)
$L_{a,jk}=-i(|j\rangle\langle k|-|k\rangle\langle j|),$ (65)

where $1\leq j<k\leq d$, with $d=8$ being the dimension of the Hilbert space.

As we explained in the main text, we need a corrective Hamiltonian that is
decomposed into at least the same operators as the error terms. In other
words, starting from the lab frame, we are looking for control pulses that
drive the C1 and C4 unwanted transitions. However, it is not a strict
requirement that the control pulse drives both error transitions (we will show
a counter-example later for the SnV-).

In the general case, the lab-frame control Hamiltonian for the SiV- has the
form:

$W_{\text{lab}}^{(n)}=(\Omega_{1}^{\text{A1}}\sigma_{1\text{A}}+\Omega_{1}^{\text{C1}}\sigma_{1\text{C}}+\Omega_{1}^{\text{C4}}\sigma_{4\text{C}}+\text{H.c.})g^{(n)}(t),$
(66)

where
$g^{(n)}(t)=(g_{1}^{(n)}\cos(\omega_{\text{d}}t)+g_{2}^{(n)}\sin(\omega_{\text{d}}t)$
and $\omega_{\text{d}}$ is the frequency of the control. The amplitudes
$g_{1/2}^{(n)}$ are expanded in a Fourier series:

$g_{1/2}^{(n)}=\sum_{k}c_{k,1/2}^{(n)}\left(1-\cos\left(\frac{2\pi
kt}{T}\right)\right),$ (67)

where $T$ is the gate time, $k$ is the order of truncation of the Fourier
expansion, and $n$ is the order of truncation of the Magnus series expansion.

We follow the same procedure of transforming our lab frame control Hamiltonian
into the interaction picture generated by $H_{0,\text{db}}$. More accurately,
we first transform $W_{\text{lab}}^{(n)}(t)$ into the interaction picture via
$R_{\text{int}}=\sum_{j}e^{i\omega_{j}t}|j\rangle\langle j|$ (where
$\omega_{j}$ are the eigenergies), then to the dark-bright frame via
$R_{\text{db}}$, and finally into the interaction picture generated by
$H_{0,\text{db}}$. After this series of transformations, the decomposition of
$W(t)$ in the final frame (and after applying the RWA) yields:

$\displaystyle W_{1}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A1}}|\sin\theta(-g_{2}\cos(\phi_{1}/2+t\Delta_{\text{c}})+g_{1}\sin(\phi_{1}/2+t\Delta_{\text{c}}))}{\sqrt{2}}L_{s,14}$
(68) $\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A1}}|\cos\theta(g_{1}\cos(\phi_{1}/2+t\Delta_{\text{c}})+g_{2}\sin(\phi_{1}/2+t\Delta_{\text{c}}))}{\sqrt{2}}L_{s,15}$
(69) $\displaystyle W_{3}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{C1}}|(g_{1}\cos(t\bar{\Delta})+g_{2}\sin(t\bar{\Delta}))+|\Omega_{1}^{\text{C4}}|(g_{1}\cos(\phi_{\text{C4}}+t\tilde{\Delta}))+g_{2}\sin(\phi_{\text{C4}}+t\tilde{\Delta}))}{\sqrt{2}}L_{s,17}$
(70) $\displaystyle W_{4}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A1}}|(g_{1}\cos(t\Delta_{\text{c}})+g_{2}\sin(t\Delta_{\text{c}}))}{\sqrt{2}}L_{s,45}$
(71) $\displaystyle W_{5}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta(|\Omega_{1}^{\text{C1}}|(g_{1}\cos\alpha-
g_{2}\sin\alpha)+|\Omega_{1}^{\text{C4}}|(g_{1}\cos\beta-
g_{2}\sin\beta))}{\sqrt{2}}L_{s,47}$ (72) $\displaystyle W_{6}$
$\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{C1}}|(g_{2}\cos\alpha+g_{1}\sin\alpha)+|\Omega_{1}^{\text{C4}}|(g_{2}\cos\beta+g_{1}\sin\beta)}{\sqrt{2}}L_{s,57}$
(73) $\displaystyle W_{7}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A1}}|\sin\theta(g_{1}\cos(\phi_{1}/2+t\Delta_{\text{c}})+g_{2}\sin(\phi_{1}/2+t\Delta_{\text{c}}))}{\sqrt{2}}L_{a,14}$
(74) $\displaystyle W_{8}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A1}}|\cos\theta(g_{2}\cos(\phi_{1}/2+t\Delta_{\text{c}})-g_{1}\sin(\phi_{1}/2+t\Delta_{\text{c}}))}{\sqrt{2}}L_{a,15}$
(75) $\displaystyle W_{9}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{C1}}|(g_{2}\cos(t\bar{\Delta})-g_{1}\sin(t\bar{\Delta}))+|\Omega_{1}^{\text{C4}}|(-g_{2}\cos(\phi_{\text{C}4}+t\tilde{\Delta})+g_{1}\sin(\phi_{\text{C}4}+t\tilde{\Delta}))}{\sqrt{2}}L_{a,17}$
(76) $\displaystyle W_{10}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A1}}|\cos(2\theta)(g_{2}\cos(t\Delta_{\text{c}})-g_{1}\sin(t\Delta_{\text{c}}))}{\sqrt{2}}L_{a,45}$
(77) $\displaystyle W_{11}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta(|\Omega_{1}^{\text{C1}}|(g_{2}\cos\alpha+g_{1}\sin\alpha)+|\Omega_{1}^{\text{C4}}|(g_{2}\cos\beta+g_{1}\sin\beta))}{\sqrt{2}}L_{a,47}$
(78) $\displaystyle W_{12}$ $\displaystyle=$
$\displaystyle\frac{\sin\theta(|\Omega_{1}^{\text{C1}}|(-g_{1}\cos\alpha+g_{2}\sin\alpha)+|\Omega_{1}^{\text{C4}}|(-g_{1}\cos\beta+g_{2}\sin\beta))}{\sqrt{2}}L_{a,57}$
(79) $\displaystyle W_{13}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{4}|\Omega_{1}^{\text{A1}}|\sin(2\theta)(g_{2}\cos(t\Delta_{\text{c}})-g_{1}\sin(t\Delta_{\text{c}}))L_{33}$
(80) $\displaystyle W_{14}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{5}}{4}|\Omega_{1}^{\text{A1}}|\sin(2\theta)(-g_{2}\cos(t\Delta_{\text{c}})+g_{1}\sin(t\Delta_{\text{c}}))L_{44}$
(81)

with $\bar{\Delta}=\Delta_{\text{c}}-\delta_{\text{es}}$,
$\tilde{\Delta}=\bar{\Delta}+\delta_{\text{gs}}$,
$\alpha=\phi_{1}/2-t\bar{\Delta}$ and
$\beta=\phi_{1}/2-\phi_{\text{C}4}-t\tilde{\Delta}$. Here $\Delta_{\text{c}}$
is the detuning of the control measured from the A1 transition, which is fixed
to be the same with the detuning of the laser field $\textbf{E}_{1}$, since we
modulate that initial laser. We have also dropped the superscript $n$ in
$g_{1}$, $g_{2}$ and $W_{i}$, that denotes the order of Magnus truncation.

The linear system of equations for the first order Magnus expansion is formed
as follows:

$\left(\begin{array}[]{ccc|ccc}w_{j=1,k=1,l}^{(1)}&\ldots&w_{j=1,k=k_{\mathrm{max}},l}^{(1)}&w_{j=1,k=1,l^{\prime}}^{(1)}&\ldots&w_{j=1,k=k_{\text{max}},l^{\prime}}^{(1)}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
w_{j=j_{\text{max}},k=1,l}^{(1)}&\ldots&w_{j=j_{\text{max}},k=k_{\text{max}},l}^{(1)}&w_{j=j_{\text{max}},k=1,l^{\prime}}^{(1)}&\ldots&w_{j=j_{\text{max}},k=k_{\text{max}},l^{\prime}}^{(1)}\end{array}\right)\begin{pmatrix}c_{k=1,l}^{(1)}\\\
\vdots\\\ c_{k=k_{\text{max}},l}^{(1)}\\\ c_{k=1,l^{\prime}}^{(1)}\\\
\vdots\\\
c_{k=k_{\text{max}},l^{\prime}}^{(1)}\end{pmatrix}=\begin{pmatrix}h_{\text{err}.,j=1}^{(1)}\\\
\vdots\\\ h_{\text{err}.,j=j_{\text{max}}}^{(1)}\end{pmatrix},$ (82)

with $l=1$ corresponding to $g_{1}$ and $l^{\prime}=2$ corresponding to
$g_{2}$. Also, we have defined
$h_{\text{err}.j}^{(1)}=-i\int_{0}^{T}dt^{\prime}H_{\text{err},j}^{(1)}(t^{\prime})$
to be the integral of the error term of the $j$-th operator. The components of
the first matrix are given by:

$w^{(1)}_{j,k,m}=\int_{0}^{T}dt^{\prime}W_{j,m}(t^{\prime})\left(1-\cos\left(\frac{2\pi
kt^{\prime}}{T}\right)\right),$ (83)

where $W_{j,m}$ corresponds to the coefficient of $g_{1}$ ($m=l$) or $g_{2}$
($m=l^{\prime}$), for each $j$ operator we decomposed the control into. Since
the control is decomposed into more operators than the errors, we set
$h_{\text{err},j}=0$, for the components of the error vector where the error
Hamiltonian has no decomposition.

Regarding the SnV-, we use $yz$-polarization which leads to two vanishing
leakage transitions, i.e. $\Omega_{1}^{\text{C2}}=0$ and
$\Omega_{2}^{\text{C4}}=0$. In this case, the error terms in the final
interaction frame have the decomposition:

$\displaystyle H_{\text{er},1}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{2}^{\text{C2}}|\cos
t\bar{\Delta}-|\Omega_{1}^{\text{C4}}|\cos\beta}{\sqrt{2}}L_{s,27}$ (84)
$\displaystyle H_{\text{er,2}}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\cos\theta(|\Omega_{2}^{\text{C2}}|\cos
t\bar{\Delta}+|\Omega_{1}^{\text{C4}}|\cos\beta)}{\sqrt{2}}L_{s,47}$ (85)
$\displaystyle H_{\text{er,3}}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\sin\theta(|\Omega_{2}^{\text{C2}}|\sin
t\bar{\Delta}+|\Omega_{1}^{\text{C4}}|\sin\beta)}{\sqrt{2}}L_{s,57}$ (86)
$\displaystyle H_{\text{er,4}}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{-|\Omega_{2}^{\text{C2}}|\sin
t\bar{\Delta}+|\Omega_{1}^{\text{C4}}|\sin\beta}{\sqrt{2}}L_{a,27}$ (87)
$\displaystyle H_{\text{er,5}}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\cos\theta(|\Omega_{2}^{\text{C2}}|\sin
t\bar{\Delta}+|\Omega_{1}^{\text{C4}}|\sin\beta)}{\sqrt{2}}L_{a,47}$ (88)
$\displaystyle H_{\text{er,6}}^{(1)}$ $\displaystyle=$
$\displaystyle-\frac{\sin\theta(|\Omega_{2}^{\text{C2}}|\cos
t\bar{\Delta}+|\Omega_{1}^{\text{C4}}|\cos\beta)}{\sqrt{2}}L_{a,57},$ (89)

where $\bar{\Delta}=\Delta-(\delta_{\text{es}}+\delta_{\text{gs}})$ and
$\beta=t(\Delta-\delta_{\text{es}}+\delta_{\text{gs}})+2\phi_{1}$. For the
resonant case, $\Delta=0$.

For the control, we start from the following lab-frame corrective Hamiltonian:

$W_{\text{lab}}=(\Omega_{1}^{\text{A2}}\sigma_{\text{2A}}+\Omega_{1}^{\text{C4}}\sigma_{4\text{C}}+\text{H.c.})g^{(n)}(t),$
(90)

where we have assumed same polarization as the original laser that drives the
A2 transition. Following a similar procedure, we find that the control
Hamiltonian in the final frame has the decomposition:

$\displaystyle W_{1}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A2}}|\sin\theta(-g_{2}\cos(t\Delta_{\text{c}})+g_{1}\sin(t\Delta_{\text{c}}))}{\sqrt{2}}L_{s,24}$
(91) $\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A2}}|\cos\theta(g_{1}\cos(t\Delta_{\text{c}})+g_{2}\sin(t\Delta_{\text{c}}))}{\sqrt{2}}L_{s,25}$
(92) $\displaystyle W_{3}$ $\displaystyle=$
$\displaystyle-\frac{|\Omega_{1}^{\text{C4}}|(g_{1}\cos\beta+g_{2}\sin\beta)}{\sqrt{2}}L_{s,27}$
(93) $\displaystyle W_{4}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A2}}|(g_{1}\cos(t\Delta_{\text{c}})+g_{2}\sin(t\Delta_{\text{c}}))}{\sqrt{2}}L_{s,45}$
(94) $\displaystyle W_{5}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{C4}}|\cos\theta(g_{1}\cos\beta+g_{2}\sin\beta)}{\sqrt{2}}L_{s,47}$
(95) $\displaystyle W_{6}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{C4}}|\sin\theta(g_{2}\cos\beta-
g_{1}\sin\beta)}{\sqrt{2}}L_{s,57}$ (96) $\displaystyle W_{7}$
$\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A2}}|\sin\theta(g_{1}\cos(t\Delta_{\text{c}})+g_{2}\sin(t\Delta_{\text{c}}))}{\sqrt{2}}L_{a,24}$
(97) $\displaystyle W_{8}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A2}}|\cos\theta(g_{2}\cos(t\Delta_{\text{c}})-g_{1}\sin(t\Delta_{\text{c}}))}{\sqrt{2}}L_{a,25}$
(98) $\displaystyle W_{9}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{C4}}|(-g_{2}\cos\beta+g_{1}\sin\beta)}{\sqrt{2}}L_{a,27}$
(99) $\displaystyle W_{10}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{A2}}|\cos(2\theta)(g_{2}\cos(t\Delta_{\text{c}})-g_{1}\sin(t\Delta_{\text{c}}))}{\sqrt{2}}L_{a,45}$
(100) $\displaystyle W_{11}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{1}^{\text{C4}}|\cos\theta(g_{2}\cos\beta-
g_{1}\sin\beta)}{\sqrt{2}}L_{a,47}$ (101) $\displaystyle W_{12}$
$\displaystyle=$
$\displaystyle-\frac{|\Omega_{1}^{\text{C4}}|(g_{1}\cos\beta+g_{2}\sin\beta)}{\sqrt{2}}L_{a,57}$
(102) $\displaystyle W_{13}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{4}|\Omega_{1}^{\text{A2}}|\sin(2\theta)(g_{2}\cos(t\Delta_{\text{c}})-g_{1}\sin(t\Delta_{\text{c}}))L_{33}$
(103) $\displaystyle W_{14}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{5}}{4}|\Omega_{1}^{\text{A2}}|\sin(2\theta)(-g_{2}\cos(t\Delta_{\text{c}})+g_{1}\sin(t\Delta_{\text{c}}))L_{44},$
(104)

where we have defined $\beta=t(\Delta-\delta_{\text{es}}+\delta_{\text{gs}})$.
The detuning of the control is set equal to the two-photon detuning (same
frequency as the $\textbf{E}_{1}$ laser field), i.e. $\Delta_{c}=\Delta$.
Notice that even though we started with a control pulse that does not have
access to the error transition C2, in the final interaction frame the linear
system of equations is well-defined, as for each error decomposition term,
there is a corresponding control decomposition.

Even though the controls are obtained in the interaction frame generated by
the ideal Hamiltonian, the fidelity of the Magnus scheme in the main text is
evaluated in the initial dark-bright (interaction) frame.

##  Appendix G Pulse corrections obtained via the DRAG method

Here, we provide further details regarding the derivation of the control
pulses based on the DRAG method. First, we briefly highlight the strategy for
deriving the controls. Following the procedure of Ref. [51], we start by
transforming our Hamiltonian that also includes the error terms in the
rotating frame. We further mention that for the derivation of the corrections,
we consider only the subspace composed of the
$\\{|1\rangle,|4\rangle,|\text{A}\rangle,|\text{C}\rangle\\}$
($\\{|2\rangle,|4\rangle,|\text{A}\rangle,|\text{C}\rangle\\}$) states for the
SiV- (for the SnV-).

Regarding the SiV- system, the Hamiltonian for this reduced subspace in the
lab frame reads:

$H=H_{\text{lab},1}+H_{\text{lab},2}+H_{0},$ (105)

where
$H_{0}=\text{diag}[\omega_{1},\omega_{1}+\delta_{\text{gs}},\omega_{A},\omega_{A}+\delta_{\text{es}}]$,
with $\delta_{\text{gs}}$ and $\delta_{\text{es}}$ being the ground and
excited states splittings and $\omega_{1}$, $\omega_{A}$ being the eigen-
energies of $|1\rangle$ and $|\text{A}\rangle$ respectively. Also,
$H_{\text{lab},1}$ and $H_{\text{lab},2}$ are given by:

$\begin{split}H_{\text{lab},1}^{(n)}=\Big{(}\Omega_{1}^{(n)}\cos(\omega_{\text{d1}}t)\Big{[}e^{i\phi_{\text{A1}}}\sigma_{\text{1A}}+\lambda_{1}\sigma_{\text{1C}}+\lambda_{12}e^{i\phi_{C4}}\sigma_{\text{4C}}\Big{]}&+\Omega_{2}^{(n)}\cos(\omega_{\text{d2}}t)\Big{[}e^{i\phi_{\text{A4}}}\sigma_{\text{4A}}-\lambda_{2}\sigma_{\text{4C}}+\lambda_{21}e^{i\phi_{\text{C1}}}\sigma_{\text{1C}}]\Big{)}f(t)\\\
&+\text{H.c.},\end{split}$ (106)

$\begin{split}H_{\text{lab},2}^{(n)}=\Big{(}\bar{\Omega}_{1}^{(n)}\sin(\omega_{\text{d1}}t)\Big{[}e^{i\phi_{\text{A1}}}\sigma_{\text{1A}}+\lambda_{1}\sigma_{\text{1C}}+\lambda_{12}e^{i\phi_{C4}}\sigma_{\text{4C}}\Big{]}&+\bar{\Omega}_{2}^{(n)}\sin(\omega_{\text{d2}}t)\Big{[}e^{i\phi_{\text{A4}}}\sigma_{\text{4A}}-\lambda_{2}\sigma_{\text{4C}}+\lambda_{21}e^{i\phi_{\text{C1}}}\sigma_{\text{1C}}]\Big{)}f(t)\\\
&+\text{H.c.}~{},\end{split}$ (107)

with $f(t)=\text{sech}(\sigma(t-t_{0}))$ and $\omega_{\text{d}1}$,
$\omega_{\text{d}2}$ the laser frequencies. The fields $\bar{\textbf{E}}_{1}$
and $\bar{\textbf{E}}_{2}$ are $\pi/2$-shifted compared to $\textbf{E}_{1}$
and E2. Starting from Eq. (105) we perform the transformation

$U_{\text{rot}}=\text{diag}[e^{i\omega_{1}t},e^{i(\omega_{1}+\omega_{\text{d1}}-\omega_{\text{d2}})t},e^{i(\omega_{1}+\omega_{\text{d1}})t},e^{i(\omega_{1}+\omega_{\text{d1}})t}],$
(108)

which leads to the rotating frame Hamiltonian, as well as the transformation

$U_{\phi}=\text{diag}[e^{-i\phi_{\text{A1}}},e^{-i\phi_{\text{A1}}},1,e^{-i\phi_{\text{A1}}}].$
(109)

Notice that the transformation $U_{\phi}$ removes the complex part
$e^{i\phi_{\text{A1}}}$ from the Rabi frequency corresponding to the A1 as
well as A4 transitions, since we fix the Rabi frequencies to be equal a priori
to satisfy the db transformation for $R_{x}$ gates. At this step, our rotating
frame Hamiltonian reads:

$\begin{split}H_{\text{rot}}&=\frac{1}{2}\Big{[}\Big{(}\Omega_{1}\sigma_{1\text{A}}+\Omega_{2}\sigma_{4\text{A}}+(\lambda_{1}\Omega_{1}+e^{-it(\delta_{\text{gs}}+\phi_{\text{C}4})}\lambda_{21}\Omega_{2})\sigma_{1\text{C}}+(e^{it(\delta_{\text{gs}}+\phi_{\text{C}4})}\lambda_{12}\Omega_{1}-\lambda_{2}\Omega_{2})\sigma_{4\text{C}}\Big{)}+\text{H.c.}\Big{]}\\\
&+\frac{-i}{2}\Big{[}\Big{(}\bar{\Omega}_{1}\sigma_{1\text{A}}+\bar{\Omega}_{2}\sigma_{4\text{A}}+(\lambda_{1}\bar{\Omega}_{1}+e^{-it(\delta_{\text{gs}}+\phi_{\text{C}4})}\lambda_{21}\bar{\Omega}_{2})\sigma_{1\text{C}}+(e^{it(\delta_{\text{gs}}+\phi_{\text{C}4})}\lambda_{12}\bar{\Omega}_{1}-\lambda_{2}\bar{\Omega}_{2})\sigma_{4\text{C}}\Big{)}+\text{H.c.}\Big{]}\\\
&-\Delta\sigma_{\text{AA}}+(\delta_{\text{es}}-\Delta)\sigma_{\text{CC}}.\end{split}$
(110)

For clarity, we mention that we have defined
$\lambda_{1}=|\Omega_{1}^{\text{C1}}|/|\Omega_{1}|$,
$\lambda_{12}=|\Omega_{1}^{\text{C4}}|/|\Omega_{1}|$,
$\lambda_{2}=|\Omega_{2}^{\text{C4}}|/|\Omega_{2}|$ and
$\lambda_{21}=|\Omega_{2}^{\text{C1}}|/|\Omega_{2}|$, where the subscripts
$k=\\{1,2\\}$ in $\Omega_{k}^{ij}$ correspond to the lasers by which the error
transitions are driven by. Finally, in order to go to the db-frame we apply
the transformation:

$R_{\text{db}}=\frac{1}{\sqrt{2}}(\sigma_{11}-\sigma_{14}+\sigma_{41}+\sigma_{44})+\sigma_{\text{AA}}+\sigma_{\text{CC}}.$
(111)

To decouple the dark state from the excited, we further set
$\Omega_{1}=\Omega_{2}$ and $\bar{\Omega}_{1}=\bar{\Omega}_{2}$. Thus, the
dark-bright (rotating) Hamiltonian reads:

$\begin{split}H_{\text{db}}=\frac{1}{\sqrt{2}}\Big{(}&-\frac{(e^{it(\delta_{\text{gs}}+\phi_{\text{C}4})}\lambda_{12}-(\lambda_{1}+\lambda_{2})-\lambda_{21}e^{-it(\delta_{\text{gs}}+\phi_{\text{C}4})})(\Omega_{1}-i\bar{\Omega}_{1})}{2}\sigma_{\text{dC}}+(\Omega_{1}-i\bar{\Omega}_{1})\sigma_{\text{bA}}\\\
&+\frac{(e^{it(\delta_{\text{gs}}+\phi_{\text{C}4})}\lambda_{12}+(\lambda_{1}-\lambda_{2})+\lambda_{21}e^{-it(\delta_{\text{gs}}+\phi_{\text{C}4})})(\Omega_{1}-i\bar{\Omega}_{1})}{2}\sigma_{\text{bC}}+\text{H.c.}\Big{)}\\\
&-\Delta\sigma_{\text{AA}}+(-\Delta+\delta_{\text{es}})\sigma_{\text{CC}}.\end{split}$
(112)

The leakage subspace $|\text{C}\rangle$ is off-resonant from the remaining
Hamiltonian by an energy cost $\delta_{\text{es}}$. Effectively, this allows
us to perform an expansion of the control fields in the parameter
$\epsilon=1/(T\delta_{\text{es}})$. More analytically, according to Ref. [51],
by multiplying $H_{\text{db}}$ by the gate time we convert it to the
dimensionless form:

$\tilde{H}_{\text{db}}=\frac{1}{\epsilon}H_{0}+\sum_{n=0}^{\infty}\epsilon^{n}\tilde{H}_{\text{db}}^{(n)}(t),$
(113)

with $H_{0}=\text{diag}[0,0,0,1]$ and $\tilde{H}_{\text{db}}^{(n)}(t)$ given
by:

$\begin{split}\tilde{H}_{\text{db}}^{(n)}(t)=\frac{1}{\sqrt{2}}\Big{(}&-\frac{(e^{it(\delta_{\text{gs}}+\phi_{\text{C}4})}\lambda_{12}-(\lambda_{1}+\lambda_{2})-\lambda_{21}e^{-it(\delta_{\text{gs}}+\phi_{\text{C}4})})(\Omega_{1}^{(n)}-i\bar{\Omega}_{1}^{(n)})}{2}\sigma_{\text{dC}}+(\Omega_{1}^{(n)}-i\bar{\Omega}_{1}^{(n)})\sigma_{\text{bA}}\\\
&+\frac{(e^{it(\delta_{\text{gs}}+\phi_{\text{C}4})}\lambda_{12}+(\lambda_{1}-\lambda_{2})+\lambda_{21}e^{-it(\delta_{\text{gs}}+\phi_{\text{C}4})})(\Omega_{1}^{(n)}-i\bar{\Omega}_{1}^{(n)})}{2}\sigma_{\text{bC}}+\text{H.c.}\Big{)}\\\
&-\Delta\sigma_{\text{AA}}+(-\Delta+\delta_{\text{es}})\sigma_{\text{CC}}.\end{split}$
(114)

Note that now in Eq. (114), the control fields $\Omega_{k}^{(n)}$ and
$\bar{\Omega}_{k}^{(n)}$, as well as the detuning, $\Delta^{(n)}$, should be
understood as dimensionless. The next step is to satisfy the target
constraints that will allow us to implement the ideal Hamiltonian of Eq. (14)
of Sec. IV.2, as well as the decoupling constraints that will suppress the
leakage to the $|\text{C}\rangle$ subspace. These constraints are imposed in
the DRAG frame, which as we mentioned in the main text is generated by the
Hermitean operator $S(t)$, via $A=e^{-iS(t)}$. To this end, the operator
$S(t)$ is expanded in power series in $\epsilon$, as
$S(t)=\sum_{n=1}\epsilon^{n}S^{(n)}(t)$, with $S^{(n)}(t)$ given by:

$S^{(n)}(t)=\sum_{j=1}s^{(n)}_{z,j}\sigma_{jj}+\sum_{j<k}s^{(n)}_{x,jk}\sigma_{x,jk}+\sum_{j<k}s^{(n)}_{y,jk}\sigma_{y,jk}.$
(115)

As a result, the decoupling and target constraints can be solved iteratively
in a consistent manner, and the set of equations for the $n$-th order can be
found in the Appendix of Ref. [51]. For transparency, we highlight how we
solve the constraints and provide the equations for the corrective
modulations.

The first step is to define the target Hamiltonian, which as given in the main
text reads:

$H_{\text{target}}=\frac{h_{x}^{(0)}}{2}\sigma_{x,\text{be}}+h_{z}^{(0)}(\sigma_{\text{bb}}-\sigma_{\text{ee}}).$
(116)

By satisfying the zero-th order constraints we ensure that
$H_{\text{D}}^{(0)}=H_{\text{db},0}$, where $H_{\text{db},0}$ is the ideal
Hamiltonian:

$H_{\text{db},0}=(\Omega_{eff}f(t)\sigma_{x,\text{be}}+\text{H.c.})-\Delta\sigma_{\text{ee}}.$
(117)

Effectively, this means that to the zero-th order, the target gate is the same
in both frames. At the same time, satisfying the zero-th order constraints
implies that certain $S^{(1)}(t)$ elements need to be restricted; these
correspond to $S^{(1)}_{k,\text{dC}}$, $S^{(1)}_{k,\text{bC}}$,
$S^{(1)}_{k,\text{AC}}$, with $k=\\{x,y\\}$. This leaves the parameters
$S^{(1)}_{z,j}$ (with $j=\\{\text{d, b, A, C}\\}$), $S^{(1)}_{k,\text{db}}$,
$S^{(1)}_{k,\text{dA}}$ and $S^{(1)}_{k,\text{bA}}$ free. We set all
$S^{(1)}_{z,j}=0$, as well as
$S^{(1)}_{k,\text{db}}=S^{(1)}_{k,\text{dA}}=0=S^{(1)}_{x,\text{bA}}$. This
choice satisfies the boundary conditions for the frame transformation $A(t)$,
and allows us to obtain the corrective fields by $S^{(1)}_{y,\text{bA}}(t)$
via the first order target constraints. In particular, for $\Delta^{(1)}=0$,
the target condition:

$\text{Tr}[H_{\text{D}}^{(1)}(\sigma_{\text{bb}}-\sigma_{\text{ee}})]=0,$
(118)

gives the following solution for $S^{(1)}_{y,\text{bA}}(t)$:

$S^{(1)}_{y,\text{bA}}(t)=\frac{\Omega_{1}^{(0)}(\lambda_{1}-\lambda_{2}+2\lambda_{12}\cos(t\delta_{\text{gs}}+\phi_{\text{C}4}))^{2}f(t)}{8\sqrt{2}\delta_{\text{es}}},$
(119)

where we have also set $\lambda_{12}=\lambda_{21}$, which arises from the
polarization definitions we have used. Then, from the target constraints:

$\displaystyle h_{x}^{(1)}$ $\displaystyle=$
$\displaystyle\text{Tr}[H_{\text{D}}^{(n)}\sigma_{x,\text{be}}]=0,$ (120)
$\displaystyle h_{y}^{(1)}$ $\displaystyle=$
$\displaystyle\text{Tr}[H_{\text{D}}^{(n)}\sigma_{y,\text{be}}]=0,$ (121)

we solve for $\Omega_{1}^{(1)}$ and $\bar{\Omega}_{1}^{(1)}$, which depend on
$S^{(1)}_{y,\text{bA}}(t)$. The expressions for the pulse corrections for the
SiV- are:

$\Omega_{1}^{(1)}=\Omega_{2}^{(1)}=\frac{\Delta\Omega_{1}^{(0)}(\lambda_{1}-\lambda_{2}+2\lambda_{1}\lambda_{2}\cos(t\delta_{\text{gs}}+\phi_{\text{C4}}))^{2}}{16\delta_{\text{es}}},$
(122)
$\bar{\Omega}_{1}^{(1)}=\bar{\Omega}_{2}^{(1)}=\frac{\Omega_{1}^{(0)}(\lambda_{1}-\lambda_{2}+2\lambda_{12}\cos(t\delta_{\text{gs}}+\phi_{\text{C4}}))}{16\delta_{\text{es}}}A(t),$
(123)

where $A(t)$ is given by:

$\begin{split}A(t)=\Big{(}&-4\delta_{\text{gs}}\lambda_{12}\sin(t\delta_{\text{gs}}+\phi_{\text{C4}})\\\
&+\left(\lambda_{1}-\lambda_{2}+2\lambda_{12}\cos(t\delta_{\text{gs}}+\phi_{\text{C4}})\right)\frac{\dot{f}(t)}{f(t)}\Big{)}\end{split}$
(124)

Lastly, we follow a similar procedure for the SnV-, and we find that the
corrections $\Omega_{1}^{(1)}$ and $\bar{\Omega}_{1}^{(1)}$ are:

$\Omega_{1}^{(1)}=\Omega_{2}^{(1)}=\frac{\Omega_{1}^{(0)}\Delta((\lambda_{1}+\lambda_{2})^{2}+2\lambda_{21}^{2}(1-\cos(2\delta_{\text{gs}}t)))}{16\delta_{\text{es}}}$
(125)
$\bar{\Omega}_{1}^{(1)}=\bar{\Omega}_{2}^{(1)}=\frac{\lambda_{21}^{2}\Omega_{1}^{(0)}\sin(t\delta_{\text{gs}})(2\delta_{\text{gs}}\cos(t\delta_{\text{gs}})+\frac{\dot{f}(t)}{f(t)}\sin(t\delta_{\text{gs}}))}{4\delta_{\text{es}}},$
(126)

where $\lambda_{21}=\Omega_{2}^{\text{C2}}/\Omega_{1}^{(0)}$.

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