††thanks: Currently at: CEMS, RIKEN, Wako, Saitama 351-0198, Japan††thanks:
Currently at: Institute of Microstructure and Properties of Advanced
materials, Beijing University of Technology, Beijing, 100124, China††thanks:
Currently at: QuTech and Kavli Institute of Nanoscience, Delft University of
Technology, Delft, The Netherlands

# A Qubit with Simultaneously Maximized Speed and Coherence

Miguel J. Carballido<EMAIL_ADDRESS>Department of Physics,
University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland  Simon
Svab  Department of Physics, University of Basel, Klingelbergstrasse 82,
CH-4056 Basel, Switzerland  Rafael S. Eggli  Department of Physics, University
of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland  Taras Patlatiuk
Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056
Basel, Switzerland  Pierre Chevalier Kwon  Department of Physics, University
of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland  Jonas Schuff
Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom
Rahel M. Kaiser  Department of Physics, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland  Leon C. Camenzind
Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056
Basel, Switzerland  Ang Li  Department of Applied Physics, TU Eindhoven, Den
Dolech 2, 5612 AZ Eindhoven, The Netherlands  Natalia Ares  Department of
Engineering Science, University of Oxford, Oxford OX1 3PJ, United Kingdom
Erik P.A.M Bakkers  Department of Applied Physics, TU Eindhoven, Den Dolech 2,
5612 AZ Eindhoven, The Netherlands  Stefano Bosco  Department of Physics,
University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland  J.
Carlos Egues  Department of Physics, University of Basel, Klingelbergstrasse
82, CH-4056 Basel, Switzerland  Instituto de Física de São Carlos,
Universidade de São Paulo, 13560-970 São Carlos, São Paulo, Brazil  Daniel
Loss  Department of Physics, University of Basel, Klingelbergstrasse 82,
CH-4056 Basel, Switzerland  CEMS, RIKEN, Wako, Saitama 351-0198, Japan
Dominik M. Zumbühl<EMAIL_ADDRESS>Department of Physics,
University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

###### Abstract

To overcome the threshold for fault-tolerant quantum computation, qubits have
to be protected from their noisy environment to attain the necessary high
fidelities Yoneda  _et al._ (2017); Hendrickx and Fuhrer (2022). Recent
experiments discovered sweet spots Tanttu  _et al._ (2019); Piot  _et al._
(2022); Hendrickx  _et al._ (2023); Bosco  _et al._ (2021a) with strongly
enhanced coherence. However, decoupling a qubit from its surroundings also
limits the control over the qubit’s state Froning  _et al._ (2021a); Gilbert
_et al._ (2023), typically leading to either coherent but slow Pla  _et al._
(2013); Veldhorst  _et al._ (2015) or fast but short-lived Hendrickx  _et al._
(2020a, b); Froning  _et al._ (2021a); Liu  _et al._ (2023) qubits. This
trade-off appears to be a severe fundamental limitation hampering the
performance of qubits. Here, we show how this can be circumvented by
demonstrating a simultaneously fast and coherent tunable regime in a hole spin
qubit. In this regime, we can triple the operation speed, while simultaneously
quadrupling the coherence time when tuning a local electric field,
demonstrating that the qubit speed and coherence scale together without
compromise. This relies on strong, quasi 1D confinement providing a local
maximum in drive strength, where charge fluctuations are decoupled and thus
the coherence is enhanced, yet the drive speed is maximal. A Ge/Si core/shell
nanowire, operated at $1.5$ K Yang  _et al._ (2020); Petit  _et al._ (2020);
Camenzind  _et al._ (2022); Huang  _et al._ (2023), provides the strong
confinement. The driving mechanism here is the strong and tunable direct
Rashba spin-orbit interaction Kloeffel  _et al._ (2011, 2018); Bosco  _et al._
(2021b); Wang  _et al._ (2021); Michal  _et al._ (2023), achieving a maximal
strength at finite electrical field due to gate-dependent heavy-hole light-
hole mixing. Breaking the speed-coherence trade-off makes it possible to boost
fidelity and speed of one- and two-qubit gates. This concept can be expanded
to planar arrays of hole or electron spin qubits as well. In this regime, the
coupling to a microwave resonator is also predicted to be both strong and
coherent Kloeffel  _et al._ (2013); Bosco  _et al._ (2022). Altogether, this
is opening a new path towards fault-tolerant quantum computation.

Main
Spins in semiconductor quantum dots (QD) have emerged as one of the leading
contenders for encoding and processing quantum information Loss and DiVincenzo
(1998); Stano and Loss (2022); Burkard  _et al._ (2023). Their success is
attributed to their competitive coherence times Laucht  _et al._ (2016);
Yoneda  _et al._ (2017) , the demonstration of robust multi-qubit operations
Hendrickx  _et al._ (2021); Philips  _et al._ (2022), coherent spin control
above $1$ K Yang  _et al._ (2020); Petit  _et al._ (2020); Camenzind  _et al._
(2022); Huang  _et al._ (2023) and their compatibility with industrial
fabrication techniques Zwerver  _et al._ (2022); Neyens  _et al._ (2023);
Weinstein  _et al._ (2023).
Among the various semiconductor systems capable of hosting spin qubits,
material systems exhibiting strong, intrinsic spin-orbit interactions (SOI),
have received increasing attention in recent years Scappucci  _et al._ (2020);
Fang  _et al._ (2023). Taking advantage of the SOI, all-electrical spin
driving can be implemented via electric dipole spin resonance (EDSR), without
the requirement for on-chip micro magnets or microwave antennas. While SOI
mediated EDSR allows for compact device architectures Nowack  _et al._ (2007);
Watzinger  _et al._ (2018), it has more importantly led to ultrafast Rabi
oscillations ranging from $80~{}$MHz in electrons Gilbert  _et al._ (2023), to
several $100~{}$MHz Froning  _et al._ (2021a) and even up to $1.2$ GHz Liu
_et al._ (2023) in holes, alas, at the expense of coherence. These remarkable
observations have thus raised a pivotal concern for the future of spin qubits
with strong SOI Wang  _et al._ (2021): Do strong couplings to the qubit
driving field inevitably lead to increased decoherence due to enhanced
couplings with undesired noise sources Massai  _et al._ (2023)?
First steps towards reducing the coupling to charge noise have been taken via
the modification of global system parameters such as the external magnetic
field orientation Tanttu  _et al._ (2019); Piot  _et al._ (2022); Hendrickx
_et al._ (2023), however, the fundamental trade-off between speed and
coherence has so far prevailed.
Here we provide first experimental evidence for compromise-free scaling,
through the demonstration of a coherence sweet spot that coincides with
maximal Rabi driving speeds. Our observations are in agreement with previous
theoretical predictions on group IV hole spin qubits Wang  _et al._ (2021);
Bosco  _et al._ (2021a); Adelsberger  _et al._ (2022); Michal  _et al._
(2023); Mauro  _et al._ (2023). Additionally, we achieve this Fast and
Coherent Tunable Operating Regime (FACTOR) all-electrically, by controlling
static gate voltages at the individual qubit level. Such local optimizations
allow us to respond to the variable electrostatic environments that each qubit
experiences, which can be comprised of electric stray fields from neighbouring
gates or non-uniform strain.
Realising a compromise-free qubit requires navigating the intricate interplay
between the tuning parameter, driving mechanism and decoherence channel.
Remarkably, certain systems naturally exhibit the conditions for a FACTOR,
such as hole spins in quasi 1D systems with strong SOI. In these systems, the
$g$-tensors often display a high level of anisotropy, which can be notably
influenced by electric fields. Consequently, random charge fluctuations couple
to the qubit energy, resulting in reduced coherence Hendrickx and Fuhrer
(2022). In order to mitigate this issue, configurations with vanishing
derivatives of the $g$-factor, with respect to voltage changes, are most
promising, as they indicate a reduced coupling of $g$ to charge noise. To
realise such a sweet spot, we exploit the properties of a spin qubit hosted
inside a squeezed, elongated hole quantum dot, subject to strong SOI, as is
naturally provided by the geometry of a Ge/Si core/shell nanowire (NW) Froning
_et al._ (2021a).
In such structures, the strong biaxial confinement causes heavy- and light-
hole (HH-LH) states to intermix, giving rise to a strong and electrically
tunable direct-Rashba spin-orbit interaction (DRSOI) Kloeffel  _et al._ (2011,
2018). At an optimal HH-LH mixture, the spin-orbit strength is expected to
reach a maximum as a function of an externally applied electric field. In the
presence of strong SOI, when the spin-orbit length $l_{\rm{SO}}$ becomes
comparable to the dot size $l_{\rm{dot}}$, the $g$-factor is reduced Kloeffel
_et al._ (2011, 2018); Bosco  _et al._ (2021b); Froning  _et al._ (2021b).
This gives rise to a minimum in $g$ at the point where SOI reaches a maximum.
This minimum in $g$ as a function of electric field corresponds to the
coherence sweet spot. By additionally employing iso-Zeeman EDSR driving of the
qubit Golovach  _et al._ (2006), the Rabi frequency $f_{\mathrm{R}}$ can be
maximised at this very same spot. This ensures an operating regime where the
qubit is both fast and long lived without compromising speed or coherence,
thus defining our FACTOR.

Coherent Spin Control at 1.5 K
We first demonstrate the operation of our Ge/Si NW hole spin qubit at $1.5$ K,
adding it to the list of previously demonstrated hot qubits Yang  _et al._
(2020); Petit  _et al._ (2020); Camenzind  _et al._ (2022); Huang  _et al._
(2023). A scanning electron micrograph showing a representative device is
presented in Fig. 1a. The device consists of a Ge/Si core/shell NW lying on
top of nine bottom gates. For details on the device fabrication we refer to
the materials and methods section. By applying positive voltages to the first
five bottom gates from the left, the intrinsic hole gas inside the NW is
depleted to form a hole double quantum dot (DQD) Froning  _et al._ (2018) with
a net-effective hole occupation $(m,n)$ on the left and right dot. The true
total hole occupation has been estimated to be in the range of several dozen
Froning  _et al._ (2018, 2021b); Ungerer  _et al._ (2023).
The device is operated at a net-effective charge-transition from $(1,1)$ to
$(2,0)$ that exhibits Pauli spin blockade (PSB). We apply a positive bias
across the NW and measure the current through the DQD, which provides spin to
charge conversion due to PSB. This allows to read out the state of the
effective spin-$1/2$ system, as shown in Fig. 1b. The characteristic DC
transport signature of PSB can be seen in Fig. 1c, where the baseline of the
bias triangle disappears in the absence of magnetic field (inset 1c).

Figure 1: Measurement of a NW hole spin qubit at 1.5 K. a, False-color
scanning electron micrograph of a representative NW-device. The Ge/Si NW is
highlighted in yellow, lying on top of nine bottom gates, and is further
connected to source and drain contacts from above, marked $S$ and $D$. The
first five bottom gates from the left were used to form a DQD, whose expected
location is indicated by the yellow and pink ovals. The colour code is
consistently used throughout the manuscript. The scale bar corresponds to
$100$ nm. b, Schematic of PSB. In the absence of magnetic field and for
detunings $\varepsilon<\varepsilon_{\rm{ST}}$, charge transport is blocked
when the system is initialised in a triplet state $T(1,1)$. c, Measurement of
bias triangles with lifted spin-blockade at $B=350$ mT ($x$-direction) and
bias of $V_{\rm{SD}}=+5$ mV. The pink star marks the qubit
initialisation/readout spot and the pink circle indicates the manipulation
point. Inset: Same bias triangle at $B=~{}0$ mT, manifesting PSB via the
suppressed current at the baseline. d, Schematic of the pulse scheme applied
to $V_{\rm RP}$, consisting of the readout/initialization stage (R/I) and the
manipulation stage for which a MW burst is applied while in Coulomb blockade.
e, Measurement of the current as a function of $f_{\rm{MW}}$ and $\mathbf{B}$
showing characteristic EDSR at a fixed microwave burst duration. f, Current as
a function of microwave burst duration and magnetic field $\mathbf{B}$ at
$f_{\rm{L}}=2.79$ GHz, showing Rabi-oscillations with a frequency
$f_{\mathrm{R}}=130$ MHz and coherence time $T_{2}^{\rm{Rabi}}=40$ ns.

In our setup, the right plunger gate RP, coloured in magenta in Fig. 1a, is
connected to a high frequency line via a bias-tee. This allows for the
application of square voltage pulses and microwave bursts to the gate, in
addition to DC voltages. The measurements were performed employing a common
two-stage pulsing protocol Maurand  _et al._ (2016); Hendrickx  _et al._
(2020a); Froning  _et al._ (2021a); Camenzind  _et al._ (2022), as
schematically shown in Fig. 1d.
The system is initialized in a spin-blockaded effective (1,1) triplet state,
by employing a specific plunger gate voltage $V_{\rm{RP}}$, represented by the
magenta stars in Figs. 1c and d. After a short waiting time, the system is
pulsed into Coulomb blockade by applying a square voltage pulse, as indicated
by the magenta circles in Figs. 1c and d. While in Coulomb blockade, a
microwave (MW) burst of duration $t_{\rm{burst}}$ is applied. Pulsing back to
the initial voltage of $V_{\rm{RP}}$ allows to record a current signal $I$,
proportional to the likelihood of a singlet configuration after coherent
manipulation. Fig. 1e shows typical EDSR measurements where the applied MW
frequency $f_{\rm{MW}}$ is swept against the external magnetic field
$\mathbf{B}$, and from which the $g$-factor is extracted. On resonance, the
spin is rotated, lifting the spin-blockade which leads to an increased
current. By varying the burst duration as a function of magnetic field
detuning at a fixed frequency of the microwave drive $f_{\rm{MW}}$, coherent
Rabi oscillations can be observed as shown in Fig. 1f. These results establish
coherent qubit control at $1.5$ K.
To gather information about the level of control over the qubit, similar scans
to those presented in Figs. 1e and f were repeated for different electrostatic
environments experienced by the qubit whilst remaining within the same charge
occupation of the DQD. Each electrostatic configuration of the qubit is
defined by the three barrier gate voltages $V_{\rm{L}}$, $V_{\rm{M}}$ and
$V_{\rm{R}}$, while the plunger gate voltages $V_{\rm{LP}}$ and $V_{\rm{RP}}$
were compensated to remain at a fixed readout point.

SOI in a Squeezed Quantum Dot
We characterize each qubit configuration by measuring the Rabi frequency
$f_{\mathrm{R}}$ and Landé $g$-factor, $g$, and show their functional
dependence on the three individual barrier voltages in Figs. 2a-c. In each of
the three studies, the voltages on the other two barrier gates are held at a
constant value, indicated by the vertical dashed lines in Figs. 2a-c. The
larger responses of $g$ and $f_{\mathrm{R}}$ to the barrier voltages
$V_{\rm{L}}$ and $V_{\rm{M}}$, compared to $V_{\rm{R}}$, suggest that the
qubit is located above the left plunger gate (LP). Moreover, the opposite
trends of $g$ and $f_{\rm{R}}$ with respect to voltage are consistent with the
theoretical description of an elongated quantum dot in the presence of SOI
Bosco  _et al._ (2021b). To understand the observations made in Figs. 2a-c, we
first explain how SOI renormalises the $g$-factor, followed by how it can be
related to $f_{\mathrm{R}}$ by choosing a specific EDSR driving mechanism.
In the case of a quasi 1D, elongated quantum dot, as it is reasonable to
assume for our NW, the longitudinal axis of gate-defined confinement is well
described by a harmonic potential resulting in a Gaussian envelope of the hole
wave function. In the presence of SOI, the spins acquire a helical texture
along the NW Kloeffel  _et al._ (2011, 2018); Bosco  _et al._ (2021b); Froning
_et al._ (2021b), provided the external magnetic field $\mathbf{B}$ acting on
the g-tensor $\mathbf{\hat{g}}$ gives rise to a Zeeman vector
$\mathbf{\hat{g}}\cdot\mathbf{B}$ with a component perpendicular to the SOI
axis $\bm{\alpha}_{\rm{SO}}$. As a result, the hole wave function averages
over different spin orientations, leading to a renormalization of the dot
$g$-factor (see Eq. 1 I ). This effect is especially relevant when the dot
extension along the NW, $l_{\rm{dot}}$, and spin-orbit length $l_{\rm{SO}}$,
which represents the distance a hole must traverse to undergo a spin rotation
due to SOI, are comparable in size. This leads to a minimum in $g$-factor
where the SOI is strongest, i.e. $l_{\rm{SO}}$ shortest. Relating $g$ to
$f_{\mathrm{R}}$, however, additionally requires an assumption on the
underlying qubit driving mechanism.

Figure 2:  SOI mediated g-factor renormalization of an elongated dot. a-c,
Experimental data of the qubit g-factor ($\medbullet$), the corresponding Rabi
frequency $f_{\mathrm{R}}$ ($\medcirc$) and the estimated gtm-EDSR
contribution $f_{\rm{gtm}}$ ($\Box$), as a function of the barrier gate
voltages $V_{\rm{L}}$, $V_{\rm{M}}$ and $V_{\rm{R}}$, respectively. For each
of the datasets a-c, the values on the other two barrier gates are fixed at
the value marked by the vertical dashed line. The yellow coloured regions
highlight the voltage ranges for which the qubit driving mechanism is
dominantly iso-EDSR, whereas the brown regions show the ranges for which the
contribution to the measured $f_{\mathrm{R}}$ originating from gtm-EDSR,
$f_{\mathrm{R}}^{\rm{gtm}}$, is $\geq 15\%$ ($5\%$ above the spread of data
points, see Supplementary Information). These colours are used consistently in
all other panels. The estimated $f_{\rm{gtm}}$ is represented by the white bar
charts. d, Schematic visualisation of how the Zeeman vector
$\mathbf{\hat{g}}_{0}\cdot\mathbf{B}$ is affected when subject to an
infinitesimal voltage change $dV$ in the presence of SOI represented by
$\alpha^{\perp}_{\rm{SO}}$ and a fixed magnetic field $|\mathbf{B}|$. Left
shows pure iso-EDSR coming from displacements of the dot potential. Right
shows the additional appearance of gtm-EDSR, when the dot is displaced and
deformed. e, Shows the extracted $\partial g/\partial V_{\rm{RP}}$, as a
function of the voltage $V_{\rm{L}}$ in blue, and f, as a function of
$V_{\rm{M}}$ in red. These derivatives are used to calculate an upper bound of
the $g$-modulated contribution to $f_{\mathrm{R}}$. g, $g$-factor as a
function of $f_{\mathrm{R}}$ exclusively showing those data points for which
iso-EDSR is the dominant driving mechanism, while the gtm-EDSR contribution is
$\leq 15\%$ of the measured $f_{\mathrm{R}}$. We fit the data using Eq. 1
which assumes iso-EDSR (fit shown by black line with yellow highlight),
yielding an intrinsic NW $g$-factor of $g_{\rm{NW}}\approx 1$.

We assume that all effects arising from SOI, induced by a change in voltage
$\Delta V$, are captured by a modulation of the $g$-tensor Crippa  _et al._
(2018) while keeping the magnetic field constant, that is
$\mathbf{\hat{g}}_{0}\cdot\mathbf{B}\xrightarrow[]{\Delta
V}\mathbf{\hat{g}}_{\Delta V}\cdot\mathbf{B}$. We represent this as a rotation
of the Zeeman vector around the perpendicular component of the spin-orbit
vector, $\bm{\alpha^{\perp}}_{\rm{SO}}$. Periodic displacements of the wave
function along the NW can result in SOI-mediated Rabi oscillations. We refer
to them as iso-Zeeman EDSR (iso-EDSR), if they conserve the modulus of the
Zeeman vector $|\mathbf{\hat{g}}\cdot\mathbf{B}|$, Fig. 2d left. In this case
$f_{\mathrm{R}}\propto 1/l_{\rm{SO}}$ Golovach  _et al._ (2006), and $g$ can
be directly related to $f_{\mathrm{R}}$ (see Eq. 1 II ),

$g~{}\overset{\rm{I}}{=}~{}g_{\rm{NW}}\cdot
e^{-{(l_{\rm{dot}}/l_{\rm{SO}})}^{2}}\overset{\rm{II}}{=}g_{\rm{NW}}\cdot
e^{-C\cdot f_{\mathrm{R}}^{2}}\quad.$ (1)

Here, $g$ denotes the measured $g$-factor as extracted from Fig. 1e,
$g_{\rm{NW}}$ is an intrinsic NW $g$-factor without renormalization due to
SOI, and $C$ is a fitting parameter. The derivation is provided in the
Supplementary Information. We further note that the functional form of Eq. 1
results from the harmonic confinement along the longitudinal axis of the NW
and is independent of the microscopic origin of the SOI.
As seen in Eq. 1, when reducing $l_{\rm{SO}}$ (thereby increasing SOI), $g$ is
suppressed and $f_{\mathrm{R}}$ increased. This behaviour can be observed in
Figs. 2a and b. To obtain a minimum in $g$ and a maximum in $f_{\mathrm{R}}$,
as seen in Fig. 2a, the governing SOI has to plateau or show a maximum, as an
implicit function of voltage. Ge/Si core-shell NWs as the one used here
Froning  _et al._ (2021a), like Ge-hut wires Watzinger  _et al._ (2018) and
Si-FinFETs Camenzind  _et al._ (2022), particularly benefit from DRSOI which
is expected to reach a local maximum at moderate electric fields below
$10~{}\rm{MV}/\rm{m}$ Kloeffel  _et al._ (2011); Froning  _et al._ (2021a);
Kloeffel  _et al._ (2018); Adelsberger  _et al._ (2022). These predicted
electric field ranges are consistent with the voltage range of $\sim 100$ mV
around the extrema of $g$ and $f_{\mathrm{R}}$ shown in Fig. 2a, assuming a
voltage drop over $\sim 50$ nm (gate-pitch).
In order to facilitate iso-EDSR, displacements of the hole wave function along
the NW are desirable, while only minimally varying the dot potential. To this
end the MW drive is chosen at gate RP, located as far as possible from the
qubit on gate LP. To see where the driving mechanism is consistent with iso-
EDSR, we measure the response of $g$ to variations $\Delta V_{\rm{RP}}$, to
extract $\partial g/\partial V_{\rm{RP}}$ at a fixed magnetic field
$\mathbf{B}$. These responses are shown in Figs. 2e and f, as a function of
$V_{\rm{L}}$ and $V_{\rm{M}}$.
If, while driving, the voltage shifts $\Delta V_{\rm{RP}}$ cause significant
variations of the target dot potential, the induced Rabi oscillations can be
significantly influenced by $g$-tensor modulated EDSR (gtm-EDSR). Such
modulations of the $g$-tensor do not conserve the modulus of the Zeeman vector
$\mathbf{\hat{g}}\cdot\mathbf{B}$, Fig. 2d right. Since the response $\partial
g/\partial V_{\rm{RP}}$ is computed from the difference in lengths of Zeeman
vectors $|\mathbf{\hat{g}}_{0}\cdot\mathbf{B}|-|\mathbf{\hat{g}}_{\Delta
V_{\rm{RP}}}\cdot\mathbf{B}|=\Delta_{|\mathbf{\hat{g}}\cdot\mathbf{B}|}$ under
$\Delta V_{\rm{RP}}$, we can only provide a rough upper bound to the
$g$-tensor modulated Rabi contribution $f_{\mathrm{R}}^{\rm{gtm}}$ which is
proportional to a transverse modulation of the Zeeman vector
$\Delta_{|\mathbf{\hat{g}}\cdot\mathbf{B}|}^{\rm{gtm}}$. These approximate
upper bounds for
$f_{\mathrm{R}}^{\rm{gtm}}~{}\propto~{}|\Delta_{|\mathbf{\hat{g}}\cdot\mathbf{B}|}^{\rm{gtm}}/\Delta
V_{\rm{RP}}|~{}\leq~{}|\Delta_{|\mathbf{\hat{g}}\cdot\mathbf{B}|}/\Delta
V_{\rm{RP}}|$ are represented by the bars in Figs. 2a and b. For the
calculation of $f_{\mathrm{R}}^{\rm{gtm}}$ and a qualitative description of
the EDSR driving mechanisms we refer to the Supplementary Information.
As seen in Figs. 2a, b, e and f, we qualitatively divide the measured data
points into two regions, driven primarily by pure iso-EDSR (yellow) and a mix
of both, iso- and gtm-EDSR (brown). If $|\partial g/\partial
V_{\rm{RP}}|\rightarrow 0$, also the component attributed to gtm-EDSR,
$f_{\mathrm{R}}^{\rm{gtm}}\rightarrow 0$. This leaves iso-EDSR as the only
available mechanism that can induce Rabi oscillations.
The pairs of $f_{\mathrm{R}}$ and $g$ classified as pure iso-EDSR are shown in
Fig. 2g where they are fitted using Eq. 1. From the fit parameter
$g_{\rm{NW}}\approx 1$ and the experimental values of $g$, the corresponding
$l_{\rm{SO}}$ were calculated yielding a range from $65$ nm to $150$ nm,
assuming $l_{\rm{dot}}=50$ nm (gate pitch). It is worth noting that this
estimation of $l_{\rm{SO}}$ does not rely on knowledge of the effective mass
$m_{\rm{eff}}$, which can be challenging to estimate in systems with HH-LH
mixing, as the one studied here. Further, the calculated values of
$l_{\rm{SO}}$ are in agreement with those obtained via magnetic field
spectroscopy Froning  _et al._ (2021b) and qubit measurements in Ref. Froning
_et al._ (2021a) assuming HH-LH mixing.

Compromise-Free Operation

Figure 3:  Compromise-free operation. a, The top plot shows the experimental
$f_{\mathrm{R}}$ and $g$ as a function of $V_{\rm{L}}$. The bottom plot shows
the coherence times $T^{\rm{Hahn}}_{2}$ (dark blue) obtained from fits of the
time-dependent readout current measured in Hahn-echo experiments at different
gate voltages $V_{\rm{L}}$. The light blue data shows the modelled coherence
times taking into account $g^{\prime}_{i}(V_{\rm{L}})$ from panels b-f using
Eq. 2, yielding $S_{\rm{G}}~{}=~{}29~{}\rm{nV}/\sqrt{\rm{Hz}}$. The vertical
dashed line at $V_{\rm{L}}=1330$ mV shows the sweet spot at which
$T^{\rm{Hahn}}_{2}$ is maximal. This spot coincides with the fastest
$f_{\mathrm{R}}$ within experimental errors. The inset shows pairs of
$f_{\mathrm{R}}$ and $g$ to the right of $V_{\rm{L}}=1330$ mV. b-f, Show
$g^{\prime}_{i}(V_{\rm{L}}):=\partial g/\partial V_{\rm{i}}(V_{\rm{L}})$, as a
function of different static voltages on the left barrier gate $V_{\rm{L}}$.
The dark blue data points represent values of $V_{\rm{L}}$ for which the
$g^{\prime}_{i}(V_{\rm{L}})$ were measured. The light blue line corresponds to
linearly interpolated data points in order to match the number of measured
data points between $g^{\prime}_{i}(V_{\rm{L}})$ and $T^{\rm{Hahn}}_{2}$,
which was fitted in panel a. The vertical dashed line marks the sweet spot at
$V_{\rm{L}}=1330$ mV for which all five $g^{\prime}_{i}(V_{\rm{L}})$ reach
near zero, defining the sweet spot. The horizontal dashed line marks the zero-
line of the $g^{\prime}_{i}(V_{\rm{L}})$. g, The FACTOR: 2D voltage space
showing $T^{\rm{Hahn}}_{2}$ as a function of gate voltages $V_{\rm{M}}$ and
$V_{\rm{L}}$. The filled circles represent the measured coherence times
$T^{\rm{Hahn}}_{2}$ extracted analogously to panel a. The data trace
corresponding to the dark blue trace on panel a is referenced by the blue
arrow on the 2D plot. An additional trace of $T^{\rm{Hahn}}_{2}$ as a function
of $V_{\rm{L}}$ for fixed $V_{\rm{M}}=630$ mV is indicated by the vertical
dark blue dashed line on the left end of the plot, as well as a trace of
$T^{\rm{Hahn}}_{2}$ as a function of $V_{\rm{M}}$ for fixed $V_{\rm{L}}=1320$
mV which is indicated by the horizontal red dashed line. The background is
obtained by a Gaussian process interpolation and serves as a guide to the eye.
The black contours show a Gaussian process interpolation of measured Rabi
frequencies, to highlight the overlap of maxima in $T_{2}^{\rm{Hahn}}$ and
$f_{\mathrm{R}}$.

Given the extremal behaviour of the g-factor and $f_{\mathrm{R}}$ from Fig.
2a, we measured the Hahn-echo decay times $T_{2}^{\rm{Hahn}}$ as a function of
$V_{\rm{L}}$ and compare them in Fig. 3a. We note that $T_{2}^{*}$ was too
short to be reliably extracted, due to its sensitivity to low-frequency noise
and the long integration times required by our transport measurements.
The applied pulse sequence involved a $\pi_{x}/2$ pulse, a refocusing $\pi$
pulse, and a $\pi_{\varphi}/2$ pulse with a sinusoidally varying phase as a
function of the free evolution time $\tau$, allowing for a more robust fit of
the Hahn decay. Assuming $1/f^{\beta}$ noise, the exponent of the Hahn-echo
decay $\alpha=1+\beta$ was defined as a global fit parameter, shared among all
data sets, and converged to $\alpha\approx 1$. This value for $\alpha$ was
extracted for the frequency range sampled by the Hahn-echo experiment from
approximately $1/\tau_{\rm{max}}=12.5$ MHz up to $f_{\mathrm{R}}$. Given the
operation temperature of $1.5$ K, a white noise spectrum could be expected, as
observed in other spin qubit experiments Camenzind  _et al._ (2022); Petit
_et al._ (2018); Huang  _et al._ (2023); Liles  _et al._ (2023). Above $1.7$
K, the qubit readout in this device significantly degrades. As seen in Fig.
3a, the coherence peaks at a gate voltage $V_{\rm{L}}$ where the g-factor is
minimal and $f_{\mathrm{R}}$ is maximal. This observation clearly indicates a
compromise-free scaling regime in which the fastest qubit operation times
coincide with the coherence sweet spot. Further, the qualitative signature of
the sweet spot remains robust regardless of the choice of noise color $\beta$,
as shown in the Supplementary Information.
Next, we analyze the response of $g$ to small voltage fluctuations of the gate
voltages $V_{i}$ as a function of $V_{\rm{L}}$, denoted as $\partial
g/\partial V_{i}(V_{\rm{L}})$. We refer to $\partial g/\partial
V_{i}(V_{\rm{L}})$ as $g^{\prime}_{i}(V_{\rm{L}})$ for brevity. Figs. 3b-f
show the $g^{\prime}_{i}(V_{\rm{L}})$ of the five gates used in our
experiments. One striking commonality among all five
$g^{\prime}_{i}(V_{\rm{L}})$ is the near-zero point crossing at the sweet spot
voltage $V_{\rm{L}}^{*}=1330$ mV, indicating minimal modulations of the Zeeman
vector magnitude $|\mathbf{\hat{g}}\cdot\mathbf{B}|$.
We suspect the noise to mainly originate from charge traps in the self-
terminated, native $\rm{SiO}_{2}$ layer, which uniformly covers the $\sim 2$
nm thick NW shell. This brings a source of noise very close to the $\rm{Ge}$
core, where the hole wave function resides. We model the noise as voltage
fluctuations on all the gates with one common noise spectrum, characterized by
a spectral density $S_{\rm{G}}$ and exponent $\beta$. Considering the
appropriate noise filter function for a Hahn-echo and $\beta=0$ (as determined
before), the characteristic decay rate can be expressed as Piot  _et al._
(2022),

$\frac{1}{T_{2}^{\rm{Hahn}}}=2\pi^{2}~{}\left(\sum_{i}\left(\frac{\partial
g}{\partial V_{i}}~{}S_{\rm{G}}\right)^{2}\right)\qquad.$ (2)

The best fit to the measured $T_{2}^{\rm{Hahn}}$, employing Eq. 2, is obtained
with a white noise spectral density of
$S_{\rm{G}}(f_{0}/f)=29~{}\rm{nV}/\sqrt{\rm{Hz}}$, with $f_{0}=12.5$ MHz. The
modeled coherence times are shown in Fig. 3a. No additional lever arms are
required in Eq. 2, as they are already captured by the
$g^{\prime}_{i}(V_{\rm{L}})$. Lastly, we repeat similar coherence measurements
for different values of $V_{\rm{M}}$ and show in Fig. 3g that the FACTOR is
not limited to a point in voltage space. In fact, the existence of a ”sweet
ridge” can be observed, which shows a weaker dependence of the coherence time
$T_{2}^{\rm{Hahn}}$ on $V_{\rm{M}}$ as compared to $V_{\rm{L}}$, over
comparable voltage ranges. As shown by the black contours in Fig. 3g, the Rabi
frequency $f_{\mathrm{R}}$ follows the trend of $T_{2}^{\rm{Hahn}}$, reaching
frequencies above $100$ MHz surrounding the coherent region in green. The
experimental data points for the contour plot of $f_{\mathrm{R}}$ are shown in
the Supplementary Information.

Conclusions
We report on the existence of a FACTOR for which the coherence sweet spot
coincides with the fastest qubit operation speeds. We demonstrate that our
FACTOR originates from the renormalisation of the $g$-factor yielding a
minimum in $g$ and thus $g^{\prime}_{i}(V_{\rm{L}})=0$, when the SOI is
maximised for a hole quantum dot subject to strong, quasi 1D confinement.
Further, maximal Rabi frequencies are achieved for iso-EDSR driving of the
qubit. We favour this driving mechanism by applying the MW drive to a remote
gate, and identify static gate voltage ranges for which the qubit is indeed
dominantly driven by iso-EDSR. Our observations are in agreement with
theoretical predictions for holes under quasi 1D confinement and a comparable
Rashba-type SOI Bosco  _et al._ (2021b); Wang  _et al._ (2021). Additionally,
our FACTOR can be tuned all-electrically at the individual qubit level,
without the need of changing global experimental parameters. Our experimental
observations thus overturn the conventional wisdom that fast qubit operations
impose a toll on qubit life times.
The conditions leading to the observed compromise-free operation might be
translated to 2D hole spin qubit platforms using lateral squeezing gate
electrodes to induce strong, quasi 1D confinement Bosco  _et al._ (2021b).
Regarding the applicability for electrons, conventional Rashba SOI could
potentially also be tuned by electric fields to reach a maximum in SOI, but
might require larger electric fields than the ones used here, as the SOI is
primarily governed by the fundamental band gap of the semiconductor obtained
in the third order of a multi-band perturbation theory Winkler (2003);
Carballido  _et al._ (2021). For systems using synthetic SOI generated by
micro-magnets, strategic placement of the QD, relative to the magnet using
static gates, might give rise to a voltage tunable maximum of the synthetic
SOI experienced by the QD Aldeghi  _et al._ (2023). Finally, it is essential
for the SOI to be strong enough to enter the strong SOI regime, for which
$l_{\rm{SO}}~{}\approx~{}l_{\rm{dot}}$ holds.
Furthermore, we have established coherent control of a hole spin qubit in a
Ge/Si NW at $1.5$ K, with qubit operation speeds and coherence times on par
with previous experiments performed at mK temperatures Froning  _et al._
(2021a). This achievement renders our platform compatible with on-chip
classical control electronics Vandersypen  _et al._ (2017).
By demonstrating the feasibility of a FACTOR in a hole spin qubit, our work
offers a new angle from which to approach fault-tolerant quantum computation
without sacrificing high qubit operation speeds.

## References

  * Yoneda  _et al._ (2017) J. Yoneda, K. Takeda, T. Otsuka, T. Nakajima, M. R. Delbecq, G. Allison, T. Honda, T. Kodera, S. Oda, Y. Hoshi, N. Usami, K. M. Itoh, and S. Tarucha, Nature Nanotechnology 13, 102 (2017).
  * Hendrickx and Fuhrer (2022) N. W. Hendrickx and A. Fuhrer, Nature Nanotechnology 17, 1040 (2022).
  * Tanttu  _et al._ (2019) T. Tanttu, B. Hensen, K. W. Chan, C. H. Yang, W. W. Huang, M. Fogarty, F. Hudson, K. Itoh, D. Culcer, A. Laucht, A. Morello, and A. Dzurak, Physical Review X 9, 021028 (2019).
  * Piot  _et al._ (2022) N. Piot, B. Brun, V. Schmitt, S. Zihlmann, V. P. Michal, A. Apra, J. C. Abadillo-Uriel, X. Jehl, B. Bertrand, H. Niebojewski, L. Hutin, M. Vinet, M. Urdampilleta, T. Meunier, Y.-M. Niquet, R. Maurand, and S. D. Franceschi, Nature Nanotechnology 17, 1072 (2022).
  * Hendrickx  _et al._ (2023) N. W. Hendrickx, L. Massai, M. Mergenthaler, F. Schupp, S. Paredes, S. W. Bedell, G. Salis, and A. Fuhrer, arXiv (2023), 10.48550/ARXIV.2305.13150.
  * Bosco  _et al._ (2021a) S. Bosco, B. Hetényi, and D. Loss, PRX Quantum 2, 010348 (2021a).
  * Froning  _et al._ (2021a) F. N. M. Froning, L. C. Camenzind, O. A. H. van der Molen, A. Li, E. P. A. M. Bakkers, D. M. Zumbühl, and F. R. Braakman, Nature Nanotechnology 16, 308 (2021a).
  * Gilbert  _et al._ (2023) W. Gilbert, T. Tanttu, W. H. Lim, M. Feng, J. Y. Huang, J. D. Cifuentes, S. Serrano, P. Y. Mai, R. C. C. Leon, C. C. Escott, K. M. Itoh, N. V. Abrosimov, H.-J. Pohl, M. L. W. Thewalt, F. E. Hudson, A. Morello, A. Laucht, C. H. Yang, A. Saraiva, and A. S. Dzurak, Nature Nanotechnology 18, 131 (2023).
  * Pla  _et al._ (2013) J. 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).
  * Veldhorst  _et al._ (2015) M. Veldhorst, C. H. Yang, J. C. C. Hwang, W. Huang, J. P. Dehollain, J. T. Muhonen, S. Simmons, A. Laucht, F. E. Hudson, K. M. Itoh, A. Morello, and A. S. Dzurak, Nature 526, 410 (2015).
  * Hendrickx  _et al._ (2020a) N. W. Hendrickx, W. I. L. Lawrie, L. Petit, A. Sammak, G. Scappucci, and M. Veldhorst, Nature Communications 11 (2020a), 10.1038/s41467-020-17211-7.
  * Hendrickx  _et al._ (2020b) N. W. Hendrickx, D. P. Franke, A. Sammak, G. Scappucci, and M. Veldhorst, Nature 577, 487 (2020b).
  * Liu  _et al._ (2023) H. Liu, K. Wang, F. Gao, J. Leng, Y. Liu, Y.-C. Zhou, G. Cao, T. Wang, J. Zhang, P. Huang, H.-O. Li, and G.-P. Guo, Nano Letters 23, 3810 (2023).
  * Yang  _et al._ (2020) C. H. Yang, R. C. C. Leon, J. C. C. Hwang, A. Saraiva, T. Tanttu, W. Huang, J. Camirand Lemyre, K. W. Chan, K. Y. Tan, F. E. Hudson, K. M. Itoh, A. Morello, M. Pioro-Ladrière, A. Laucht, and A. S. Dzurak, Nature 580, 350 (2020).
  * Petit  _et al._ (2020) L. Petit, H. G. J. Eenink, M. Russ, W. I. L. Lawrie, N. W. Hendrickx, S. G. J. Philips, J. S. Clarke, L. M. K. Vandersypen, and M. Veldhorst, Nature 580, 355 (2020).
  * Camenzind  _et al._ (2022) L. C. Camenzind, S. Geyer, A. Fuhrer, R. J. Warburton, D. M. Zumbühl, and A. V. Kuhlmann, Nature Electronics 5, 178 (2022).
  * Huang  _et al._ (2023) J. Y. Huang, R. Y. Su, W. H. Lim, M. Feng, B. van Straaten, B. Severin, W. Gilbert, N. D. Stuyck, T. Tanttu, S. Serrano, J. D. Cifuentes, I. Hansen, A. E. Seedhouse, E. Vahapoglu, N. V. Abrosimov, H.-J. Pohl, M. L. W. Thewalt, F. E. Hudson, C. C. Escott, N. Ares, S. D. Bartlett, A. Morello, A. Saraiva, A. Laucht, A. S. Dzurak, and C. H. Yang, arXiv (2023), 10.48550/ARXIV.2308.02111.
  * Kloeffel  _et al._ (2011) C. Kloeffel, M. Trif, and D. Loss, Physical Review B 84, 195314 (2011).
  * Kloeffel  _et al._ (2018) C. Kloeffel, M. J. Rančić, and D. Loss, Physical Review B 97, 235422 (2018).
  * Bosco  _et al._ (2021b) S. Bosco, M. Benito, C. Adelsberger, and D. Loss, Physical Review B 104, 115425 (2021b).
  * Wang  _et al._ (2021) Z. Wang, E. Marcellina, A. R. Hamilton, J. H. Cullen, S. Rogge, J. Salfi, and D. Culcer, npj Quantum Information 7 (2021), 10.1038/s41534-021-00386-2.
  * Michal  _et al._ (2023) V. P. Michal, J. C. Abadillo-Uriel, S. Zihlmann, R. Maurand, Y.-M. Niquet, and M. Filippone, Physical Review B 107, l041303 (2023).
  * Kloeffel  _et al._ (2013) C. Kloeffel, M. Trif, P. Stano, and D. Loss, Physical Review B 88, 241405 (2013).
  * Bosco  _et al._ (2022) S. Bosco, P. Scarlino, J. Klinovaja, and D. Loss, Physical Review Letters 129, 066801 (2022).
  * Loss and DiVincenzo (1998) D. Loss and D. P. DiVincenzo, Physical Review A 57, 120 (1998).
  * Stano and Loss (2022) P. Stano and D. Loss, Nature Reviews Physics 4, 672 (2022).
  * Burkard  _et al._ (2023) G. Burkard, T. D. Ladd, A. Pan, J. M. Nichol, and J. R. Petta, Reviews of Modern Physics 95, 025003 (2023).
  * Laucht  _et al._ (2016) A. Laucht, R. Kalra, S. Simmons, J. P. Dehollain, J. T. Muhonen, F. A. Mohiyaddin, S. Freer, F. E. Hudson, K. M. Itoh, D. N. Jamieson, J. C. McCallum, A. S. Dzurak, and A. Morello, Nature Nanotechnology 12, 61 (2016).
  * Hendrickx  _et al._ (2021) N. W. Hendrickx, W. I. L. Lawrie, M. Russ, F. van Riggelen, S. L. de Snoo, R. N. Schouten, A. Sammak, G. Scappucci, and M. Veldhorst, Nature 591, 580 (2021).
  * Philips  _et al._ (2022) S. G. J. Philips, M. T. Madzik, S. V. Amitonov, S. L. de Snoo, M. Russ, N. Kalhor, C. Volk, W. I. L. Lawrie, D. Brousse, L. Tryputen, B. P. Wuetz, A. Sammak, M. Veldhorst, G. Scappucci, and L. M. K. Vandersypen, Nature 609, 919 (2022).
  * Zwerver  _et al._ (2022) A. M. J. Zwerver, T. Kräähenmann, T. F. Watson, L. Lampert, H. C. George, R. Pillarisetty, S. A. Bojarski, P. Amin, S. V. Amitonov, J. M. Boter, R. Caudillo, D. Correas-Serrano, J. P. Dehollain, G. Droulers, E. M. Henry, R. Kotlyar, M. Lodari, F. Lüthi, D. J. Michalak, B. K. Mueller, S. Neyens, J. Roberts, N. Samkharadze, G. Zheng, O. K. Zietz, G. Scappucci, M. Veldhorst, L. M. K. Vandersypen, and J. S. Clarke, Nature Electronics 5, 184 (2022).
  * Neyens  _et al._ (2023) S. Neyens, O. Zietz, T. Watson, F. Luthi, A. Nethwewala, H. George, E. Henry, A. Wagner, M. Islam, R. Pillarisetty, R. Kotlyar, K. Millard, S. Pellerano, N. Bishop, S. Bojarski, J. Roberts, and J. S. Clarke, “Probing single electrons across 300 mm spin qubit wafers,”  (2023).
  * Weinstein  _et al._ (2023) A. J. Weinstein, M. D. Reed, A. M. Jones, R. W. Andrews, D. Barnes, J. Z. Blumoff, L. E. Euliss, K. Eng, B. H. Fong, S. D. Ha, D. R. Hulbert, C. A. C. Jackson, M. Jura, T. E. Keating, J. Kerckhoff, A. A. Kiselev, J. Matten, G. Sabbir, A. Smith, J. Wright, M. T. Rakher, T. D. Ladd, and M. G. Borselli, Nature 615, 817 (2023).
  * Scappucci  _et al._ (2020) G. Scappucci, C. Kloeffel, F. A. Zwanenburg, D. Loss, M. Myronov, J.-J. Zhang, S. De Franceschi, G. Katsaros, and M. Veldhorst, Nature Reviews Materials 6, 926 (2020).
  * Fang  _et al._ (2023) Y. Fang, P. Philippopoulos, D. Culcer, W. A. Coish, and S. Chesi, Materials for Quantum Technology 3, 012003 (2023).
  * Nowack  _et al._ (2007) K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and L. M. K. Vandersypen, Science 318, 1430 (2007).
  * Watzinger  _et al._ (2018) H. Watzinger, J. Kukučka, L. Vukušić, F. Gao, T. Wang, F. Schäffler, J.-J. Zhang, and G. Katsaros, Nature Communications 9 (2018), 10.1038/s41467-018-06418-4.
  * Massai  _et al._ (2023) L. Massai, B. Hetényi, M. Mergenthaler, F. J. Schupp, L. Sommer, S. Paredes, S. W. Bedell, P. Harvey-Collard, G. Salis, A. Fuhrer, and N. W. Hendrickx, arXiv (2023), 10.48550/ARXIV.2310.05902, arXiv:2310.05902 [cond-mat.mes-hall] .
  * Adelsberger  _et al._ (2022) C. Adelsberger, M. Benito, S. Bosco, J. Klinovaja, and D. Loss, Physical Review B 105, 075308 (2022).
  * Mauro  _et al._ (2023) L. Mauro, E. A. Rodríguez-Mena, M. Bassi, V. Schmitt, and Y.-M. Niquet, arXiv (2023), 10.48550/ARXIV.2312.09840, arXiv:2312.09840 [cond-mat.mes-hall] .
  * Froning  _et al._ (2021b) F. N. M. Froning, M. J. Rančić, B. Hetényi, S. Bosco, M. K. Rehmann, A. Li, E. P. A. M. Bakkers, F. A. Zwanenburg, D. Loss, D. M. Zumbühl, and F. R. Braakman, Physical Review Research 3, 013081 (2021b).
  * Golovach  _et al._ (2006) V. N. Golovach, M. Borhani, and D. Loss, Physical Review B 74, 165319 (2006).
  * Froning  _et al._ (2018) F. N. M. Froning, M. K. Rehmann, J. Ridderbos, M. Brauns, F. A. Zwanenburg, A. Li, E. P. A. M. Bakkers, D. M. Zumbühl, and F. R. Braakman, Applied Physics Letters 113 (2018), 10.1063/1.5042501.
  * Ungerer  _et al._ (2023) J. H. Ungerer, P. Chevalier Kwon, T. Patlatiuk, J. Ridderbos, A. Kononov, D. Sarmah, E. P. A. M. Bakkers, D. Zumbühl, and C. Schönenberger, Materials for Quantum Technology 3, 031001 (2023).
  * Maurand  _et al._ (2016) R. Maurand, X. Jehl, D. Kotekar-Patil, A. Corna, H. Bohuslavskyi, R. Laviéville, L. Hutin, S. Barraud, M. Vinet, M. Sanquer, and S. D. Franceschi, Nature Communications 7 (2016), 10.1038/ncomms13575.
  * Crippa  _et al._ (2018) A. Crippa, R. Maurand, L. Bourdet, D. Kotekar-Patil, A. Amisse, X. Jehl, M. Sanquer, R. Laviéville, H. Bohuslavskyi, L. Hutin, S. Barraud, M. Vinet, Y.-M. Niquet, and S. D. Franceschi, Physical Review Letters 120, 137702 (2018).
  * Petit  _et al._ (2018) L. Petit, J. Boter, H. Eenink, G. Droulers, M. Tagliaferri, R. Li, D. Franke, K. Singh, J. Clarke, R. Schouten, V. Dobrovitski, L. Vandersypen, and M. Veldhorst, Physical Review Letters 121, 076801 (2018).
  * Liles  _et al._ (2023) S. D. Liles, D. J. Halverson, Z. Wang, A. Shamim, R. S. Eggli, I. K. Jin, J. Hillier, K. Kumar, I. Vorreiter, M. Rendell, J. H. Huang, C. C. Escott, F. E. Hudson, W. H. Lim, D. Culcer, A. S. Dzurak, and A. R. Hamilton, arXiv (2023), 10.48550/ARXIV.2310.09722.
  * Winkler (2003) R. Winkler, “Spin-orbit coupling effects in two-dimensional electron and hole systems,”  (2003).
  * Carballido  _et al._ (2021) M. J. Carballido, C. Kloeffel, D. M. Zumbühl, and D. Loss, Physical Review B 103, 195444 (2021).
  * Aldeghi  _et al._ (2023) M. Aldeghi, R. Allenspach, and G. Salis, Applied Physics Letters 122 (2023), 10.1063/5.0139670.
  * Vandersypen  _et al._ (2017) L. M. K. Vandersypen, H. Bluhm, J. S. Clarke, A. S. Dzurak, R. Ishihara, A. Morello, D. J. Reilly, L. R. Schreiber, and M. Veldhorst, npj Quantum Information 3 (2017), 10.1038/s41534-017-0038-y.

Methods
Device Fabrication. The QD device featured a set of nine bottom gates, each
with a width of $\sim 20$ nm and a pitch of $\sim 50$ nm. The gates were
fabricated on an intrinsic Si $(100)$ chip with $290$ nm of thermal
$\rm{SiO}_{2}$ using electron beam lithography (EBL). After cold development,
the bottom gates were metallized with $1$ nm/$9$ nm of Ti/Pd, respectively. To
provide electrical insulation between the bottom gates and the NW, $175$
cycles ($\sim 200$ nm) of $\rm{Al}_{2}\rm{O}_{3}$ were grown by atomic layer
deposition at approximately $225^{\circ}$ C using atomic layer deposition. In
an effort to improve the quality of the gate dielectric, the chip underwent
annealing in a $20$ mbar forming gas atmosphere
($\rm{N}_{2}~{}92\%,~{}\rm{H}_{2}~{}8\%$) for 15 minutes at $300^{\circ}$ C,
prior to NW deposition. Details on the impact of the annealing process on gate
hysteresis and qubit coherence are described in the Supplementary Information.
A single Ge/Si core/shell NW was deterministically placed in a perpendicular
orientation to the nine bottom gates. The NW has a core radius of $\sim 10$ nm
and a shell thickness of $\sim 2.5$ nm. The exact in-plane angle, however,
remains unknown. Subsequently, ohmic contacts were patterned by EBL and
metallized with Ti/Pd layers of $0.3$ nm/$50$ nm of Ti/Pd, respectively,
following a $10$ second dip in buffered hydrofluoric acid to locally remove
the native $\rm{SiO}_{2}$ layer in the defined contact region. Fig. 1a
presents a scanning electron micrograph of an analogously fabricated device
from the same batch, representative of the measured device.
Measurement Apparatus. The experimental setup featured a variable temperature
insert (VTI) in a liquid helium bath with the sample mounted below the $1$K
pot (base temperature $1.5$ K). The VTI was equipped with a solenoid magnet
controlled by an Oxford Instruments IPS magnet power supply. DC voltages were
supplied by a Basel Precision Instruments digital-to-analog converter (LNHR
$927$) and filtered on a dedicated filter PCB (second-order RC low-pass
filter, cutoff frequency $8$ kHz). Fast gate pulses and IQ control pulses were
generated on a Tektronix AWG $5204$. A Rohde & Schwarz SGS100A Vector Signal
Generator was used to generate the qubit control pulses through IQ modulation.
The gate- and control pulses were combined using a Wainwright WDKX11 diplexer
and delivered to the sample PCB using attenuated coaxial lines. A bias-tee on
the sample PCB was used to combine the high frequency pulses with a DC bias.
The DC current through the NW was amplified by a Basel Precision Instruments
current-to-voltage converter (LSK389A) with a gain of $10^{9}$ and measured
using a National Instruments DAQ card (USB-6363). The Vector Signal Generator
output was pulse-modulated by a Zurich Instruments MFLI lock-in amplifier at
$77.777$ Hz to enhance the signal-to-noise ratio of the qubit measurements.
Data Analysis. The $g$-factors were measured as described in Fig. 1e, and were
extracted for each considered electrostatic configuration defined by the
barrier voltages $V_{\rm{L}}$, $V_{\rm{M}}$ and $V_{\rm{R}}$. The positions of
the resonance condition, with respect to $f_{\rm{MW}}$ at fixed $\bf{B}$, are
obtained by fitting each column to a Gaussian. The slope of a linear fit to
the center positions of the Gaussians then yields the $g$-factor. The Rabi
frequencies $f_{\mathrm{R}}$ were extracted from fits to
$I(t)=I_{\rm{offset}}+I_{0}\sin(2\pi
f_{\mathrm{R}}t_{\rm{burst}}+~{}\phi_{0})\exp(-t_{\rm{burst}}/T_{2}^{\rm{Rabi}})$.
Here, $I_{\rm{offset}}$ is an offset, $I_{0}$ the amplitude, $\phi_{0}$ a
phase shift and $T_{2}^{\rm{Rabi}}$ the characteristic decay time. The
characteristic decay times of each individual Hahn-echo experiment
$T_{2}^{\rm{Hahn}}$ were obtained from a global fit of all echo-experiments
using $I(t)~{}=~{}I_{\rm{offset}}~{}+~{}I_{0}\sin(2\pi
f_{\varphi}\tau_{\rm{wait}}~{}+~{}\phi_{0})\exp(-(\tau/T_{2}^{\rm{Hahn}})^{\alpha})$,
and one shared parameter $\alpha=\beta+1$. Furthermore $f_{\varphi}$ describes
the frequency at which the phase of the pulse $\pi_{\varphi}/2$ was
artificially varied as a function of the free evolution time $\tau$.
Measurement Details. The derivatives $\partial g/\partial
V_{i}(V_{j}):=g^{\prime}_{i}(V_{j})$ presented in Fig. 3b-f were extracted in
three different ways at fixed $f_{\rm{MW}}$: i)
$\bm{g^{\prime}_{i}(V_{\rm{L}})}$, for $\bm{i}\in\\{\rm{LP,M,R}\\}$ : The
readout point was defined by fixing all gate voltages. The derivatives were
then obtained by recording the $g$-factors at manually varied voltages $\Delta
V_{i}=2-4$ mV, without loosing readout. ii)
$\bm{g^{\prime}_{\rm{RP}}~{}(V_{j})}$, for $\bm{j}\in\\{\rm{L,M}\\}$: For a
fixed readout point, the depth $\Delta V_{\rm{CP}}$, by which the system was
pulsed into Coulomb blockade, was varied by up to $10$ mV while all other
voltages were held constant. Only the DC voltage on $V_{\rm{RP}}$ was
algorithmically adjusted for each $\Delta V_{\rm{CP}}$ by a linear correction
factor in order to keep the readout point fixed. As in i), the derivatives
were computed by fitting the slope of the recorded $g$-factor versus the
variation of $\Delta V_{\rm{CP}}$. iii) $\bm{g^{\prime}_{\rm{L}}(V_{\rm{L}})}$
: We first computed the derivative of the recorded $g(V_{\rm{L}})$ presented
in Fig. 2a with respect to $V_{\rm{L}}$, yielding
$g^{\prime}_{\rm{L}}(\tilde{V}_{\rm{L}})$. Here, $\tilde{V}_{\rm{L}}$ is used
to highlight that for each value of $V_{\rm{L}}$, the voltage $V_{\rm{LP}}$
was compensated as otherwise the readout point would have been lost over the
considered range of $V_{\rm{L}}$ due to the large cross-capacitance.
Therefore, in order to obtain the true derivative
$g^{\prime}_{\rm{L}}(V_{\rm{L}})$, the influence of $V_{\rm{LP}}$ was
subtracted to first order via
$g^{\prime}_{\rm{L}}(V_{\rm{L}})=g^{\prime}_{\rm{L}}(\tilde{V}_{\rm{L}})-g^{\prime}_{\rm{LP}}(V_{\rm{L}})\cdot\Delta
V_{\rm{LP}}/\Delta V_{\rm{L}}$. Here, $g^{\prime}_{\rm{LP}}(V_{\rm{L}})$ is
taken from i) and the compensation $\Delta V_{\rm{LP}}/\Delta
V_{\rm{L}}\approx-0.49$.
All qubit measurements were performed at a microwave frequency of $2.79$ GHz
at $P_{\rm{MW}}=-13.4$ dBm power and $V_{\rm{IQ}}=300$ mV amplitude IQ
voltage, corresponding to an AC excitation of $V_{\rm{ac}}=7.8$ mV at the
driving gate RP. The Rabi chevron shown in Fig. 1f was taken near the optimal
operating regime at $V_{\rm{L}}=1320$ mV, $V_{\rm{M}}=660$ mV,
$V_{\rm{R}}=1020$ mV. The EDSR resonance shown in Fig. 1e was recorded at a
fixed $t_{\rm{burst}}=4$ ns.
The Hahn-echo experiments were taken at $2$ seconds integration time and
depending on amplitude of the transport current, up to 50 averages were taken
of each trace to improve the signal to noise ratio.

Data Availability
The data supporting the plots of this paper are available at the Zenodo
repository at https://doi.org/10.5281/zenodo.10223162?preview=1.

Acknowledgments
We thank Y.-M. Niquet, S. Geyer and F.N.M. Froning for the fruitful
discussions. This work was supported by the NCCR SPIN, the Swiss Nanoscience
Institute (SNI), the Georg H. Endress Foundation, the Swiss National Science
Foundation (SNSF), the EU H2020 European Microkelvin Platform (EMP) project
(Grant No. $824109$) and the Topologically Protected and Scalable Quantum Bits
(TOPSQUAD) project (Grant No. $862046$). L.C.C. acknowledges support from a
Swiss NSF mobility fellowship (P$2$BSP$2-$200127).

Author Contributions
M.J.C., S.S., T.P. and D.M.Z. conceived of the project and planned the
experiments. M.J.C., S.S. and P.C.K. fabricated the device. A.L. and
E.P.A.M.B. synthesized the NWs. S.B. developed the theoretical framework for
squeezed quantum dots with inputs from D.L.. R.M.K. studied the gate
dielectric quality. M.J.C. and T.P. designed the experimental setup with
inputs from L.C.C.. M.J.C., S.S. and T.P. executed the experiments. J.S.
contributed to the extended data sets supervised by N.A.. M.J.C., R.S.E., T.P.
and S.B. developed the requirements for a FACTOR and analyzed the data with
input from J.C.E. and D.M.Z.. D.M.Z. supervised the project. M.J.C. wrote the
manuscript with inputs from all the authors.


competitions, M6 was overall a success. Following the M5 forecasting
competition, we set up to empirically investigate an interesting and specific
problem: how well forecasters and investors can forecast various tradable
assets and set their investment weights. Makridakis’ forecasting competitions
have always been in the forefront of forecasting research, pushing boundaries,
setting research agendas, and shaping the field. The M6 competition was no
different - its impact to the community is already evident.

###  7.1 Discussion

Compared to past forecasting competitions (Makridakis et al., 2021), we
implemented three key changes in our competition design. First, the M6
competition was a live competition, meaning that participants collected the
most up-to-date data and submitted forecasts in real time. Their forecasts
were evaluated against the actuals as time progressed, and then they submitted
their forecasts for the next period. This cycle was repeated twelve times
(over almost one calendar year) to achieve certain levels of robustness.
Second, the nature of the forecasting task differed compared to past
competitions. In the past, the objective was to submit forecasts for the
future values of each target variable. In the M6 competition, the forecasting
task involved forecasting the relative performance of different tradable
assets. Third, this competition did not stop in evaluating the performance of
the forecasting task, but also directly evaluated the implications of the
forecasts by measuring the performance of the investment decisions.

One of the key results of the competition was the inability of participants to
significantly outperform the forecasting benchmark (that assumed equal
probabilities) in predicting the probabilities for the returns quintiles of
the various assets. Despite that, participants were able to form efficient
portfolios that balanced returns with risk. We observed a considerable
disconnect between forecasting performance and investment performance, where
there was almost zero correlation between the two if all participants were
taken into account. Our empirical results confirmed the EMH, with only one
team outperforming the investment benchmark (which referred to an equal-
weighted portfolio across all assets with long positions) across all 4
quarters, while none of the teams managed to outperform the investment
benchmark across all 12 months of the competition. Another key finding that
reiterates the results of past forecasting competitions is that the use of
forecast combinations (or aggregation, in this case through “wisdom of
crowds”) is a simple but efficient approach to improve performance. Finally,
again in-line with insights from past M competitions, simple and standard
methods can yield good performance overall.

The disconnect of the performance between the two tasks, the forecasting and
the investment challenge, was an interesting finding that was discussed during
the presentation of the results in the 2023 International Symposium on
Forecasting. Some attendees argued that it is likely that participants simply
focused on making informed investments and simply avoided producing any
forecasts. We disagree with this argument. We strongly believe that any
decision, in this case the investment weights for the different assets,
requires a forecast. This forecast is sometimes implicit and, thus, not
properly quantified. However, it is still a forecast: one opts for a high
weight and long position in a particular asset only because they implicitly
forecast that asset will perform better compared to other available assets. In
fact, by construction, M6 enabled the submission of either qualitative (i.e,
pure judgmental) or quantitative forecasts, however seemingly some
participants bypassed the forecasting task completely, failing to efficiently
record what a forecast, which implicitly or explicitly informed their
investment decisions, could look like. Our results in section 5 clearly show
that if participants have used a realistic risk model (as the one we present
in this paper) then there would be a clear link between forecasting and
investment performance. Submissions with forecasting performance in the best
or second-best quintile could have yielded a higher investment performance (as
measured by the IR) and higher return.

###  7.2 Limitations

One of the limitations of this competition was its duration. The competition
involved twelve non-overlapping submission points, each 28 days after the
previous one. As a result, the competition ran for almost one year. This had a
negative impact on both the total number of participating teams as well as
engagement. In fact, we anticipated this negative impact and we attempted to
mitigate it by offering rich monetary awards for the top-performing teams
(both overall and per quarter). However, still, only a small percentage of the
teams updated their forecasts regularly during the duration of the
competition, while many teams simply submitted once or twice at the very
beginning of the competition. Could we have designed the competition such that
it lasted less than a year to increase engagement and decrease drop-outs?
Probably not. On the contrary, we believe that, in evaluating the EMH, it
would be even better to have the competition running for a longer time period,
possibly 2-5 years, to be able to include significant cycles in the economy.
As a result, we believe that while neither the competition’s duration nor the
achieved participation were perfect, we managed to achieve a good balance
between these two aspects.

The second limitation of the M6 competition design has to do with the selected
one hundred assets to form the investment universe. As with any empirical
exercise that uses a finite and finely defined set of data, one can argue that
the results of the competition are limited to its data. In an ideal scenario,
we could have included all possible investable assets available worldwide.
This, though, would render the exercise impractical from the participants’
point of view, as the forecasting and investment task would be too complex.
Another consideration would be to just allow participants to focus (and
provide forecasts of) only specific subsets of the one hundred assets (for
instance, stock only or ETFs only). We decided against doing so, as this would
inevitably have led us to have more distinct evaluation categories (resulting
in a further split on the prizes) as the forecasting performance of one subset
(such as stocks) would not be comparable to that of the other subset (such as
ETFs), not allowing us to rank all participants together.

The third limitation of the M6 competition was that, on top of the “duathlon”
prizes, we offered prizes for each of the tasks/challenges separately. This
meant that participants could focus exclusively on one of the challenges
(forecasting or investment), receive significant awards, and ignore the other
challenge. For instance, the top-performing teams in the forecasting task did
not perform very well in the investment task. As a consequence, this limited
our learning opportunities with regards to the actual connectedness between
forecasting and investment performance. Our post-competition analysis
suggested that if the top-performers in one challenge made use of state-of-
the-art risk models, then they could have performed much better in the other
challenge compared to their actual performance.

The final limitation of the M6 competition has to do with the limited
opportunities it offers for reproducing or even replicating the forecasts and
investment decisions of the participating teams. Given the nature of the
competition (financial forecasting), and in order to attract participation
from expert financial analysts, we decided to only ask for a brief, bird’s eye
view, description of the participants’ forecasting and investment approach.
The main argument was that an expert analyst would not participate in such a
competition if they had to fully “expose” their approach as this would render
them “obsolete”. Making the submission of a detailed description of the teams’
approaches, or even the submission of the respective code, a prerequisite for
participation would allow us to replicate the submissions but, at the same
time, would decrease participation from experts and would potentially render
submissions based on pure judgment not viable.

###  7.3 Directions for future research

We believe that the live aspect of the M6 forecasting competition offered a
unique take on forecasting competitions. We, as the organizers, did not need
to conceal the data or their source and did not need to define a suitable
hold-out sample. At the same time, the participants had access to all the
information related to the data and were able to also apply, apart from
quantitative approaches, judgmental approaches either exclusively or in
conjunction with formal models. An additional advantage is that live
competitions will allow us to be in a position to evaluate the domain
expertise of the participants on top of the technical expertise. We would like
to see more forecasting competitions moving away from concealed data and hold-
out samples towards live set-ups using real-time data. Associated challenges
include data availability and the imminence of the task (forecasts need to be
produced very shortly after the actual values have been published).

In this and the past M forecasting competitions, participants had to submit
the outputs of their approaches (forecasts and decisions). An alternative way
forward would be to ask the participants to submit their solutions directly
(i.e., code or executable). While this would limit the scope of their
approaches to strictly quantitative ones, it would offer a considerable
advantage. Competition organizers would be able to run the participants’
solutions directly, being in a position to evaluate their effectiveness over
longer time periods, in a rolling origin manner. Also, the submission of
solutions instead of outputs would guarantee the reproducibility of the
submitted forecasts and decisions.

Three of the most cited M forecasting competitions, M1, M3 and M4, offered a
diverse forecasting challenge, encompassing data from various domains (micro,
macro, industry, finance, demographic, etc.) and a variety of frequencies
(yearly, quarterly, monthly, weekly, daily, hourly). Such competitions allow
us to identify methods and approaches that are robust and efficient over a
variety of settings and contexts. The latter two competitions, M5 and M6,
focused on specific contexts, that of retail and financial forecasting. We
believe that focusing on particular industries will allow us to better
identify the “horses for the course”. In that sense, we can see future
competitions focusing on specific industries such as energy forecasting,
pharmaceutical forecasting, and medical forecasting. Equally, more
“generalist” competitions push the boundaries of innovation towards the
development of generic robust forecasting approaches.

Finally, we would like to reiterate the value of the M forecasting
competitions in bridging theory and practice via robust and wide-scale
empirical evaluations. As we move forward, we invite companies and industries
that face challenges with their forecasting tasks to consider M forecasting
competitions as a platform of innovation that will offer them a diverse pool
of efficient forecasting solutions that is unique to their data and
challenges.

## References

  * Alexander (2002) Alexander, C. (2002).  Principal component models for generating large garch covariance matrices.  Economic Notes, 31, 337–360.
  * Andersen et al. (2001) Andersen, T. G., Bollerslev, T., Diebold, F. X., & Ebens, H. (2001).  The distribution of realized stock return volatility.  Journal of Financial Economics, 61, 43–76.
  * Armour (2023) Armour, B. (2023).  Active funds continue to fall short of their passive peers.  https://www.morningstar.com/etfs/active-funds-continue-fall-short-their-passive-peers.  Accessed: 2023-09-01.
  * Best (2023) Best, R. (2023).  Top 5 positions in warren buffett’s portfolio.  https://www.investopedia.com/articles/investing/022816/top-5-positions-warren-buffetts-portfolio.asp.  Accessed: 2023-09-01.
  * Black (1992) Black, F. (1992).  Global portfolio optimization.  Financial Analysts Journal, 48, 28–44.
  * Bollerslev (1986) Bollerslev, T. (1986).  Generalized autoregressive conditional heteroskedasticity.  Journal of Econometrics, 31, 307–327.
  * Bollerslev (2009) Bollerslev, T. (2009).  Glossary to ARCH (GARCH), in T. Bollerslev, T., Russel, J., Watson, M., Volatility and Time Series Econometrics: Essays in Honor of Robert F. Engle. Oxford University Press, London.
  * Bollerslev et al. (2018) Bollerslev, T., Hood, B., Huss, J., & Pedersen, L. H. (2018).  Risk Everywhere: Modeling and Managing Volatility.  Review of Financial Studies, 31, 2729–2773.
  * Brockwell (1991) Brockwell, P. J. (1991).  Time series : theory and methods.  Springer series in statistics (2nd ed.).  New York: Springer-Verlag.
  * Buffett (2023) Buffett, W. E. (2023).  Letter 2022 to berkshire shareholders.
  * Chau et al. (2020) Chau, M., Lin, C.-Y., & Lin, T.-C. (2020).  Wisdom of crowds before the 2007–2009 global financial crisis.  Journal of Financial Stability, 48, 100741.
  * Cheng et al. (2021) Cheng, M., Swanson, N. R., & Yang, X. (2021).  Forecasting volatility using double shrinkage methods.  Journal of Empirical Finance, 62, 46–61.
  * Connor (2019) Connor, G. (2019).  The three types of factor models: A comparison of their explanatory power.  Financial Analysts Journal, 51, 42–46.
  * Dai et al. (2021) Dai, M., Jia, Y., & Kou, S. (2021).  The wisdom of the crowd and prediction markets.  Journal of Econometrics, 222, 561–578.
  * Doung & Swanson (2015) Doung, D., & Swanson, N. R. (2015).  Empirical evidence on the importance of aggregation, asymmetry, and jumps for volatility prediction.  Journal of Econometrics, 187, 606–621.
  * Engle (2002) Engle, R. (2002).  Dynamic conditional correlation - a simple class of multivariate garch models.  Journal of Business and Economic Statistics, 20, 339–350.
  * Engle (2009) Engle, R. (2009).  Anticipating Correlations: A New Paradigm for Risk Management.  Princeton University Press, New Jersey.
  * Engle & Mezrich (1996) Engle, R., & Mezrich, J. (1996).  Garch for groups.  Risk, 9, 36–40.
  * Engle (1982) Engle, R. F. (1982).  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation.  Econometrica, 50, 987.
  * Engle & Kroner (1995) Engle, R. F., & Kroner, K. F. (1995).  Multivariate simultaneous generalized arch.  Econometric Theory, 11, 122–150.
  * Fama (1969) Fama, E. F. (1969).  Papers and Proceedings of the twenty-eight annual meeting of the American Finance Association.  New York.
  * Frazzini et al. (2018) Frazzini, A., Kabiller, D., & Pedersen, L. H. (2018).  Buffett’s alpha.  Financial Analysts Journal, 74, 35–55.
  * French (2023) French, K. R. (2023).  Kenneth R. French Data Library.  https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
  * Garman & Klass (1980) Garman, M. B., & Klass, M. J. (1980).  On the estimation of security price volatilities from historical data.  The Journal of Business, 53, 67–78.
  * Ghysels et al. (2007) Ghysels, E., Sinko, A., & Valkanov, R. (2007).  Midas regressions: further results and new directions.  Econometric Reviews, 26, 53–90.
  * Goldstein & Wooff (2007) Goldstein, M., & Wooff, D. (2007).  Bayes Linear Statistics: Theory and Methods.  John Wiley & Sons, New York.
  * Gottschlich & Hinz (2014) Gottschlich, J., & Hinz, O. (2014).  A decision support system for stock investment recommendations using collective wisdom.  Decision Support Systems, 59, 52–62.
  * Graham (1949) Graham, B. (1949).  The Intelligent Investor: The Definitive Book on Value Investing.  New York: Harper Business.
  * Grinold & Kahn (1999) Grinold, R. C., & Kahn, R. N. (1999).  Active Portfolio Management (PB).  McGraw Hill Professional, New York.
  * Hafner & Reznikova (2012) Hafner, C. M., & Reznikova, O. (2012).  On the estimation of dynamic conditional correlation models.  Computational Statistics & Data Analysis, 56, 3533–3545.
  * Hansen & Lunde (2005) Hansen, P. R., & Lunde, A. (2005).  A forecast comparison of volatility models: does anything beat a GARCH(1,1)?  Journal of Applied Econometrics, 20, 873–889.
  * Jondeau et al. (2007) Jondeau, E., Poon, S.-H., & Rockinger, M. (2007).  Financial Modeling Under Non-Gaussian Distributions.  Springer Science & Business Media, New York.
  * Lassance (2022) Lassance, N. (2022).  Maximizing the out-of-sample sharpe ratio.  Available at SSRN, .
  * Liao & Fan (2011) Liao, Y., & Fan, J. (2011).  High dimensional covariance matrix estimation in approximate factor models.  Annals of Statistics, 39, 3320–3356.
  * Lichtenstein et al. (1982) Lichtenstein, S., Fischhoff, B., & Phillips, L. D. (1982).  Calibration of probabilities: The state of the art to 1980\.  In Judgment under Uncertainty: Heuristics and Biases (pp. 306–334).  Cambridge University Press.
  * Makridakis et al. (1982) Makridakis, S., Andersen, A., Carbone, R., Fildes, R., Hibon, M., Lewandowski, R., Newton, J., Parzen, E., & Winkler, R. (1982).  The accuracy of extrapolation (time series) methods: Results of a forecasting competition.  Journal of Forecasting, 1, 111–153.
  * Makridakis et al. (1993) Makridakis, S., Chatfield, C., Hibon, M., Lawrence, M., Mills, T., Ord, K., & Simmons, L. F. (1993).  The m2-competition: A real-time judgmentally based forecasting study.  International Journal of Forecasting, 9, 5–22.
  * Makridakis et al. (2021) Makridakis, S., Fry, C., Petropoulos, F., & Spiliotis, E. (2021).  The future of forecasting competitions: Design attributes and principles.  INFORMS Journal on Data Science, 1, 96–113.
  * Makridakis & Hibon (2000) Makridakis, S., & Hibon, M. (2000).  The M3-Competition: results, conclusions and implications.  International Journal of Forecasting, 16, 451–476.
  * Makridakis et al. (2020) Makridakis, S., Spiliotis, E., & Assimakopoulos, V. (2020).  The M4 Competition: 100,000 time series and 61 forecasting methods.  International Journal of Forecasting, 36, 54–74.
  * Makridakis et al. (2022) Makridakis, S., Spiliotis, E., & Assimakopoulos, V. (2022).  The M5 competition: Background, organization, and implementation.  International Journal of Forecasting, 38, 1325–1336.
  * Malkiel (1973) Malkiel, B. G. (1973).  A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing.  New York: Norton and Company.
  * Markowitz (1959) Markowitz, H. (1959).  Portfolio Selection: Efficient Diversification of Investments.  Wiley, New York.
  * Molnár (2012) Molnár, P. (2012).  Properties of range-based volatility estimators.  International Review of Financial Analysis, 23, 20–29.
  * Montero-Manso et al. (2020) Montero-Manso, P., Athanasopoulos, G., Hyndman, R. J., & Talagala, T. S. (2020).  FFORMA: Feature-based forecast model averaging.  International Journal of Forecasting, 36, 86–92.
  * Parkinson (1980) Parkinson, M. (1980).  The extreme value method for estimating the variance of the rate of return.  The Journal of business (Chicago, Ill.), 53, 61–65.
  * Pedersen (2015) Pedersen, L. H. (2015).  Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined.  Princeton University Press.
  * Petropoulos et al. (2022) Petropoulos, F., Apiletti, D., Assimakopoulos, V., Babai, M. Z., Barrow, D. K., Ben Taieb, S., Bergmeir, C., Bessa, R. J., Bijak, J., Boylan, J. E., Browell, J., Carnevale, C., Castle, J. L., Cirillo, P., Clements, M. P., Cordeiro, C., Cyrino Oliveira, F. L., De Baets, S., Dokumentov, A., Ellison, J., Fiszeder, P., Franses, P. H., Frazier, D. T., Gilliland, M., Gönül, M. S., Goodwin, P., Grossi, L., Grushka-Cockayne, Y., Guidolin, M., Guidolin, M., Gunter, U., Guo, X., Guseo, R., Harvey, N., Hendry, D. F., Hollyman, R., Januschowski, T., Jeon, J., Jose, V. R. R., Kang, Y., Koehler, A. B., Kolassa, S., Kourentzes, N., Leva, S., Li, F., Litsiou, K., Makridakis, S., Martin, G. M., Martinez, A. B., Meeran, S., Modis, T., Nikolopoulos, K., Önkal, D., Paccagnini, A., Panagiotelis, A., Panapakidis, I., Pavía, J. M., Pedio, M., Pedregal, D. J., Pinson, P., Ramos, P., Rapach, D. E., Reade, J. J., Rostami-Tabar, B., Rubaszek, M., Sermpinis, G., Shang, H. L., Spiliotis, E., Syntetos, A. A., Talagala, P. D., Talagala, T. S., Tashman, L., Thomakos, D., Thorarinsdottir, T., Todini, E., Trapero Arenas, J. R., Wang, X., Winkler, R. L., Yusupova, A., & Ziel, F. (2022).  Forecasting: theory and practice.  International Journal of Forecasting, 38, 705–871.
  * Prado et al. (2021) Prado, R., Ferreira, M. A. R., & West, M. (2021).  Time Series: Modeling, Computation, and Inference, Second Edition.  CRC Press.
  * Rogers & Satchell (1991) Rogers, L. C. G., & Satchell, S. E. (1991).  Estimating variance from high, low and closing prices.  The Annals of applied probability, 1, 504–512.
  * Smyl (2020) Smyl, S. (2020).  A hybrid method of exponential smoothing and recurrent neural networks for time series forecasting.  International Journal of Forecasting, 36, 75–85.
  * Surowiecki (2005) Surowiecki, J. (2005).  The Wisdom of Crowds.  Anchor Books.
  * Swanson & Xiong (2018) Swanson, N. R., & Xiong, W. (2018).  Big data analytics in economics: what have we learned so far, and where should we go from here?  Canadian journal of Economics, 3, 695–746.
  * Tashman (2000) Tashman, L. J. (2000).  Out-of-sample tests of forecasting accuracy: an analysis and review.  International Journal of Forecasting, 16, 437 – 450.
  * Triantafyllopoulos (2021) Triantafyllopoulos, K. (2021).  Bayesian Inference of State Space Models: Kalman Filtering and Beyond.  Springer Texts in Statistics.  Cham: Springer International Publishing AG.
  * West (1992) West, M. (1992).  Modelling agent forecast distributions.  J. R. Stat. Soc., 54, 553–567.
  * West & Crosse (1992) West, M., & Crosse, J. (1992).  Modelling probabilistic agent opinion.  J. R. Stat. Soc., 54, 285–299.
  * Yang & Zhang (2000) Yang, D., & Zhang, Q. (2000).  Drift‐independent volatility estimation based on high, low, open, and close prices.  The Journal of business (Chicago, Ill.), 73, 477–492.
  * Yardley & Petropoulos (2021) Yardley, E., & Petropoulos, F. (2021).  Beyond error measures to the utility and cost of the forecasts.  Foresight: The International Journal of Applied Forecasting, (pp. 36–45).

## Appendix A: Construction of the M6 competition universe of assets

The 50 stocks were selected so that each GICS (Global Industry Classification
Standard) sector is represented in M6 in a similar proportion with that in the
S&P500 index. In this regard, the number of assets to be sampled per sector
was defined as shown in Table A1.

Table A1: Stocks included in the S&P500 and the M6 investment universe per sector. Sector | S&P500 | M6
---|---|---
Count | Proportion (%) | Count | Proportion (%)
Communication Services | 27 | 5.3 | 3 | 6.0
Consumer Discretionary | 63 | 12.5 | 6 | 12.0
Consumer Staples | 32 | 6.3 | 3 | 6.0
Energy | 21 | 4.2 | 2 | 4.0
Financial | 65 | 12.9 | 7 | 14.0
Health Care | 64 | 12.7 | 6 | 12.0
Industrial | 74 | 14.7 | 7 | 14.0
Information Technology | 74 | 14.7 | 7 | 14.0
Materials | 28 | 5.5 | 3 | 6.0
Real Estate | 29 | 5.7 | 3 | 6.0
Utilities | 28 | 5.5 | 3 | 6.0

In November 2021, the 9 key features presented in Subsection 2.2 were computed
for each of the stocks, taking into account the average price of the stock and
its volatility, the daily and compound returns, the volatility of the returns,
as well as the trading volume. These features where then used to group the
stocks of each sector into diverse clusters and make sure that the selected
assets would sufficiently represent the market.

To do that, the silhouette method was first employed to define the optimal
number of clusters per sector. Then, based on the population size of each
cluster, an appropriate number of stocks was randomly sampled per cluster. As
an example, Figure A showcases that the 21 stocks of the “Energy” sector were
grouped into two clusters of 5 and 16 stocks, respectively. Given that the
examined sector should contribute 2 stocks in total to the M6 universe, the
first cluster should provide $(5/21)*2=0.48\approx 0$ stocks, while the second
$(16/21)*2=1.52\approx 2$ stocks. By randomly sampling two stocks from the
second cluster, the COP and XOM stocks were included among the M6 assets.

Figure A: Example illustrating the clustering process taking place for the
case of the “Energy” GICS sector.

ETFs were selected qualitatively with the objective to capture the overall
returns of major markets (e.g. US, UK, Germany, Japan, China, India,
Australia, Brazil), involve short- and long-term treasury and corporate bond
options, allow investing in major sectors (e.g. energy, infrastructures,
technology, health) Provide precious metals and commodity options (e.g. gold
and silver), and include equity style options (e.g. value, momentum and
minimum volume for the United States of America and Europe). The selected ETFs
can be categorized according to their type as shown in Table A2.

Table A2: ETFs included in the M6 investment universe per type. ETF Type | Count
---|---
Large Cap | 15
Small Cap | 2
Sector | 14
Equity Style | 6
Government | 5
Credit | 4
Precious Metals | 2
Diversified Commodities | 1
Volatility | 1

The 100 assets that were included in the M6 investment universe are presented
in Table A3.

Table A3: Assets included in the M6 investment universe. The M6 Investment
Universe (Stocks and ETFs)
---
ABBV | ACN | AEP | AIZ | ALLE | AMAT | AMP | AMZN | AVB | AVY
AXP | BDX | BF-B | BMY | BR | CARR | CDW | CE | CHTR | CNC
CNP | COP | CTAS | CZR | DG | DPZ |  DRE999On October 3, 2022 the DRE price stopped being updated due to the company being acquired by PLD. Therefore, participants had to assume a return of zero for DRE (any investment to DRE would have no effect on portfolio returns and the rank of the asset in terms of returns would have to be forecast assuming no price change). | DXC | EWA | EWC
EWG | EWH | EWJ | EWL | EWQ | EWT | EWU | EWY | EWZ |  FB101010On June 9, 2022 the FB identifier changed into META.
FTV | GOOG | GPC | GSG | HIG | HIGH.L | HST | HYG | IAU | ICLN
IEAA.L | IEF | IEFM.L | IEMG | IEUS | IEVL.L | IGF | INDA | IUMO.L | IUVL.L
IVV | IWM | IXN | JPEA.L | JPM | KR | LQD | MCHI | MVEU.L | OGN
PG | PPL | PRU | PYPL | RE | REET | ROL | ROST | SEGA.L | SHY
SLV | SPMV.L | TLT | UNH | URI | V | VRSK | VXX | WRK | XLB
XLC | XLE | XLF | XLI | XLK | XLP | XLU | XLV | XLY | XOM

## Appendix B: Connection of forecasts and investment decisions

In order to quantify the degree to which the forecasts of the participating
teams were connected with the corresponding investment decisions, we
introduced a correlation measure, $r_{\text{con}}$, computed as follows:

  * •

For each submission, we multiply the investment weight of each asset with the
corresponding forecasts (probability that the asset will be ranked within the
first, second, third, fourth and fifth quintile). By doing that, we end up
with 100 vectors, each containing 5 elements, {Rank1, Rank2, Rank3, Rank4,
Rank5}, that represent the expected value of the investment. Elements of high
positive and negative values suggest that the uncertainty around the forecast
was relatively low or that the invested amount (either long or short position)
was relatively large. Accordingly, elements of close to zero values suggest
that the uncertainty around the forecast was relatively high or that the
invested amount was relatively small.

  * •

We compute the average of the {Rank1, Rank2, Rank3, Rank4, Rank5} elements by
aggregating their values across all assets and submissions made by a team
throughout the competition.

  * •

We calculate the correlation, $r_{\text{con}}$, between the averages computed
in the previous step and a vector containing the following integer numbers:
{1,2,3,4,5}.

It becomes evident that if the forecasts of a team are connected with its
investment decisions, $r_{\text{con}}$ should be close to unity (larger
amounts of capital are invested at higher ranked assets and lower - or even
negative - amounts of capital are invested at lower ranked assets). For
instance, the following vector {-0.036, -0.022, 0.004, 0.019, 0.073} would
score $r_{\text{con}}=0.96$, while the following one
{0.003,-0.019,-0.019,-0.007,0.027} would score $r_{\text{con}}=0.29$

In this regard, the submissions of the teams where classified into the
following categories:

  * •

“Well connected” ($r_{\text{con}}\geq 0.75$);

  * •

“Connected” ($0.50\leq r_{\text{con}}<0.75$);

  * •

“Weekly connected” ($0.25\leq r_{\text{con}}<0.50$);

  * •

“Disconnected” ($-0.25\leq r_{\text{con}}<0.25$);

  * •

“Opposite connection” ($r_{\text{con}}\leq-0.25$);

  * •

“NA” (Teams whose forecasts were identical to those of the benchmark and,
therefore, $r_{\text{con}}$ could not be computed).

From the 163 teams included in the “Global” leaderboard, 39 fell into the
“well connected” category, 20 into the “connected” category, 19 were
classified as “weakly connected”, 42 as “disconnected”, while 16 had an
“opposite connection”. This result suggests that the majority of the teams
developed separate approaches for preparing their submissions for the two
challenges of the M6 competition, thus often making investments that can
barely be justified based on the provided forecasts. Figure B1 further
supports this finding, indicating that although some teams have shorted the
assets that fell according to their predictions in the worst category (Rank1)
and invested more capital to the assets that fell according to their
predictions in the best category (Rank5), in most of the cases, similar
investments were made across all five asset classes.

Figure B1: Distribution of the values computed for the {Rank1, Rank2, Rank3,
Rank4, Rank5} elements (expected value of investment) for the 163 teams
included in the “Global” leaderboard.

Figure B2 provides further insights on the impact that the degree of
connection between the forecasts and the investment decisions had on the
performance of the teams. As seen, “well connected” submissions tend to
involve better portfolios than the “disconnected” and “opposite connection”
portfolios. However, “weekly connected” submissions perform similarly well and
the IR values largely overlap across the individual classes. The differences
in terms of performance are even smaller when it comes to RPS. In fact, we
observe that “well connected” submissions involve some of the worst sets of
forecasts, in contrast to the “weekly connected” submissions that perform much
better on average.

Figure B2: Distribution of the IR and RPS scores achieved by the 163 teams
included in the “Global” leaderboard, grouped based on the degree the
forecasts are connected with the investment decisions.

## Appendix C: Technical details of the risk model

In this appendix, we discuss some technical details associated with the
estimation of our investment risk model. Before discussing estimation of the
key building blocks of the model (i.e., volatility forecasts) that are used in
portfolio optimization when creating pseudo-true portfolios for comparison
with M6 participant portfolios, however, we first briefly discuss the
historical data set used for this purpose. Our data set includes daily prices
for the cross section of M6 assets, collected from from Yahoo Finance. More
specifically, we analyze 10 calendar years of daily data, ending on the start
date of the competition, namely February 2, 2022. The data comprise Open,
Close, High and Low prices, as well as Adjusted Prices, accounting for
splits/corporate actions, dividends etc. When building our dataset, we
attempted to select a broad cross section of underlying asset classes for our
ETF universe, and unfortunately this meant that we are unable to obtain a
balanced panel of data; several ETF securities were launched subsequent to our
data start date. Additionally there are several days with missing data, due to
market holidays and other reasons. We _forward_ filled such missing data with
the last available set of prices when data were missing, effectively assigning
a return of zero to market holidays, for example.

#### Modeling asset level volatility

To model asset level volatility we use Heterogeneous Exponential Realized
Volatility (HEXP) models, as described in Bollerslev et al. (2018). We select
these models for several reasons. First, the approach taken by the authors is
to fit a ‘global’ model for the entire panel of assets, which is inherently
parsimonious and robust, as it uses the same set of parameters for each
underlying asset (see above discussion). Such global models reflect a Bayesian
philosophy, where the (scaled) time series of each asset is regarded as
exchangeable, resulting in significant reductions in estimation error.
Empirical evidence summarized in Bollerslev et al. (2018) indicates that these
models are appropriate for the wide range of underling asset classes,
including those examined here. Second, the model can be easily fitted using
OLS, making the approach accessible to competitors without sophisticated
econometric knowledge and software. Third, the model offers a simple and
intuitive route to account for volatility spillovers and common dynamics in
volatility across asset / asset classes. Because the model uses smooth,
exponentially weighted averages with several look back periods, it avoids
potential variance estimation issues to which its closely related predecessor
(i.e., the heterogeneous autoregression (HAR) model) is prone111111For further
discussion of the HAR model, see Doung & Swanson (2015), and the references
cited therein. In addition, note that the HEXP model also has some features
that draw on the MIDAS model discussed in Ghysels et al. (2007).. Of further
note is that the HEXP model described in Bollerslev et al. (2018) uses a
database of intra-day realized volatility estimates from a cross section of
futures contracts. We do not assume that competition participants have access
to such data, and instead use daily volatility estimates based on our intra-
day high/low/open/close data. Evidence presented in Molnár (2012) suggests
that such estimates are reasonably interchangeable with those obtained from
intra-day data.

Recall, that competitors are asked to produce portfolios which maximize the
Sharpe Ratio over the 20 trading days following each submission date. We
therefore build our risk model to make direct single step variance estimates
with a 20 day horizon. Separately, we fit a model with a 1 day horizon as a
filter prior to constructing our covariance matrix estimates. The HAR
structure of our model is readily adaptable to both requirements.

To fit the variance model we begin by calculating historic variances in three
ways, adopting the methodology described in Yang & Zhang (2000). We call this
the YZ estimator in the sequel. Namely, for each asset define:

$\displaystyle\\{O_{t},C_{t},H_{t},L_{t}\\}$
$\displaystyle=\\{Open,Close,High,Low\\}\text{ Prices, day t.}$ $\displaystyle
o_{t}$ $\displaystyle=log(O_{t})-log(C_{t-1})$ $\displaystyle u_{t}$
$\displaystyle=log(H_{t})-log(O_{t})$ $\displaystyle d_{t}$
$\displaystyle=log(L_{t})-log(O_{t})$ $\displaystyle c_{t}$
$\displaystyle=log(C_{t})-np.log(O_{t})$ $\displaystyle V_{P}$
$\displaystyle=\frac{1}{T}\sum_{t=1}^{T}\frac{1}{4log2}(u_{t}-d_{t})^{2}$
$\displaystyle V_{RS}$
$\displaystyle=\frac{1}{T}\sum_{t=1}^{T}[u_{t}(u_{t}-c_{t})+d_{t}(d_{t}-c_{t})]$

Here, $V_{P}$ and $V_{RS}$ are Parkinson (1980) and Rogers & Satchell (1991)
variance estimators. When a variance estimate is required for a given day
(these values are used as a basis for the exponentially weighted moving
average values which are the regressors in our model, and as dependent
variables in our 1 day horizon model described below) we use the Garman-Klass
estimator:

$V_{GK}=V^{\prime}_{O}-0.383V^{\prime}_{C}+1.364V_{P}+0.019V_{RS},$

with $V^{\prime}_{O}=o_{t}^{2}$, $V^{\prime}_{C}=c_{t}^{2}$, and with $V_{P}$
and $V_{RS}$ defined as above, setting $T=1$. When a variance estimate is
required for a 20 day period (these form the dependent variables in our
models) we use the YZ estimator which accounts for drift in the asset price
over the relevant time period. The model also requires a long-run volatility
estimate for each asset. For this we utilize the YZ estimator based on an
expanding window of data commencing with the first recorded price for each
asset. The YZ estimator is defined as121212Refer to Yang & Zhang (2000) for
derivation of $V$.:

$V=V_{O}+kV_{C}+(1-k)V_{RS}$

with,

$\displaystyle V_{O}$
$\displaystyle=\frac{1}{T-1}\sum_{t=1}^{T}(o_{t}-\bar{o})^{2}$ $\displaystyle
V_{C}$ $\displaystyle=\frac{1}{T-1}\sum_{t=1}^{T}(c_{t}-\bar{c})^{2}$
$\displaystyle\bar{o}$ $\displaystyle=(1/T)\sum_{t=1}^{T}o_{t}$
$\displaystyle\bar{c}$ $\displaystyle=(1/T)\sum_{t=1}^{T}c_{t}$ $\displaystyle
k=\frac{.34}{1.34+\frac{(T+1)}{(T-1)}}$

At each estimation date, we follow Bollerslev et al. (2018), and form 4
(centered) exponentially weighted averages of daily variance, called
$ExpRV^{K}_{T}$, with centers of mass (CoM) at 1, 5, 25, and 125 days. Namely,
we define:

$\displaystyle ExpRV^{K}_{T}$ $\displaystyle=\sum_{t=1}^{T}w_{t}RV_{t},$
$\displaystyle w_{t}$ $\displaystyle=\frac{e^{-\lambda t}}{\sum e^{-\lambda
t}},$ $\displaystyle\lambda$ $\displaystyle=CoM[1,5,25,125],$

with $\lambda=log(1+1/CoM)$. We also calculate a universe ‘average’ volatility
factor for our entire investment universe (using assets available at each
historic point in time) and re-scale this back to the native scale of each
asset as described in Bollerslev et al. (2018). For our $h$ step ahead
forecast ($h=[1,20]$) we then stack all assets and all historic $h$ day time
windows in to one vector, and run a single LS regression on the entire data
set. We estimate the model as at each competition entry point. The LS model
is:

$\displaystyle(\hat{V}_{t:t+h}-RV^{LR}_{t})$
$\displaystyle=\kappa_{1}(ExpRV^{1}_{t}-RV^{LR}_{t})+\kappa_{2}(ExpRV^{5}_{t}-RV^{LR}_{t})$
$\displaystyle+\kappa_{3}(ExpRV^{25}_{t}-RV^{LR}_{t})+\kappa_{4}(ExpRV^{125}_{t}-RV^{LR}_{t})+\kappa_{5}(ExpGlRV^{5}_{t}-RV^{LR}_{t})+e_{t}.$

#### Modeling asset level volatility covariance

There are two key components to our estimates of the covariance structure of
our asset volatility. First, we assume that M6 asset returns can be described
using a factor model. To do this we define factors as linear combinations of
standardized returns on M6 assets. Using returns standardized using univariate
volatility models is common practice in financial risk management (see Engle
(2002)). For example, Alexander (2002) use such returns as the basis for
principal component analysis, which is used to estimate factors. We adopt a
similar approach, except that we define our factors based on prior knowledge,
rather than identifying them from the data. For further details, refer to
Connor (2019), who describe several possible approaches to building factor
models. These authors note that for ‘asset allocation’ type analysis,
regression models with pre-specified factors tend to be used in practice.
Building a bottom up fundamental model for a portfolio of equities would
indeed require significant effort, and we assume that the data collection,
estimation and testing of such a model was unrealistic for M6 participants. We
therefore adopt a purely regression based approach to this problem, solely
based on the use of historical returns.

More specifically, we define our factors in three hierarchical layers. The
layers are fitted sequentially on residuals from the previous layer(s). This
enables the use of a very simple identity prior or shrinkage target for our
dynamic regression at each stage. For the top level of the hierarchy we use a
single factor which we call the ‘M6 Market factor’ (M6M), as an equally
weighted combination of all 50 M6 ETF assets. In some ways this is analogous
to a market portfolio for the competition, although obviously it is not market
capitalization weighted or appropriate for use outside the context of M6. To
construct a market capitalization weighted average we would have needed to
either choose some subset of the assets, or obtain incremental data, which we
chose not to do. We also decided to exclude individual equity securities, as
equities already have significant underlying representation in our ETF
securities.

The next layer of the factor model hierarchy is designed to capture the
differential dynamics of various asset classes and style factors, which are
known from the literature to have significant effects on asset pricing. We
again define these using returns on various linear combinations of ETF assets
form the M6 universe.

Finally, our level 3 factors are defined to be congruent with the equity
market industry ETFs which form a part of the M6 Universe. At this stage, we
again adopt a simple identity prior for asset loadings (i.e. we do not attempt
to use prior information regarding individual equities to identify them with
particular industries, although might be useful). The full set of factors are
defined as in Table C1, with precise weightings available on request from the
authors.

Level | Factor | Definition
---|---|---
1 | M6M | M6 Market Factor
2 | USE | US Equity factor
2 | EUE | European equity factor
2 | AE | Asian equity factor
2 | TERM | Yield curve slope factor
2 | CREDIT | Corporate debt - Govt. debt
2 | MOM | Momentum factor
2 | VAL | Value factor
2 | SIZ | Size factor
3 | TECH | Technology factor
3 | FIN | Financial factor
3 | HEALTH | Healthcare factor
3 | ENERGY | Energy factor
3 | DISC | Consumer discretionary factor
3 | IND | Industrial factor
3 | COMMS | Communications factor
3 | UTIL | Utilities factor
3 | STAPLES | Consumer staples factor
3 | MATERIAL | Materials factor
Table C1: Factor Definitions for M6 Risk Model Factors

The second step of our approach is to estimate time varying loadings for each
asset in these pre-specified factors. We do so by using a parsimonious variant
of the BEKK model described above, with an additional shrinkage component,
along the lines suggested in Hafner & Reznikova (2012).

More specifically, to estimate asset covariance we proceed as follows. At each
submission date we take the in-sample 1 day horizon fitted values from our
univariate variance model for each asset. We then standardise the observed
asset return for that day using the standard deviations corresponding to these
variance estimates. We then use these daily standardized returns to build the
hierarchical factor model, using the BEKK approach to estimating the rolling
covariance between the M6M factor and each individual asset 131313 We note
that the M6 assets are traded both in London and in New York, and thus there
is the potential for imperfect time synchronicity to affect our results.. As
the constant term in the BEKK model, we use the long run whole period
covariance between each asset and M6M (estimated via an OLS regression of the
standardized asset returns on the factor plus a constant). We specify our BEKK
model using two _scalar_ parameters, namely $\gamma$, the weight on the time
$t-1$ error term, and $\omega$, the weight on the long run covariance
estimate. The weight on the time $t-1$ covariance estimate is defined as
$(1-\omega-\gamma)$. The BEKK model used for this and the following layers of
the factor model can be written as:

$\boldsymbol{\Sigma_{t}}=\omega\boldsymbol{\Sigma_{0}}+\gamma\mathbf{e}_{t-1}\mathbf{e}_{t-1}^{\prime}+(1-\omega-\gamma)\boldsymbol{\Sigma_{t-1}},$
(18)

where $\Sigma_{t}$ denotes our time varying covariance matrix and $\Sigma_{0}$
is our constant long run covariance matrix, the $e_{t-1}$ are previous period
errors, and all other terms in the above expression are constants. We then
collect the residuals from this exercise, and re-standardize them to have mean
zero and standard deviation of 1. We again use a BEKK model (with the same
$\omega$ and $\delta$) to estimate time varying covariance between all the
assets and each factor, with shrinkage towards the identity matrix.

The third step of our approach is to again collect the residuals from the new
BEKK model, re-standardize and run the BEKK model once more with the ‘level 3’
industry factors as regressors.

Finally, to calculate the asset specific risks, we utilize yet another BEKK
model, again using the same $\omega$ and $\gamma$ values, with the long run
specific variance calculated from the entire sample period. Also, the factor
covariance matrix is modeled using a similar structure, again with an identity
matrix as the long run shrinkage target and the same BEKK parameters as
previously141414At this step, one can introduce a separate set of parameters
for the factor covariance matrix, which might be expected to be more stable
than the factor/asset loading process. This is left to future research,
however.. To choose $\omega$ (the weight on the previous day’s cross error
cross product) and $\gamma$ (the weight on the long run shrinkage target) we
run a grid search, with a test data set comprising the 36 20 day trading
periods ending on November 30, 2021. For each date in the period, we estimate
our model and compute the log-likelihood of the entire set of M6 asset return
observations for following 20 day out of sample period. Prior experience with
similar models leads us to test values of $\omega$ from the set
$[.030,.020,.010,.005]$ and $\gamma$ from the set $[.0025,.0050,.0075]$. The
values which maximize the log-likelihood for the test data are $\omega=.01$
and $\gamma=.005$. As expected, these values place substantial weight on the
shrinkage target. We use these values to estimate the covariance matrix for
the rest of the competition.