figurec

# An exchange-based surface-code quantum computer architecture in silicon

Charles D. Hill<EMAIL_ADDRESS>School of Physics, The University of
Melbourne, Parkville, 3010, Australia  School of Mathematics and Statistics,
The University of Melbourne, Parkville, 3010, Australia  Muhammad Usman
<EMAIL_ADDRESS>Centre for Quantum Computation and Communication, School
of Physics, The University of Melbourne, Parkville, 3010, Australia  School of
Computing and Information Systems, Melbourne School of Engineering, The
University of Melbourne, Parkville, 3010, Australia  Lloyd C.L. Hollenberg
<EMAIL_ADDRESS>Centre for Quantum Computation and Communication,
School of Physics, The University of Melbourne, Parkville, 3010, Australia

###### Abstract

Phosphorus donor spins in silicon offer a number of promising characteristics
for the implementation of robust qubits. Amongst various concepts for scale-
up, the shared-control concept takes advantage of 3D scanning tunnelling
microscope (STM) fabrication techniques to minimise the number of control
lines, allowing the donors to be placed at the pitch limit of $\geq$30 nm,
enabling dipole interactions. A fundamental challenge is to exploit the faster
exchange interaction, however, the donor spacings required are typically 15 nm
or less, and the exchange interaction is notoriously sensitive to lattice site
variations in donor placement. This work presents a proposal for a fast
exchange-based surface-code quantum computer architecture which explicitly
addresses both donor placement imprecision commensurate with the atomic-
precision fabrication techniques and the stringent qubit pitch requirements.
The effective pitch is extended by incorporation of an intermediate donor
acting as an exchange-interaction switch. We consider both global control
schemes and a scheduled series of operations by designing GRAPE pulses for
individual CNOTs based on coupling scenarios predicted by atomistic tight-
binding simulations. The architecture is compatible with the existing
fabrication capabilities and may serve as a blueprint for the experimental
implementation of a full-scale fault-tolerant quantum computer based on donor
impurities in silicon.

###### pacs:

Valid PACS appear here

##  I Introduction

Quantum computing based on spin qubits formed by phosphorus donors in silicon
Kane (1998) is an attractive approach for large scale implementation of
quantum information processing. Some of the milestones achieved to date
include single shot spin readout Morello  _et al._ (2010), the demonstration
of single qubits based on both electron Pla  _et al._ (2012) and nuclear Pla
_et al._ (2013) spins, the fabrication of donor based devices in silicon based
on scanning tunnelling microscope (STM) techniques Fuechsle  _et al._ (2012);
Weber  _et al._ (2012), the post-fabrication pinpointing of their locations in
silicon with the exact lattice site precision Usman  _et al._ (2016), and a
direct two-electron SWAP operation He  _et al._ (2019). With ongoing
experimental efforts focused on increasing the number of qubits in quantum
devices and achieving control with high fidelities, the challenges associated
with scale-up and the design of a universal quantum computer architecture
incorporating quantum error correction come into sharper focus.

The development of topological quantum error correction (TQEC) codes such as
the surface code has provided a scheme for error correction with a relatively
high threshold that is commensurate with experiments Bravyi and Kitaev (1998);
Dennis  _et al._ (2002); Raussendorf  _et al._ (2007); Wang  _et al._ (2011).
While the physical requirements of the surface code are relatively
straightforward to contemplate a two dimensional array of nearest-neighbour
coupled qubits. However, for all physical qubit platforms, even with
assumptions about quantum interconnects Nguyen  _et al._ (2017), the
challenges inherent in the spatial arrangement of gates, and temporal
characterisation and control complexity for large numbers of independent
qubits to carry out TQEC are formidable. Since Kane’s original concept for a
1D qubit array Kane (1998), a number of proposals have been presented
addressing scalability issues, particularly with respect to the requirements
of incorporating quantum error correction Hill  _et al._ (2015); Pica  _et
al._ (2016); Gorman  _et al._ (2016); Tosi  _et al._ (2017); Cai  _et al._
(2019). In Ref. Hill  _et al._ (2015), a surface-code architecture was
reported for impurity spins in silicon which was based on the dipole
interactions between the P impurities. This work presented a detailed design
introducing shared control, however it was limited to dipole couplings which
are of the order of kHz. The difficulty of providing fast, available couplings
in solid state architectures has led to several proposals. Pica et al. Pica
_et al._ (2016) proposed a surface code architecture, in which electrons were
shuttled between neighbouring qubits. Gorman et al addressed the problem of
coupling by mechanically moving a set of probe qubits in order to establish
the required couplings Gorman  _et al._ (2016). Tosi et al Tosi  _et al._
(2017) proposed the use of long range couplings provided by a flip-flop qubit,
a combination of electronic and nuclear spin states that can be controlled
with a microwave electric fields. For donor spins in silicon, the
incorporation of exchange interaction in surface-code based error correction
schemes is still an open question.

Figure 1: 3D Surface-code Architecture: The schematic diagram plots the layout
of the proposed surface-code architecture based on phosphorus (P) donor qubits
in silicon. The architecture is based on previously published scheme Hill  _et
al._ (2015), however it is updated to exploit the fast exchange interaction
between P donor electron spins. The data qubits are separated by 32 nm and
additional coupler qubits (orange dots) are incorporated in-between data
qubits to control (turn on/off) interaction between them. The qubit plane is
addressed by top and bottom gates shown by the blue and gray stripes.

The introduction of shared control Hill  _et al._ (2015) in donor qubit
architecture design space reduces the spatial complexity and dovetails
naturally with the repetitive spatio-temporal control requirements of surface
code TQEC. Assuming a high level of qubit uniformity and a fundamental qubit
pitch of $\geq$ 30 nm, corresponding to the fundamental contol line pitch
limit in these devices, CNOT gates were based on the donor electron spin
dipole interaction with a phase-matched electron loading protocol to rectify
timing variations associated with the hyperfine interaction. Ideally, one
would use the exchange interaction, however, the severe spacing requirements
($\leq$ 15nm) and variations in the exchange coupling work against the design
of a direct exchange based 2D array for TQEC. Here, we address these problems
by introducing an intermediate donor acting as an exchange coupler. The qubit
donors containing quantum data can be spaced comfortably with respect to a
control line pitch of 35 nm, and phase matched loading at qubit donors is no
longer required. Atomic level simulations, with typical placement variations
expected in STM fabrication, indicate CNOT gate times at O($\upmu$sec) are
possible and the overall scheme has potential to meet the stringent control
requirements of the surface code.

##  II Results & Discussions

###  II.1 Overview of the Architecture

Figure 1 schematically illustrates the layout of the exchange-based surface-
code architecture proposed in this work. The architecture, as its predecessor
dipole-based architecture Hill  _et al._ (2015), is based on three-dimensional
layout. In Figure 1 (a) The colored dots indicate 2D arrangement of donor
atoms, interleaved with black squares representing SET islands for
loading/unloading and readout of electron to/from qubits. The nuclear spins on
donors define the qubit states as shown in Figure 1 (b). The 2D qubit plane is
sandwiched between the top and bottom layout of wires forming source and
drain. The exponential decay of the exchange interaction with the separation
between the donor atoms is well known in the literature, as is the sensitivity
of the interaction to valley interference effects Cullis and Marko (1970);
Wellard  _et al._ (2003); Gonzalez-Zalba  _et al._ (2014); Hu  _et al._
(2005); Sarma  _et al._ (2005); Wellard  _et al._ (2004); Kettle  _et al._
(2006, 2004); Koiller  _et al._ (2004); Song and Sarma (2016); Wellard and
Hollenberg (2005); Testolin  _et al._ (2007); Saraiva  _et al._ (2015); Pica
_et al._ (2014); Koiller  _et al._ (2005); Voisin  _et al._ (2020); Usman
(2021). This results in a tension between donor separation and exchange
strength to design a fast CNOT gate while maintaining sufficient distance
between the atoms to allow for control wires, also known as pitch problem. In
the previous dipole-based architecture Hill  _et al._ (2015), the separation
between the adjacent donor atoms was taken to be 30 nm, defined by the gate-
leakage pitch limit for STM control-lines. At such distances, the exchange
interaction is effectively zero. In our scheme we introduce a coupler donor
which switches the exchange on and off by loading and unloading an electron to
that position (Figure 1 (c)).

Figure 2: Coupler-mediated CNOT gate: A schematic circuit diagram showing
conceptual triple-donor CNOT gate construction illustrated for the case
$\lvert 1\rangle\lvert 1\rangle\rightarrow\lvert 1\rangle\lvert 0\rangle$. In
our convention, arrows with single (double) heads label nuclear (electron)
spins, and down (up) direction of arrows define $\lvert 1\rangle$ ($\lvert
0\rangle$). The CNOT gate comprises three phosphorus donor qubit: target (T),
control (C), and coupler (c). (a-f) The spin configurations of electron and
nuclear spins on three qubits are shown at various stages of the CNOT circuit
operation.

The design of a robust two-qubit CNOT gate is a fundamental component of any
quantum computer architecture. Figure 2 plots the schematic diagram (center
circuit) of our design for a two-qubit CNOT gate based on the coupler qubit,
digitally controlling the exchange interaction between the control and target
data qubits. This mechanism allows the placement of control and target qubits
at distances commensurate with the pitch limit of STM control lines and yet
achieve MHz to GHz exchange interactions mediated via the coupler qubit. The
operation sequence of the proposed CNOT gate is explained in steps (a) to (f)
as shown in the diagram. We have indicated both the nuclear and electron qubit
spins on each qubit by plotting single and double head arrows, respectively.
As shown in (a), we assume that the gate is initialised as both electron spins
on control and target qubits in down-spin configuration and the nuclear spins
encode the qubit information. In the second step, (b), the coupler qubit is
loaded with an electron in down-spin configuration. Next, (c), the nuclear and
the electron spins of the target and control qubits are swapped. The CNOT
operation is performed between the target and control qubits (d), and then the
electron/nuclear spins are swapped again (e) to store the information back in
the nuclear spins. Finally, (f) brings the circuit back to the initial
condition by unloading the electron from the coupler qubit. This will turn off
the interaction between the target and control qubits.

Figure 3: Exchange distributions for triple donor protocol: (a) The possible
spatial locations are shown within the $\pm a_{0}$ placement precision for the
target (T), coupler (c) and control (C) dopants. Each dopant atom could be
placed on one of the possible nine locations, resulting in 81 values for
exchange interaction $J_{\rm Tc}$ and $J_{\rm cC}$. However, due to silicon
crystal symmetry, only 15 configurations are distinct. (b, c) The distinct
values of exchange interactions $J_{\rm Tc}$ and $J_{\rm cC}$ are plotted for
14 nm and 18 nm separations selected between target/coupler and
coupler/control, respectively.

###  II.2 Exchange strength and distribution

The current state-of-the-art scanning tunnelling microscope (STM) based
atomic-precision fabrication technology Fuechsle  _et al._ (2012) has
demonstrated donor placement with $\pm a_{0}$ accuracy, where $a_{0}$ is the
lattice constant of silicon. However, even such small variations in the donor
position may lead to considerably large variations in exchange interaction
Cullis and Marko (1970); Wellard  _et al._ (2003); Gonzalez-Zalba  _et al._
(2014); Hu  _et al._ (2005); Sarma  _et al._ (2005); Wellard  _et al._ (2004);
Kettle  _et al._ (2006, 2004); Koiller  _et al._ (2004); Song and Sarma
(2016); Wellard and Hollenberg (2005); Testolin  _et al._ (2007); Saraiva  _et
al._ (2015); Pica  _et al._ (2014); Koiller  _et al._ (2005); Voisin  _et al._
(2020); Usman (2021), placing stringent requirement on uniformity assumptions
in the design of control schemes for large-scale architectures Testolin  _et
al._ (2007); Usman (2021). In the past, strategies have been developed to
mitigate the impact of exchange variations, which include the design of robust
composite pulse schemes such as BB1 Hill (2007), exchange characterisation
Testolin  _et al._ (2007), the application of electric fields Wang  _et al._
(2016) and the placement of donor atoms along the [110] crystal direction
Voisin  _et al._ (2020). In this work, we propose the application of a small
strain field (5%) which allows full control of exchange interaction variation
for both in-plane and out-of-plane donor position variations Usman (2021).

Fig. 3 (a) plots a schematic illustration of donor positions for target,
coupler and control qubits. Each qubit is indicated by the target donor
position and the possible locations under $\pm a_{0}$ placement imprecision,
which is commensurate with the precision placement of donor atoms by STM
fabrication techniques. This results in 81 possible configurations between
target and coupler, and likewise another 81 possible configurations between
coupler and control. We note that due to the symmetry of the silicon crystal
lattice, only 15 configurations out of the 81 possibilities are distinct.

The calculation of exchange interaction is performed based on the atomistic
tight-binding wave functions of donor electrons in silicon Usman  _et al._
(2015a, b) and the Heiltler-London theory Wellard and Hollenberg (2005). Fig.
3 (c) and (d) plots the computed exchange values for the 15 distinct donor
configurations between the target and coupler and between the coupler and
control, respectively. As an example, the separations between the target and
the coupler qubits is selected as 14 nm, and between the coupler and control
qubits as 18 nm. These separations allow a pitch of 32 nm which is consistent
with the reported STM control-line requirements ($\geq$ 30 nm) Hill  _et al._
(2015). We note that the two separations are purposely selected to be slightly
different (18 nm and 14 nm), to minimise frequency band overlaps which will
allow efficient design of control pulses addressing individual donor pairs.

Figure 3(c) and (d) show a relatively small variation in exchange interaction
(about a factor of 5 or less), when compared to roughly three orders of
magnitude variation reported for similar donor position uncertainties in
unstrained silicon substrate Voisin  _et al._ (2020). This considerably
suppressed variation in exchange strength has important implication for the
fidelity of CNOT gate which sharply decreases when the exchange distribution
is large Testolin  _et al._ (2007). Furthermore, full exchange control can be
achieved in strained silicon system by an external in-plane electric field
which can provide a tuning of factor or ten or more for donor separations
above 14 nm Usman (2021).

The application of strain offers another direct benefit in terms of CNOT gate
operation times as the interaction time is inversely proportional to exchange
strength. Figure 4 plots exchange strength for various donor separations along
the [100] and [110] directions for both unstrained and strained silicon
environments. From these plots, a two-fold impact of strain is evident. First,
the application of strain significantly boosts the strength of exchange
interaction, as also reported in the literature Wellard and Hollenberg (2005);
Koiller  _et al._ (2002, 2004); Sarma  _et al._ (2005); Kettle  _et al._
(2006). For example, our calculations show that donors placed at 20 nm
separation in strained silicon will have roughly the same exchange
interactions as the donor pairs which are 12-14 nm separations in the
unstrained silicon. This implies that donors can be placed much larger
distances in strained system without sacrificing exchange interaction or CNOT
interaction times, which is important to meet the pitch requirements of a
large-scale architecture. From our calculations, we estimate O($\upmu$sec)
interaction times for donor separations of upto 25 nm in strained silicon
case, which is drastically faster when compared to O($m$sec) interaction times
for unstrained silicon substrates. Secondly, the exchange interaction in
strained environment is highly uniform, i.e., nearly same strength along the
[100] and [110] directions. The uniformity of exchange strength with respect
to donor placement orientation ([100] and [110]) will be useful in the design
of a planar 2D surface-code architecture such as proposed in this work (Figure
1).

Figure 4: Exchange enhancement: (a, b) Exchange interactions ($J$) between two
P atoms separated along the [100] and [110] directions are plotted for both
unstrained (diamond symbols) and 5% strained (square symbols) silicon
substrates. The $J$ values are presented in the exchange term of the effective
spin Hamiltonian ($J\vec{\sigma^{e}_{1}}\cdot\vec{\sigma^{e}_{2}}$), in which
case $J$ = $\frac{E_{T}-E_{S}}{4}$, where $E_{T}-E_{S}$ is the singlet-triplet
splitting. The conversion of energy to frequency is based on 1 meV $\sim$ 242
MHz.

###  II.3 GRAPE Pulse Engineering

The configurations of donor separations as shown in Figure 3 lead to a
distribution of corresponding interaction strengths, $J_{Tc}$ and $J_{cC}$.
Typically, at the selected spacing of 14-18 $\mathrm{nm}$ these coupling
strengths are larger than the hyperfine interaction, $A$, and so do not fall
into the regime described in Figure 2. Conceptually, the same operations are
being applied, however since all three electrons are strongly interacting, the
control pulses do not lend themselves to such a simple description. In order
to quantitatively determine control pulses required, we calculated pulses for
the electron to electron CNOT gate from control to target electrons using
numerically optimized GRAPE sequences.

Figure 5: Engineered Pulse Control: Schematic showing the strategy for
developing control pulses for a large array of donors. (a) The placement of
donors gives rise to different transition frequencies (b) Several of these
frequencies will overlap between distinct donor triples. (c) From these donor
triples, we identify sets of potential candidates triples for concurrent
pulses - spatially separated and either non-overlapping transitions in
frequency space, or with frequencies amenable to a broadband pulse (d) Optimal
pulses are found numerically using GRAPE which concurrently applies a CNOT to
all donor triples in that set. Difference colors indicate optimised pulse
sequences for different frequency combinations.

Since a wide range of exchange interaction strengths would be present in our
architecture, our strategy for implementing these pulses started from a simple
electron spin Hamiltonian (in the absence of an $AC$ control pulse applied):

$\displaystyle H_{\rm en}$ $\displaystyle=$ $\displaystyle
g\mu_{B}B(Z_{T}+Z_{C}+Z_{c})+g_{n}\mu_{n}B(Z_{nT}+Z_{nC}+Z_{nc})$ (1)
$\displaystyle+A_{T}\sigma_{T}\cdot\sigma_{nT}+A_{C}\sigma_{C}\cdot\sigma_{nC}+A_{c}\sigma_{c}\cdot\sigma_{nc}$
$\displaystyle+J_{Tc}\sigma_{T}\cdot\sigma_{c}+J_{cC}\sigma_{c}\cdot\sigma_{C}$

where $T$, $C$, and $c$ subscripts refer to the electron spins corresponding
to target, coupler and control qubits respectively, and the corresponding
$nT$, $nC$, and $nc$ refer to the nuclear spins. Here, and throughout the
paper, $X$, $Y$ and $Z$ are the Pauli spin operators, and $\sigma\cdot\sigma$
the exchange interaction between spins. Using the approximation that nuclear
spins remain static during this evolution, the electron spin Hamiltonian can
be reduced to the more tractable,

$\displaystyle H_{\rm e}$ $\displaystyle=$
$\displaystyle(g\mu_{B}B+A_{T})Z_{T}+(g\mu_{B}B-A_{C})Z_{C}+(g\mu_{B}B+A_{c})Z_{c}$
(2)
$\displaystyle+J_{Tc}\sigma_{T}\cdot\sigma_{c}+J_{cC}\sigma_{c}\cdot\sigma_{C}$

We wish to control the electron spins with a transverse $AC$ field,

$\displaystyle H_{\rm AC}$ $\displaystyle=$ $\displaystyle
g\mu_{B}B_{AC}\cos{\omega_{r}t}\left(X_{T}+X_{c}+X_{C}\right)$ (3)
$\displaystyle+g\mu_{B}B_{AC}\sin{\omega_{r}t}\left(Y_{T}+Y_{c}+Y_{C}\right)$

where typically $\omega$ is chosen to be resonant with a transition between
two of the eigenstates of $H$ given in Eqn. (2).

Not every transition between every pair of eigenstates is allowed. As an
illustrative example, if $J_{Tc}\gg A$ and $J_{Tc}\gg J_{cC}$ then a
transverse field of the form of Eqn. (3) would not excite transitions between
the singlet and triplet eigenstates due to symmetry considerations. Note,
however, that over a long time period, even though an individual transition
might not be able to be individually addressed, the symmetry can be broken
because the central spin interacts with both neighbours. Such disallowed
transitions can be identified numerically by considering the off-diagonal
elements of $H_{AC}$ given in Eqn. (3) written in the eigenbasis of $H_{e}$
given in Eqn. (2). In addition, two transitions can lie close in frequency,
and not able to be individually addressed in experiment. These two
considerations given rise to a viable set of control frequencies, $\omega$
which significantly excite transition between eigenstates of $H_{e}$ and can
be effectively addressed in experiment.

We performed GRAPE numerical optimization to determine gate pulse sequences
for the CNOT gate between electron spins. To do this, we considered each of
the different resonant frequencies which excite transitions between
eigenstates of the system as different control parameters. At each time-step,
it was possible to vary the strength of the $AC$ field applied, as well as the
phase of the applied microwave field. Using gradient ascent, we optimized the
trace fidelity,

$F(U)=\mathrm{Tr}\left[U_{C}U_{G}\right]$ (4)

where $U_{C}$ is the perfect CNOT gate applied between electronic spin states
1 and 3 and leaving the second electronic spin unchanged. $U_{G}$ is the
obtained evolution obtained from a given GRAPE pulse sequence.

We repeated GRAPE for each of the 225 different pairs of strengths of exchange
interactions $J_{Tc}$ and $J_{cC}$, obtaining a numerically optimized CNOT
pulse sequence in each case. Almost all pulse sequences resulted in a high
fidelity CNOT gate, accurate to $0.1\%$ accuracy. Only six CNOT gates had
lower fidelities. We note that there are 225 different triples of qubits. To
operate on each of these triples independently would require 225 different
pulse schemes - such as those calculated by GRAPE. However, many of these
pulses can, in principle, be applied in parallel. This can be applied in
parallel if (i) pulses have disjoint frequencies, which do not overlap, (ii)
broadband pulses can be applied to implement the gate on triples with near-
lying frequencies.

Pulses with disjoint frequencies can be operated in parallel, since an out of
resonance field will not excite transitions in off-resonant spins. The larger
the number of triples with non-overlapping frequencies, the more operations
that can be applied in parallel because they have disjoint frequencies. A
rough estimate of the number of triples (CNOT gates) that can be made is as
follows: If any two triples have a probability of 30% (40%) of having a
transition with an overlapping frequencies, then approximately 12 (9) of the
225 CNOT gates can be chosen to operate in parallel. Further tuning of
exchange interactions can be performed by the application of external electric
fields Usman (2021), which could allow more frequencies to be operated in
parallel.

##  III Summary

We have introduced a new concept for the incorporation of fast exchange
interaction in surface-code architecture scheme for donor spin qubits in
silicon. The proposal is underpinned by the design of a CNOT gate in which the
coupling between target and control data qubits in mediated by an additional
coupler qubit which can selectively turn on/off exchange interaction between
data qubits. The introduction of coupler qubit allows data qubits to be placed
at large separations ($\geq$ 30 nm) commensurate with the requirements of a
large-scale architecture. We also discuss the application of a small strain
field ( 5%) which provides important benefits such as significant enhancement
in exchange strength leading to O($\upmu$sec) interaction times, suppressed
exchange variation arising from the donor placement inaccuracy and uniformity
in exchange interactions along the [100] and [110] crystal directions. We
consider a both global control as well as targeted GRAPE control based on
mapping frequency distributions arising from exchange variations. The work
here is a step on the path to the design and implementation of a large-scale
error-corrected quantum computer architecture based on atomic spin qubits in
silicon.

Acknowledgements: This work was supported by the Australian Research Council
(ARC) funded Center for Quantum Computation and Communication Technology
(CE170100012). Computational resources were provided by the National Computing
Infrastructure (NCI) and Pawsey Supercomputing Center through National
Computational Merit Allocation Scheme (NCMAS). This research was undertaken
using the LIEF HPC-GPGPU Facility hosted at the University of Melbourne. This
Facility was established with the assistance of LIEF Grant LE170100200.

Author contributions: All authors contributed in the development of the
concept, planning, data analysis and writing of the manuscript.

Conflict of Interest: The authors declare no competing financial or non-
financial interests. A patent application has been filed based on aspects of
the architecture design.

Data availability: The data that support the findings of this study are
available within the article. Further information can be provided upon
reasonable request.

## References

  * Kane (1998) B. E. Kane, Nature 393, 133 (1998).
  * Morello  _et al._ (2010) A. Morello, J. J. Pla, F. A. Zwanenburg, K. W. Chan, K. Y. Tan, H. Huebl, M. Mottonen, C. D. Nugroho, C. Yang, J. A. van Donkelaar, A. D. C. Alves, D. N. Jamieson, C. C. Escott, L. C. L. Hollenberg, R. G. Clark, and A. S. Dzurak, Nature 467, 687 (2010).
  * Pla  _et al._ (2012) J. J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, D. N. Jamieson, A. S. Dzurak, and A. Morello, Nature 489, 541 (2012).
  * Pla  _et al._ (2013) J. Pla, K. Y. Tan, J. P. Dehollain, W. H. Lim, J. J. L. Morton, F. A. Zwanenburg, D. N. Jamieson, A. S. Dzurak, and A. Morello, Nature 496, 334 (2013).
  * Fuechsle  _et al._ (2012) M. Fuechsle, J. A. Miwa, S. Mahapatra, H. Ryu, S. Lee, O. Warschkow, L. C. L. Hollenberg, G. Klimeck, and M. Y. Simmons, Nature Nanotechnology 7, 242 (2012).
  * Weber  _et al._ (2012) B. Weber, S. Mahapatra, H. Ryu, S. Lee, A. Fuhrer, T. C. G. Reusch, D. L. Thompson, W. C. T. Lee, G. Klimeck, L. C. L. Hollenberg, and M. Y. Simmons, Science 335, 64 (2012).
  * Usman  _et al._ (2016) M. Usman, J. Bocquel, J. Salfi, B. Voisin, A. Tankasala, R. Rahman, M. Y. Simmons, S. Rogge, and L. Hollenberg, Nature Nanotechnology 11, 763 (2016).
  * He  _et al._ (2019) Y. He, S. K. Gorman, D. Keith, L. Kranz, J. G. Keizer, and M. Y. Simmons, Nature 571, 371 (2019).
  * Bravyi and Kitaev (1998) S. Bravyi and A. Kitaev, arXiv: quant-ph/9811052 (1998).
  * Dennis  _et al._ (2002) E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, J. Math. Phys. 43, 4452 (2002).
  * Raussendorf  _et al._ (2007) R. Raussendorf, J. Harrington, and K. Goyal, New J. Phys. 9, 199 (2007).
  * Wang  _et al._ (2011) D. S. Wang, A. G. Fowler, and L. C. L. Hollenberg, Phys. Rev. A 83, 020302 (2011).
  * Nguyen  _et al._ (2017) T. Nguyen _et al._ , Sci. Rep. 7, 13386 (2017).
  * Hill  _et al._ (2015) C. D. Hill, E. Peretz, S. Hile, M. House, M. Fuechsle, S. Rogge, M. Y. Simmons, and L. Hollenberg, Science Advances 1, e1500707 (2015).
  * Pica  _et al._ (2016) G. Pica, B. W. Lovett, R. N. Bhatt, T. Schenkel, and S. A. Lyon, Phys. Rev. B 93, 035306 (2016).
  * Gorman  _et al._ (2016) J. Gorman, N. Nickerson, P. Ross, J. Morton, and S. Benjamin, npj Quantum Information 2, 15019 (2016).
  * Tosi  _et al._ (2017) G. Tosi, F. Mohiyaddin, V. Schmitt, S. Tenberg, R. Rahman, G. Klimeck, and A. Morello, Nature Comm. 8, 450 (2017).
  * Cai  _et al._ (2019) Z. Cai _et al._ , Quantum 3 (2019).
  * Cullis and Marko (1970) P. R. Cullis and J. R. Marko, Phys. Rev. B 1 (1970).
  * Wellard  _et al._ (2003) C. J. Wellard, L. C. L. Hollenberg, F. Parisoli, L. M. Kettle, H.-S. Goan, J. A. L. McIntosh, and D. N. Jamieson1, Phys. Rev. B 68, 195209 (2003).
  * Gonzalez-Zalba  _et al._ (2014) M. F. Gonzalez-Zalba, A. Saraiva, M. J. Calderon, D. Heiss, B. Koiller, and A. J. Ferguson, Nanoletters 14, 5672 (2014).
  * Hu  _et al._ (2005) X. Hu, B. Koiller, and S. D. Sarma, Phys. Rev. B 71, 235332 (2005).
  * Sarma  _et al._ (2005) S. D. Sarma, R. de Sousa, X. Hu, and B. Koiller, Solid Stat. Comm. 133, 737 (2005).
  * Wellard  _et al._ (2004) C. J. Wellard, L. Hollenberg, L. M. Kettle, and H.-S. Goan, J. Phys.: Cond. Matt. 16, 5697 (2004).
  * Kettle  _et al._ (2006) L. Kettle, H.-S. Goan, and S. C. Smith, Phys. Rev. B 73, 115205 (2006).
  * Kettle  _et al._ (2004) L. M. Kettle, H.-S. Goan, S. C. Smith, L. C. L. Hollenberg, and C. J. Wellard, J. Phys.: Cond. Matt. 16, 1011 (2004).
  * Koiller  _et al._ (2004) B. Koiller, R. B. Capaz, X. Hu, and S. D. Sarma, Phys. Rev. B 70, 115207 (2004).
  * Song and Sarma (2016) Y. Song and S. D. Sarma, Appl. Phys. Lett. 109, 253113 (2016).
  * Wellard and Hollenberg (2005) C. J. Wellard and L. C. L. Hollenberg, Phys. Rev. B 72, 085202 (2005).
  * Testolin  _et al._ (2007) M. J. Testolin, C. Hill, C. J. . Wellard, and L. C. L. Hollenberg, Phys. Rev. A 76, 012302 (2007).
  * Saraiva  _et al._ (2015) A. L. Saraiva, A. Baena, M. J. Calderon, and B. Koiller, J. Phys.: Cond. Matt. 27, 154208 (2015).
  * Pica  _et al._ (2014) G. Pica, B. W. Lovett, R. N. Bhatt, and S. A. Lyon, Phys. Rev. B 89, 235306 (2014).
  * Koiller  _et al._ (2005) B. Koiller, X. Hu, R. Capaz, A. Martins, and S. D. Sarma, An Acad Bras Cienc 77, 201 (2005).
  * Voisin  _et al._ (2020) B. Voisin, J. Bocquel, A. Tankasala, M. Usman, J. Salfi, R. Rahman, M. Y. Simmons, L. Hollenberg, and S. Rogge, Nature Communications 11, 6124 (2020).
  * Usman (2021) M. Usman, Computational Materials Science 193, 110280 (2021).
  * Hill (2007) C. D. Hill, Phys. Rev. Lett. 98, 180501 (2007).
  * Wang  _et al._ (2016) Y. Wang, A. Tankasala, L. Hollenberg, G. Klimeck, M. Y. Simmons, and R. Rahman, NPJ Quantum Information 2, 16008 (2016).
  * Usman  _et al._ (2015a) M. Usman, R. Rahman, J. Salfi, J. Bocquel, B. Voisin, S. Rogge, G. Klimeck, and L. C. L. Hollenberg, J. Phys.: Cond. Matt. 27, 154207 (2015a).
  * Usman  _et al._ (2015b) M. Usman, C. D. Hill, R. Rahman, G. Klimeck, M. Y. Simmons, S. Rogge, and L. C. L. Hollenberg, Phys. Rev. B 91, 245209 (2015b).
  * Koiller  _et al._ (2002) B. Koiller, X. Hu, and S. D. Sarma, Phys. Rev. B 66, 115201 (2002).