# The $\mathrm{{}^{103}Rh}$ NMR Spectroscopy and Relaxometry of the Rhodium
Formate Paddlewheel Complex

Harry Harbor Collins  School of Chemistry, University of Southampton, SO17
1BJ, UK  Mohamed Sabba  School of Chemistry, University of Southampton, SO17
1BJ, UK  Gamal Moustafa  School of Chemistry, University of Southampton, SO17
1BJ, UK  Bonifac Legrady  School of Chemistry, University of Southampton, SO17
1BJ, UK  Murari Soundararajan  School of Chemistry, University of Southampton,
SO17 1BJ, UK  Markus Leutzsch  Max-Planck-Institut für Kohlenforschung,
Kaiser-Wilhelm-Platz 1, Mülheim an der Ruhr, 45470, Germany  Malcolm H. Levitt
<EMAIL_ADDRESS>School of Chemistry, University of Southampton, SO17 1BJ, UK
http://www.Second.institution.edu/~Charlie.Author.

###### Abstract

The NMR spectroscopy of spin-1/2 nuclei with low gyromagnetic ratio is
challenging, due to the low NMR signal strength. Methodology for the rapid
acquisition of $\mathrm{{}^{103}Rh}$ NMR parameters is demonstrated for the
case of the rhodium formate “paddlewheel" complex
$\mathrm{Rh_{2}(HCO_{2})_{4}}$. A scheme is described for enhancing the
$\mathrm{{}^{103}Rh}$ signal strength by polarization transfer from
$\mathrm{{}^{1}H}$ nuclei and which also greatly reduces the interference from
ringing artifacts, a common hurdle for the direct observation of low-$\gamma$
nuclei. The $\mathrm{{}^{103}Rh}$ relaxation time constants $T_{1}$ and
$T_{2}$ are measured within 20 minutes by using 1H-detected experiments. The
field-dependence of the $\mathrm{{}^{103}Rh}$ $T_{1}$ is measured. The high-
field relaxation is dominated by the chemical shift anisotropy (CSA)
mechanism. The $\mathrm{{}^{103}Rh}$ shielding anisotropy is found to be very
large: $|\Delta\sigma|=9900\pm 540\mathrm{\,ppm}$. This estimate is compared
with density functional theory calculations.

††preprint: AIP/123-QED

##  I Introduction

Rhodium paddlewheel complexes have attracted significant attention due to
their unique properties and diverse applications where they have played roles
as catalysts and potential anticancer agents.Erck  _et al._ (1974); Fandzloch
_et al._ (2020); Ohata and T. Ball (2018); Lin and Turro (2021); Hrdina (2021)
These complexes consist of two rhodium atoms bridged by four carboxylate
ligands, forming a lantern-like structure, with some resemblance to the
paddlewheels of a river boat. A typical example is rhodium formate,
$\mathrm{Rh_{2}(HCO_{2})_{4}}$, see figure 1.

Nuclear magnetic resonance (NMR) is a powerful probe of the properties of
rhodium complexes. $\mathrm{{}^{103}Rh}$ carries the distinction of being one
of only 4 (with 19F, 31P, and 89Y) spin-1/2 nuclei with a natural abundance of
100%. Nevertheless, it has been relatively neglected by spectroscopists:
$\mathrm{{}^{103}Rh}$ is a member of what Mann dubbed "the Cinderella
nuclei"Mann (1991) \- transition metals with spin-1/2 but very low
magnetogyric ratio $\gamma$. The NMR of $\mathrm{{}^{103}Rh}$ is associated
with multiple experimental challenges leading to a relative scarcity of
experimental data. However, many of these challenges have been successfully
overcome by the creative application of modern NMR methodology, such as
heteronuclear multiple-quantum (HMQC) NMR Caló  _et al._ (2021). However,
although HMQC experiments allow the rapid acquisition of $\mathrm{{}^{103}Rh}$
NMR spectra in suitable cases, it is not possible to estimate
$\mathrm{{}^{103}Rh}$ spin-lattice and spin-spin relaxation time constants
through HMQC experiments. For this purpose, experiments exploiting
$\mathrm{{}^{103}Rh}$ magnetization are needed.

In this work, we utilise a variant of the PulsePol polarisation transfer
technique Schwartz  _et al._ (2018); Tratzmiller (2021); Sabba  _et al._
(2022) to enhance the $\mathrm{{}^{103}Rh}$ NMR spectroscopy of the rhodium
formate paddlewheel complex in solution. We report (i) NMR methodology for the
acquisition of directly detected $\mathrm{{}^{103}Rh}$ spectra with effective
ringing filtration; (ii) NMR methodology for the rapid measurement of
$\mathrm{{}^{103}Rh}$ $T_{1}$ and $T_{2}$ relaxation time constants over a
range of magnetic field strengths. We observe a strong field dependence of the
$\mathrm{{}^{103}Rh}$ $T_{1}$, which is qualitatively consistent with a
dominant chemical shift anisotropy relaxation mechanism. We estimate the
$\mathrm{{}^{103}Rh}$ shielding anisotropy by using information from
${}^{13}\mathrm{C}$ and $\mathrm{{}^{103}Rh}$ relaxation experiments in
solution, and from ${}^{13}\mathrm{C}$ solid-state NMR.

Figure 1:  Molecular structure of the rhodium formate paddlewheel complex
ligated by solvent tetrahydrofuran (THF) molecules at the axial sites. This
work exploits the ${}^{3}J_{\mathrm{RhH}}$ scalar couplings for polarisation
transfer between the 103Rh and 1H nuclei.

##  II Experimental

###  II.1 Sample

Experiments were performed on a saturated ($\sim$10 mM) solution of rhodium
formate ($\mathrm{Rh_{2}(HCO_{2})_{4}}$) dissolved in 500 $\,\mu\mathrm{L}$
deuterated tetrahydrofuran (THF-d8) contained in a Wilmad LPV 5 mL sample
tube. The rhodium formate was synthesised from rhodium chloride using a
reported procedureRempel  _et al._ (1972) and dried extensively under heated
vacuum. The resulting rhodium formate solid was green in colour and dissolved
in THF to produce a green solution.

###  II.2 Solution NMR

$\mathrm{{}^{1}H}$ and $\mathrm{{}^{103}Rh}$ Spectra were acquired at a
magnetic field strength of 9.4 T using a standard commercial Bruker 5 mm NMR
BBO probe ($\mathrm{{}^{1}H}$/2H/109Ag-31P) equipped with a z-gradient with a
maximum strength of 50 G cm-1.

Proton resonances are referenced to the absolute frequency 400.14300 MHz;
whereas $\mathrm{{}^{103}Rh}$ resonances are referenced to an absolute
frequency that is proportional to the protons
($\Xi\left({}^{103}\rm{Rh}\right)$ = 3.16$\%$) per the most common convention
Carlton (2008).

Although the probe could be tuned to $\mathrm{{}^{103}Rh}$ beyond the
manufacturer specifications, it was set to mismatched (overcoupled) conditions
to reduce ringdown times Chingas (1983); Buess and Petersen (1978);
Gerothanassis (1987); Fukushima and Roeder (1979). The radiofrequency
amplitudes on the 1H and $\mathrm{{}^{103}Rh}$ channels were both adjusted to
give an intentionally matched nutation frequency of $\omega_{\rm
nut}/(2\pi)\simeq 4\mathrm{\,kHz}$, corresponding to a 90∘ pulse duration of
$62.5\,\mu\mathrm{s}$.

Additional isolation of the rf channels by electronic filters was found to be
necessary - without the filters, noise on the $\mathrm{{}^{103}Rh}$ channel
was significant enough to preclude observation of other nuclei. At the
preamplifier output we installed: a 30 MHz lowpass filter (Chemagnetics) on
the $\mathrm{{}^{103}Rh}$ channel, a 400 MHz bandpass filter (K&L Microwave)
on the 1H channel, and a 61 MHz bandpass filter (FSY Microwave) on the 2H lock
channel.

To measure relaxation times as a function of magnetic field, the experiments
used rapid sample shuttling from inside the 9.4 T magnet bore to regions of
lower field outside the magnet bore. The shuttling was performed using a
motorised fast shuttling system based on the design by Kiryutin Kiryutin  _et
al._ (2016). The shuttling time was kept constant at 1 second.

The pulse sequences described below use the following elements:

####  II.2.1 Composite pulses

Composite pulses were used to minimize the effects of rf field inhomogeneity
and are denoted by shaded black rectangles in the pulse sequence diagrams. All
composite pulses are implemented using the symmetrized BB1 composite pulse
scheme Wimperis (1994); Cummins, Llewellyn, and Jones (2003) in which a simple
pulse $\beta_{\phi}$ (where $\beta$ is the flip angle and $\phi$ is the phase)
is replaced by:

$(\beta/2)_{\phi}180_{\phi+\theta_{W}}360_{\phi+3\theta_{W}}180_{\phi+\theta_{W}}(\beta/2)_{\phi}$
(1)

Where $\theta_{W}=\arccos{(-\beta/(4\pi))}$. For the $\pi/2$ and $\pi$ flip
angles used in this paper, this corresponds to the following sequences:

$90_{\phi}\rightarrow
45_{\phi}180_{\phi+97.18}360_{\phi+291.54}180_{\phi+97.18}45_{\phi}$ (2)
$180_{\phi}\rightarrow
90_{\phi}180_{\phi+104.48}360_{\phi+313.43}180_{\phi+104.48}90_{\phi}$ (3)

Figure 2:  DualPol pulse sequence used for
$\mathrm{{}^{1}H}$-$\mathrm{{}^{103}Rh}$ cross polarisation, and consisting of
simultaneous PulsePol sequences Schwartz  _et al._ (2018) on the two channels.
Each PulsePol sequence is a repeating sequence of two R-elements. Each
R-element has duration $\tau_{R}$, and is given by a composite 180∘ pulse
Levitt and Freeman (1979) with delays of duration $\tau$ between the pulses.
The R-element duration should be short compared to the inverse of the relevant
J-couplings. The black rectangles indicate BB1 composite $\pi$-pulses
(equation 3).

####  II.2.2 DualPol Polarization Transfer Sequence

The transfer of polarisation between $\mathrm{{}^{103}Rh}$ and 1H was achieved
using the pulse sequence shown in figure 2. This consists of repeating
PulsePol sequences Schwartz  _et al._ (2018); Tratzmiller (2021), applied
simultaneously to the 1H and $\mathrm{{}^{103}Rh}$ radiofrequency channels.
The PulsePol sequence consists of six phase-shifted radiofrequency pulses and
four intervals $\tau$, and was originally developed for polarization transfer
between electron and nuclear spins in the context of nitrogen-vacancy diamond
magnetometry Schwartz  _et al._ (2018). It has also been shown to be effective
for singlet-to-magnetization conversion Tratzmiller (2021); Sabba  _et al._
(2022), and has been interpreted in terms of symmetry-based recoupling theory
Sabba  _et al._ (2022). For convenience, we refer to the "dual PulsePol"
sequence in figure 2 as “DualPol".

DualPol is an unusual example of a solution-state polarization transfer
sequence combining (i) multiple-pulse averaging Haeberlen and Waugh (1968);
Zuiderweg (1990) and (ii) hard pulses separated by delays. The sequence
provides robust polarization transfer even in the strong-coupling regime,
where the standard INEPT sequence breaks down Baishya and Khetrapal (2014);
Williamson  _et al._ (2000); Koskela, Kilpeläinen, and Heikkinen (2005); Gil,
Espinosa, and Parella (2011); Koskela, Kilpeläinen, and Heikkinen (2003). That
particular feature is not essential for the results described here. However,
it is advantageous in other circumstances, as will be discussed in a future
publication.

The repeating sequences of PulsePol and DualPol are composed of three-pulse
elements of the form $90_{y}180_{x}90_{x}$, with the pulses separated by
intervals $\tau$, and variants thereof. Each three-pulse sequence is therefore
a “windowed" version of a composite 180∘ pulse Levitt and Freeman (1979). We
therefore call this three-pulse sequence a “R-element", using notation
originally introduced in the context of broadband heteronuclear decoupling
Levitt and Freeman (1981), and later adapted for symmetry-based recoupling
sequences in solid-state NMR Carravetta  _et al._ (2000), and symmetry-based
singlet-triplet conversion sequences in solution NMR Sabba  _et al._ (2022).
In the case of DualPol, there is no special constraint or matching condition
on the duration $\tau_{R}$ of the R-element, except that it should be much
shorter than the period of the relevant J-coupling,
$\tau_{R}\ll|^{3}J_{\mathrm{RhH}}|^{-1}$. Under these conditions, the average
Hamiltonian Haeberlen and Waugh (1968) generated by the DualPol sequence, for
a heteronuclear 2-spin system, has the form

$\overline{H}^{(1)}\simeq\kappa_{\mathrm{DP}}\times 2\pi
J_{IS}\left(I_{x}S_{x}+I_{y}S_{y}\right)$ (4)

where the nuclides $\mathrm{{}^{1}H}$ and $\mathrm{{}^{103}Rh}$ are referred
to as $I$ and $S$, respectively. The numbering convention for the average
Hamiltonian terms starts with $1$ for the lowest-order approximation, in
common with the symmetry-based recoupling literature Carravetta  _et al._
(2000). The DualPol scaling factor is given, under suitable approximations, by
$\kappa_{\mathrm{DP}}\simeq\frac{1}{2}$ in the limit of strong radiofrequency
pulses. Equation 4 corresponds to an anisotropic Hartmann-Hahn Hamiltonian
Glaser and Quant (1996), indicating that the DualPol sequence exchanges
$z$-magnetization components between the $I$-spins and $S$-spins. The theory
and performance of the DualPol sequence will be discussed in more depth in a
future paper.

In the experiments described here, all DualPol sequences used an R-element
duration of $\tau_{R}=5\mathrm{\,ms}$ and a repetition number of $n=10$. The
total duration of each DualPol sequence was $T=2n\tau_{R}=100\mathrm{\,ms}$.

Figure 3:  Proton destruction filter for the removal of residual proton
magnetisation. The gradient strengths are given by G1=100$\%$ and G2=-61.8$\%$
with respect to the maximum gradient strength 50 G cm-1. Each gradient has a
duration of 2 ms. The black rectangle indicates a BB1 composite $\pi/2$ pulse
(equation 2).

####  II.2.3  1H Destruction Filter

The $\mathrm{{}^{1}H}$ destruction filter is shown in figure 3. The filter has
the net effect of dephasing residual proton transverse and longitudinal
magnetisation (which may be generated by accidental excitation, and recovery
during the decay interval respectively).

Figure 4:  Proton z-filter for the selection of proton z-magnetisation, using
bipolar gradients. The gradient strengths are given by G1=$40\%$ and
G2=-40$\%$ with respect to the maximum gradient strength of 50 G cm-1. Each
gradient pulse has a duration of 2 ms. The black rectangle indicates a BB1
composite $\pi$-pulse (equation 3).

####  II.2.4 1H z-filter

The z-filter for the selection of longitudinal $\mathrm{{}^{1}H}$
magnetisation is shown in figure 4. This employs a bipolar gradient scheme in
order to reduce spectral distortions by eddy currents or residual gradient
fields Wider, Dotsch, and Wuthrich (1994).

###  II.3 Solid-state NMR

Solid state CPMAS 13C NMR was performed using a 4 mm Bruker probe at 14.1 T
and $\sim$303 K.

###  II.4 Computational Chemistry

Quantum chemical geometry optimisation and shielding tensor calculations for
the rhodium formate complex axially ligated by solvent THF molecules were
performed using the ORCA program package version 5.0.3 Neese  _et al._ (2020).
$\mathrm{{}^{103}Rh}$ shielding tensors were computed at the TPSSh/SARC-ZORA-
TZVPP level of theory.

Figure 5:  $\mathrm{{}^{1}H}$ spectrum of a $\sim$10 mM solution of rhodium
formate in THF-d8, acquired at 9.4 T and at 298K in a single scan. Exponential
line broadening (0.75 Hz) was applied.

##  III Results

###  III.1 NMR Spectra

####  III.1.1 Solution-state 1H Spectrum

The rhodium formate $\mathrm{{}^{1}H}$ spectrum features a single formate
$\mathrm{{}^{1}H}$ resonance split into a 1:2:1 triplet by coupling to the
pair of magnetically equivalent $\mathrm{{}^{103}Rh}$ nuclei (figure 5). The
three-bond $\mathrm{{}^{1}H}$-$\mathrm{{}^{103}Rh}$ J-coupling is estimated to
be $|^{3}J_{\mathrm{RhH}}|=4.7\pm 0.1$ Hz.

Figure 6:  Pulse sequence for the acquisition of $\mathrm{{}^{1}H}$ enhanced
$\mathrm{{}^{103}Rh}$ spectra. A 16-step phase cycle is used where
$\phi_{1}=[{-x,x,-x,x}]$, $\phi_{2}=[{x,x,-x,-x}]$,
$\phi_{3}=[x,x,x,x,y,y,y,y,-x,-x,-x,-x,-y,-y,-y,-y]$ and the receiver
$\phi_{rec}$ = [x,-x,x,-x,y,-y,y,-y,-x,x,-x,x,-y,y,-y,y] all of which combine
to suppress ringing artefacts on the $\mathrm{{}^{103}Rh}$ channel.  Figure 7:
(a) $\mathrm{{}^{103}Rh}$ spectrum of a $\sim$10 mM solution of rhodium
formate in THF-d8 scaled 2.5 times, acquired using 128 scans at 9.4 T and at
298 K using the pulse sequence in figure 6. (b) $\mathrm{{}^{1}H}$-decoupled
$\mathrm{{}^{103}Rh}$ spectrum acquired using 128 scans at 9.4 T and at 298 K
using the pulse sequence in figure 6 with continuous-wave $\mathrm{{}^{1}H}$
decoupling during signal acquisition. Acquisition time for each spectrum was 1
hour. Exponential line broadening (1 Hz) was applied to each spectrum.

####  III.1.2 Solution-state 103Rh Spectra

The sequence shown in figure 6 was used for the acquisition of directly-
detected $\mathrm{{}^{103}Rh}$ spectra, enhanced by polarization transfer from
$\mathrm{{}^{1}H}$ nuclei. After an initial pair of 90∘ pulses, used for the
suppression of ringing artefacts (see below), the DualPol sequence transfers
z-magnetization from the $\mathrm{{}^{1}H}$ to the $\mathrm{{}^{103}Rh}$
nuclei, exploiting the form of the DualPol average Hamiltonian (equation 4).
The resultant $\mathrm{{}^{103}Rh}$ z-magnetization is converted into
observable transverse magnetization by a final 90∘ pulse. The
$\mathrm{{}^{103}Rh}$ NMR signal is enhanced by a factor of up to
$|\gamma_{I}/\gamma_{S}|\sim 31$, relative to that induced by a single 90∘
pulse applied to $\mathrm{{}^{103}Rh}$ nuclei in thermal equilibrium.

Ringing artifacts are strongly suppressed by a phase-cycled pair of 90∘ pulses
on the proton channel, before the polarization transfer takes place. The signs
of the $\mathrm{{}^{1}H}$ magnetization and the $\mathrm{{}^{103}Rh}$ receiver
are simultaneously inverted in successive scans. Since the phases of the
ringing are correlated with the phases of the pulses on the
$\mathrm{{}^{103}Rh}$ channel, the ringing is strongly suppressed in the
$\mathrm{{}^{103}Rh}$ spectrum. Further suppression of ringing is achieved by
additional phase cycling of the PulsePol blocks. The sign of the
$\mathrm{{}^{103}Rh}$ magnetization is invariant under global phase shifts of
the DualPol sequence, while the ringing contribution is phase-correlated and
largely cancels out. Similar logic has been used to design excitation schemes
for ringing suppression in homonuclear NMR experiments Zhang, Wu, and Mehring
(1990); Wang  _et al._ (2021).

The rhodium formate $\mathrm{{}^{103}Rh}$ spectrum features a single
$\mathrm{{}^{103}Rh}$ resonance split into a 1:4:6:4:1 pentet by couplings to
the four equivalent $\mathrm{{}^{1}H}$ nuclei on the formate ligands (figure
7(a). The three-bond $\mathrm{{}^{1}H}$-$\mathrm{{}^{103}Rh}$ J-coupling is
estimated to be $|^{3}J_{\mathrm{RhH}}|=4.7\pm 0.1\mathrm{\,Hz}$, in agreement
with the $\mathrm{{}^{1}H}$ spectrum. The $\mathrm{{}^{103}Rh}$ resonances
collapse into a single peak centred at 7516 ppm upon $\mathrm{{}^{1}H}$
decoupling (figure 7(b)).

The 103Rh resonances are broadened by the short 103Rh $T_{2}$ (see figure-13).

The $\mathrm{{}^{103}Rh}$ chemical shift is temperature-dependent (see Figure
8). The temperature-dependence of the $\mathrm{{}^{103}Rh}$ chemical shift is
approximately linear over the relevant temperature range, with a gradient of
$\sim 1.48\mathrm{\ ppm\,K^{-1}}$. This is in general agreement with
observations on similar Rh complexes Carlton (2008); Caló  _et al._ (2021).

Figure 8:  $\mathrm{{}^{103}Rh}$ chemical shift of rhodium formate dissolved
in THF-d8 at 9.4 T, as a function of temperature. The chemical shifts are
referenced to $\Xi\left({}^{103}\rm{Rh}\right)$ = 3.16$\%$.

####  III.1.3 Solid-state ${}^{13}\mathrm{C}$ NMR

The chemical shift anisotropy (CSA) of the formate ${}^{13}\mathrm{C}$ nuclei
was estimated by magic-angle-spinning NMR experiments on rhodium formate solid
(figure 9).

The estimated eigenvalues of the traceless, symmetric (rank-2) part of the
shielding tensor are as follows: $\sigma^{(2)}_{xx}=65.1\mathrm{\,ppm}$,
$\sigma^{(2)}_{yy}=5.5\mathrm{\,ppm}$, and
$\sigma^{(2)}_{zz}=-70.7\mathrm{\,ppm}$. This corresponds to the following
Frobenius norm of the rank-2 13C shielding tensor:

$\displaystyle||\bm{\sigma}^{(2)}||(^{13}\mathrm{C})$
$\displaystyle=\\{(\sigma^{(2)}_{xx})^{2}+(\sigma^{(2)}_{yy})^{2}+(\sigma^{(2)}_{zz})^{2}\\}^{1/2}$
$\displaystyle=96.3\pm 1.0\mathrm{\ ppm}$ (5)

Figure 9:  Rhodium formate 13C$\\{^{1}\mathrm{H}\\}$ solid-state CPMASMeier
(1992) NMR spectrum obtained at a spinning frequency of 4 kHz acquired using
2048 scans at 14.1 T and at 303K. The chemical shift was referenced to
admantane. The contact time was 160 $\mu$s. The recycle delay was 3 s.
$\sim$150 mg of sample was used. Further details of the pulse sequence are
provided in the Supporting Information.  Figure 10:  Sequence used for the
indirect measurement of rhodium T1 through $\mathrm{{}^{1}H}$ NMR signals.
Phase cycles are given by $\phi_{1}=[{x,x,-x,-x}]$, $\phi_{2}=[{-x,x,-x,x}]$,
$\phi_{3}=[x,x,x,x,y,y,y,y,-x,-x,-x,-x,-y,-y,-y,-y]$ and the receiver
$\phi_{rec}=[x,-x,-x,x,y,-y,-y,y,-x,x,x,-x,-y,y,y,-y]$. The optional shuttling
of the sample to low field, and back again, during the interval
$\tau_{\mathrm{relax}}$, is indicated.

###  III.2 Relaxation Times

####  III.2.1 $\mathrm{{}^{1}H}$-Detected $\mathrm{{}^{103}Rh}$ $T_{1}$

$\mathrm{{}^{103}Rh}$ $T_{1}$ relaxation time constants were measured
indirectly through $\mathrm{{}^{1}H}$ NMR signals using the sequence shown in
figure 10. DualPol is used to transfer z-magnetization from the
$\mathrm{{}^{1}H}$ nuclei to the $\mathrm{{}^{103}Rh}$ nuclei, and allowed to
relax towards equilibrium during the relaxation interval
$\tau_{\mathrm{relax}}$. For field-dependent relaxation measurements, the
sample is shuttled to a region of lower magnetic field during this interval,
and back again. A proton destruction filter is applied to eliminate any
residual proton magnetisation, such as that generated during
$\tau_{\mathrm{relax}}$ through longitudinal relaxation towards equilibrium.
Remaining $\mathrm{{}^{103}Rh}$ z-magnetisation, selected for by the two 90∘
pulses, is now transferred back to $\mathrm{{}^{1}H}$ z-magnetisation by a
second DualPol block and is selected for by a proton z-filter. A final
$\mathrm{{}^{1}H}$ 90∘ pulse generates observable $\mathrm{{}^{1}H}$
transverse magnetization. The sequence is repeated with variation of
$\tau_{\mathrm{relax}}$ in order to follow the equilibration of longitudinal
$\mathrm{{}^{103}Rh}$ magnetization.

The trajectory of indirectly-detected $\mathrm{{}^{103}Rh}$ z-magnetization in
a field of $9.4~{}\mathrm{T}$ is shown in figure 11(a). The trajectory fits
well to a single-exponential decay with time constant
$T_{1}(^{103}\mathrm{Rh})=0.483\pm 0.002\mathrm{\ s}$. A trajectory in the low
magnetic field of $\mathrm{1\ mT}$ is shown in figure 11(b). This was produced
by shuttling the sample to low magnetic field during the interval
$\tau_{\mathrm{relax}}$. The relaxation process is much slower in low field,
with a time constant of $T_{1}(^{103}\mathrm{Rh})=28.2\pm 1.2\mathrm{\ s}$.

The rhodium $T_{1}^{-1}$ increases approximately quadratically with the
magnetic field strength $B$, as shown in figure 11(c). The field-dependent
relaxation rate constant is a reasonable fit to the quadratic function
$T_{1}^{-1}(B)=T_{1}^{-1}(0)+aB^{2}$, where $T_{1}^{-1}(0)=0.065\pm
0.04\mathrm{\,s^{-1}}$ and $a=0.023\pm 0.001\mathrm{\,s^{-1}\,T^{-2}}$.

Figure 11:  (a) Decay curve for $\mathrm{{}^{103}Rh}$ longitudinal
magnetization at a field of 9.4 T, obtained using the pulse sequence in figure
10, but without shuttling the sample to low field. The data was acquired in
$\sim$20 minutes. The integrals are normalised against the $\mathrm{{}^{1}H}$
spectrum obtained by a single $\mathrm{{}^{1}H}$ 90∘ pulse applied to a system
in thermal equilibrium at 9.4 T. The data fits well to an exponential decay
with time constant $T_{1}=0.483\pm 0.002\mathrm{\,s}$. (b) Decay curve for
$\mathrm{{}^{103}Rh}$ longitudinal magnetization at a field of 1 mT, obtained
using the pulse sequence in figure 10, including the shuttling of the sample
to low field. The data fits well to an exponential decay with time constant
$T_{1}=28.2\pm 1.2\mathrm{\,s}$. (c) 103Rh relaxation rate constant
$T_{1}^{-1}$ as a function of magnetic field strength. The dashed line shows
the quadratic function $T_{1}^{-1}(B)=T_{1}^{-1}(0)+aB^{2}$, where
$T_{1}^{-1}(0)=0.065\pm 0.038\mathrm{\,s^{-1}}$ and $a=0.023\pm
0.001\mathrm{\,s^{-1}\,T^{-2}}$.  Figure 12:  Sequence used for the indirect
measurement of rhodium T2 with detection on protons. Phase cycles are given by
$\phi_{1}=[{x,x,-x,-x}]$, $\phi_{2}=[{-x,x,-x,x}]$,
$\phi_{3}=[x,x,x,x,y,y,y,y,-x,-x,-x,-x,-y,-y,-y,-y]$ and the receiver
$\phi_{rec}$=[x,-x,-x,x,y,-y,-y,y,-x,x,x,-x,-y,y,y,-y]. The black rectangle
indicates a BB1 composite $\pi$-pulse (equation 3).  Figure 13:  Decay curve
for 103Rh transverse magnetization at a field of 9.4 T, obtained using the
pulse sequence in figure 12. The data fits well to an exponential decay with
time constant $T_{2}=0.181\pm 0.001\mathrm{\,s}$. The integrals are normalised
against the $\mathrm{{}^{1}H}$ spectrum obtained by a single
$\mathrm{{}^{1}H}$ 90∘ pulse applied to a system in thermal equilibrium at 9.4
T.

####  III.2.2 $\mathrm{{}^{1}H}$-Detected $\mathrm{{}^{103}Rh}$ $T_{2}$

The sequence shown in figure 12 was used to measure the $\mathrm{{}^{103}Rh}$
spin-spin relaxation time constant $T_{2}$ in high magnetic field.

Conversion of $\mathrm{{}^{1}H}$ z-polarization to $\mathrm{{}^{103}Rh}$
z-polarization is achieved via DualPol. $\mathrm{{}^{103}Rh}$ transverse
magnetisation is generated by a 90∘ pulse and allowed to decay during the
subsequent spin echo of duration $\tau_{\mathrm{echo}}$. The ensuing 90∘
$\mathrm{{}^{103}Rh}$ pulse returns the remaining transverse
$\mathrm{{}^{103}Rh}$ magnetisation back to longitudinal $\mathrm{{}^{103}Rh}$
polarisation. A $\mathrm{{}^{1}H}$ destruction filter destroys any residual
$\mathrm{{}^{1}H}$ magnetisation before another DualPol cross-polarisation
block transfers $\mathrm{{}^{103}Rh}$ z-magnetisation back to
$\mathrm{{}^{1}H}$ z-magnetization. The $\mathrm{{}^{1}H}$ z-filter selects
for $\mathrm{{}^{1}H}$ z-magnetization before the $\mathrm{{}^{1}H}$ signal is
induced by the final 90∘ $\mathrm{{}^{1}H}$ pulse. The pulse sequence is
repeated varying the echo delay $\tau_{\mathrm{echo}}$ in order to follow the
decay of $\mathrm{{}^{103}Rh}$ transverse magnetization.

The trajectory of indirectly-detected $\mathrm{{}^{103}Rh}$ transverse
magnetization in a field of $9.4~{}\mathrm{T}$ is shown in figure 13. The
trajectory fits well to a single-exponential decay with time constant
$T_{2}($$\mathrm{{}^{103}Rh}$ $\\!)=0.181\pm 0.001\mathrm{\ s}$. Note that the
measured value of $T_{2}$ is much smaller than $T_{1}$ under the same
conditions.

####  III.2.3 ${}^{13}\mathrm{C}$ inversion-recovery

As discussed below, the rotational correlation time $\tau_{c}$ of the rhodium
formate complex may be estimated by a study of the 13C longitudinal
relaxation. This data was obtained by an indirect detection method exploiting
the scalar-coupled formate protons, as described in the Supporting
Information. The inversion-recovery data fits well to a single-exponential
recovery with a time constant of $2.64\pm 0.13\mathrm{\,s}$ for a solution in
THF-d8, in a magnetic field of $9.4$ T. However, as described below, the
inversion-recovery curve for the 13C magnetization is best analyzed using a
bi-exponential relaxation model.

##  IV Discussion

As shown in figure 11(c), the $\mathrm{{}^{103}Rh}$ relaxation rate constant
$T_{1}^{-1}$ has a quadratic dependence on magnetic field $B$, with an
additional zero-field contribution of $T_{1}^{-1}(0)=0.0653\pm
0.0383\mathrm{\,s}^{-1}$. The quadratic field dependence is consistent with a
dominant chemical shift anisotropy (CSA) relaxation mechanism, as is commonly
observed for the $\mathrm{{}^{103}Rh}$ NMR of rhodium complexes Cocivera  _et
al._ (1982); Carlton (2008).

It is difficult to estimate the $\mathrm{{}^{103}Rh}$ chemical shift
anisotropy by solid-state NMR. The small magnetogyric ratio of
$\mathrm{{}^{103}Rh}$ and the very large CSA value make solid-state
$\mathrm{{}^{103}Rh}$ NMR very difficult. Our attempts to use the PROSPR
method Jaroszewicz  _et al._ (2021) to observe the $\mathrm{{}^{103}Rh}$
spectrum indirectly in the solid state, by saturation transfer to the
$\mathrm{{}^{1}H}$ nuclei, were also unsuccessful. This is likely due to the
very small dipole-dipole couplings between 1H and 103Rh nuclei in this
complex, which greatly inhibits dipolar-mediated polarization transfer in the
solid state.

The symmetry of the complex indicates that the $\mathrm{{}^{103}Rh}$ CSA
tensors should have uniaxial symmetry ($\eta=0$) with their unique principal
axis along the Rh-Rh bond. This property is assumed in the following
discussion.

Although the $\mathrm{{}^{103}Rh}$ CSA may not be measured directly, it is
possible to estimate it by a combination of field-dependent
$\mathrm{{}^{103}Rh}$ and ${}^{13}\mathrm{C}$ $T_{1}$ measurements. The
compact cage structure of the rhodium formate complex (figure 1) suggests
that, to a good approximation, the complex tumbles in solution as a near-rigid
body, with a common rotational correlation time $\tau_{c}$ for all spin
interactions. This approximation allows a correlation time estimate from
${}^{13}\mathrm{C}$ NMR to be applied in the context of $\mathrm{{}^{103}Rh}$
NMR.

A ${}^{13}\mathrm{C}$ nucleus of rhodium formate experiences two strong
anisotropic interactions: the ${}^{13}\mathrm{C}$-$\mathrm{{}^{1}H}$ dipole-
dipole coupling with the directly-bonded hydrogen nucleus, and the
${}^{13}\mathrm{C}$ chemical shift anisotropy. For point nuclei (i.e. ignoring
the spatial spread of the nuclear wavefunctions), the
${}^{13}\mathrm{C}$-$\mathrm{{}^{1}H}$ dipole-dipole coupling constant is
given by $b_{CH}=-(\mu_{0}/4\pi)\hbar\gamma_{C}\gamma_{H}r_{CH}^{-3}$, where
$r_{CH}$ is the ${}^{13}\mathrm{C}$-$\mathrm{{}^{1}H}$ internuclear distance
Levitt (2007). Quantum chemical calculationsNeese  _et al._ (2020) (see SI)
predict an internuclear ${}^{13}\mathrm{C}$-$\mathrm{{}^{1}H}$ distance of
1.097 Å, corresponding to a dipole-dipole coupling constant of
$b_{CH}=-2\pi\times 22.8$ kHz. However, solid-state NMR studies have shown
that the true dipole-dipole coupling is weakened by the angular spread of the
1H wavefunctions, associated with the zero-point librational motion of the C-H
bonds Nakai, Ashida, and Terao (1989). In the calculations below, we therefore
assume a 13C-1H dipole-dipole coupling constant of $b_{CH}=-2\pi\times(20.4\pm
0.5)$ kHz.

For isolated 13C-1H spin systems in the extreme narrowing approximation (fast
tumbling), the theoretical recovery of 13C longitudinal magnetization
$M_{z}(t)$ after perturbation from equilibrium at time $t=0$ is expected to
follow the biexponential curve

$\displaystyle M_{z}(t)=M_{z}^{\mathrm{eq}}+$
$\displaystyle\left(M_{z}(0)-M_{z}^{\mathrm{eq}}\right)\times$
$\displaystyle\frac{1}{2}\\!\left(\exp\\{-(\tfrac{1}{2}b_{CH}^{2}+\tfrac{1}{5}\omega_{\mathrm{CSA}}^{2})\tau_{c}\\}\right.$
$\displaystyle+\left.\exp\\{-(\tfrac{3}{2}b_{CH}^{2}+\tfrac{1}{5}\omega_{\mathrm{CSA}}^{2})\tau_{c}\\}\right)$
(6)

where $M_{z}^{\mathrm{eq}}$ is the thermal equilibrium 13C magnetization, and
$\omega_{\mathrm{CSA}}$ is defined as follows:

$\omega_{\mathrm{CSA}}=-\gamma_{C}B^{0}||\bm{\sigma}^{(2)}||$ (7)

where $||\bm{\sigma}^{(2)}||$ is the norm of the 13C shielding tensor, as
defined in equation III.1.3. The biexponential form of equation IV is due to
1H-${13}$C cross-relaxation during the magnetization recovery Solomon (1955);
Alger, Freeman, and Grant (1972); Kowalewski and Mäler (2018).

In a magnetic field of 9.4 T, the 13C CSA, as estimated by 13C solid-state NMR
(section III.1.3), corresponds to an interaction strength of
$\omega_{\mathrm{CSA}}\simeq 2\pi\times(9.7\pm 0.1)\mathrm{\,kHz}$. By fitting
the experimental 13C inversion-recovery trajectory to an equation of the form
in eq. IV, we obtain the following estimate of the rotational correlation time
for the rhodium formate paddlewheel complex in THF-d8 solution at
$298\mathrm{\ K}$: $\tau_{c}\simeq 24.5\pm 1.5\mathrm{\,ps}$. The
$\mathrm{{}^{103}Rh}$ relaxation may now be analyzed using the estimate of
$\tau_{c}$ from the ${}^{13}\mathrm{C}$ data. As shown in figure 11, the
$\mathrm{{}^{103}Rh}$ $T_{1}^{-1}$ relaxation rate constant is well-described
by the function $T_{1}^{-1}(B)=T_{1}^{-1}(0)+aB^{2}$, with the field-
independent term $T_{1}^{-1}(0)=0.065\pm 0.038\mathrm{\,s^{-1}}$, and the
quadratic coefficient $a=0.023\pm 0.001\mathrm{\,s^{-1}\,T^{-2}}$.

The quadratic field-dependent term may be ascribed to the CSA mechanism. In
the extreme narrowing approximation (fast tumbling), the CSA contribution to
the $T_{1}^{-1}$ relaxation rate constant for $\mathrm{{}^{103}Rh}$ is given
by Kowalewski and Mäler (2018)

$\left(T_{1}(^{103}\mathrm{Rh})\right)^{-1}_{\rm{CSA}}=\frac{2}{15}B_{0}^{2}\gamma_{\mathrm{Rh}}^{2}\Delta\sigma^{2}\tau_{c}$
(8)

where the shielding anisotropy $\Delta\sigma$ is defined as follows Kowalewski
and Mäler (2018):

$\displaystyle\Delta\sigma$
$\displaystyle=\frac{3}{2}(\sigma_{ZZ}-\sigma_{\mathrm{iso}})=-\frac{3}{2}\delta^{\mathrm{aniso}}$
(9)

Equation 8 implies that the quadratic field-dependent coefficient $a$ for the
$\mathrm{{}^{103}Rh}$ $T_{1}^{-1}$ relaxation rate constant is given by

$a=\frac{2}{15}\gamma_{\mathrm{Rh}}^{2}\Delta\sigma^{2}\tau_{c}$ (10)

The experimental estimate of the quadratic coefficient $a=0.023\pm
0.001\mathrm{\,s^{-1}\,T^{-2}}$ may be combined with the correlation time
estimate $\tau_{c}\simeq 24.5\pm 1.5\mathrm{\,ps}$ to obtain the following
experimental estimate of the $\mathrm{{}^{103}Rh}$ shielding anisotropy:
$|\Delta\sigma|=9900\pm 540\mathrm{\ ppm}$.

This is a very large number. Although prior estimates of the
$\mathrm{{}^{103}Rh}$ CSA are scarce in the literature, CSA values for heavy
spin-1/2 nuclei are sometimes of a similar magnitude Dechter and Kowalewski
(1984); Paluch  _et al._ (2019); Sparks and Ellis (1986); Venkatesh  _et al._
(2020); Benn, Michael Büch, and Reinhardt (1985); Bagno and Bini (2010);
Perras  _et al._ (2017); Venkatesh  _et al._ (2018, 2022); Venkatesh, Perras,
and Rossini (2021), with closely related platinum (II) compounds displaying
195Pt CSA values on the order of 10,000 ppm Dechter and Kowalewski (1984);
Sparks and Ellis (1986); Venkatesh  _et al._ (2018); Venkatesh, Perras, and
Rossini (2021). To our knowledge, the only other measurements of
$\mathrm{{}^{103}Rh}$ CSAs, in very different Rh(III) compounds, were on the
order of $\sim$500-1500 ppm Adams  _et al._ (1987); Phillips  _et al._ (2006).
This dramatic range is also typicalBenn, Michael Büch, and Reinhardt (1985);
Venkatesh  _et al._ (2022, 2020) for heavy spin-1/2 nuclei.

Using ORCA Neese  _et al._ (2020); Stoychev  _et al._ (2018); Stoychev, Auer,
and Neese (2018), $\mathrm{{}^{103}Rh}$ shielding tensors were computed at the
TPSSh/SARC-ZORA-TZVPP level of theory using implicit solvation (CPCMBarone and
Cossi (1998); Garcia-Ratés and Neese (2020) for THF), the zeroth-order regular
approximation (ZORA)Bouten  _et al._ (2000); van Wüllen (1998) for the
inclusion of relativistic effects, GIAOs, the RI approximationStoychev, Auer,
and Neese (2018) , and the tau-dependent correction as suggested by
DobsonDobson (1993); Reimann  _et al._ (2015); Schattenberg and Kaupp (2021)
(see Supporting Information). The result is summarised in table 1.

The calculated CSA is somewhat smaller than the experimental estimate.
Underestimation of CSAs calculated using the ZORA method has been reported for
other heavy spin-1/2 nuclei Alkan and Dybowski (2018); Autschbach (2004);
Malkin  _et al._ (2013), where better agreement might be obtained with higher-
order four-component relativistic calculations Autschbach (2004) or by
accounting for the relativistic breakdown of the relationship between spin-
rotation and the paramagnetic contribution to the anisotropy Malkin  _et al._
(2013).

Method |  $|\Delta\sigma|$ /ppm
---|---
Calculated | 7070
Experimental estimate | 9900 $\pm$ 540
Table 1: Estimates of the $\mathrm{{}^{103}Rh}$ shielding tensor anisotropy
$\Delta\sigma$ of Rh formate, defined in equation 9. The computational
estimate is given by quantum chemical calculation using ORCA Neese  _et al._
(2020). The experimental estimate is from the analysis of field-dependent
$\mathrm{{}^{103}Rh}$ relaxation in solution, as described in this paper.

The origin of the zero-field contribution $T_{1}^{-1}(0)$ to the
$\mathrm{{}^{103}Rh}$ relaxation rate constant is currently unknown. As
discussed in the Supporting Information, the
$\mathrm{{}^{103}Rh}$-$\mathrm{{}^{103}Rh}$ and
$\mathrm{{}^{103}Rh}$-$\mathrm{{}^{1}H}$ dipole-dipole couplings are much too
weak to account for this term. In the literature, the low-field relaxation of
heavy spin-1/2 nuclei is often attributed to a spin-rotation mechanism.
However, to our knowledge, this conclusion has not been supported by any
theoretical or computational studies.

The experimental estimate of the $\mathrm{{}^{103}Rh}$ $T_{2}$ is much shorter
than the estimate of $T_{1}$ under the same conditions ($T_{2}=0.181\pm
0.001\mathrm{\ s}$ as against $T_{1}=0.483\pm 0.002\mathrm{\,s}$, in a field
of $9.4\mathrm{\,T}$. We tentatively attribute the short $T_{2}$ value to the
modulation of the isotropic chemical shift by ligand exchange at the axial
positions. Other decoherence mechanisms, such as diffusion in the presence of
inhomogeneous magnetic fields, are expected to be too weak to account for the
observed $T_{2}$ value in this case.

In conclusion, this paper has demonstrated methodology for the indirect
estimation of $\mathrm{{}^{103}Rh}$ $T_{1}$ and $T_{2}$ values by
magnetization transfer to and from $\mathrm{{}^{1}H}$ nuclei using the DualPol
pulse sequence. Field-dependent $\mathrm{{}^{103}Rh}$ $T_{1}$ measurements
indicate a very large chemical shift anisotropy for the $\mathrm{{}^{103}Rh}$
sites in the rhodium formate paddlewheel complex. The field-independent
contribution to the $\mathrm{{}^{103}Rh}$ relaxation rate constant is not
fully understood at the current time.

A limitation of the methodology described here is the prerequisite of a spin
system with direct scalar couplings between 103Rh nuclei and a proton, which
is not present in all rhodium complexes. This limitation may be addressed via
the use of a relay nucleus, such as 13C at natural abundance.Caló  _et al._
(2021); Benn and Rufińska (1988); McFarlene, McFarlane, and Rycroft (1976);
Weske  _et al._ (2015)

###### Acknowledgements.

We acknowledge funding from the European Research Council (grant
786707-FunMagResBeacons), and EPSRC-UK (grants EP/P009980/1, EP/P030491/1,
EP/V055593/1). M.L. acknowledges financial support by the Max-Planck-
Gesellschaft and the Max-Planck-Institut für Kohlenforschung. We thank
Alexander A. Auer for advice on quantum chemical calculations. We thank
Professor Brian E. Mann for advice and historical insights on rhodium NMR. We
thank Alexey Kiryutin for sharing his designs for the sample shuttle.

## Author Declarations

### Conflict of interest

The authors have no conflicts to disclose.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Quark: An Integer RISC-V Vector Processor for Sub-Byte Quantized DNN
Inference

MohammadHossein AskariHemmat1, Théo Dupuis1, Yoan Fournier1, Nizar El Zarif1,
Matheus Cavalcante2,
Matteo Perotti2, Frank Gürkaynak2, Luca Benini2, François Leduc-Primeau1, Yvon
Savaria1, Jean-Pierre David1  1Electrical Engineering Department, École
Polytechnique de Montréal, Québec, Canada  2Integrated Systems Laboratory, ETH
Zürich, Switzerland

###### Abstract

In this paper, we present Quark, an integer RISC-V vector processor
specifically tailored for sub-byte DNN inference. Quark is implemented in
GlobalFoundries’ 22FDX FD-SOI technology. It is designed on top of Ara, an
open-source 64-bit RISC-V vector processor. To accommodate sub-byte DNN
inference, Quark extends Ara by adding specialized vector instructions to
perform sub-byte quantized operations. We also remove the floating-point unit
from Quarks’ lanes and use the CVA6 RISC-V scalar core for the re-scaling
operations that are required in quantized neural network inference. This makes
each lane of Quark 2 times smaller and 1.9 times more power efficient compared
to the ones of Ara. In this paper we show that Quark can run quantized models
at sub-byte precision. Notably we show that for 1-bit and 2-bit quantized
models, Quark can accelerate computation of Conv2d over various ranges of
inputs and kernel sizes.

###### Index Terms:

RISC-V, Vector ISA, Quantization, Machine Learning, Efficiency

##  I Introduction

Vector processors are gaining traction for machine learning computations,
thanks to their efficiency when executing tensor operations, combined with the
flexibility of fully programmable instruction processing. Many Deep Neural
Network (DNN) tasks can perform well with far lower precision than is
typically used for operands of generic digital signal processing computations
[1, 2, 3, 4]. Quantized models operate on much smaller processing and power
budgets than their full precision counterparts, making them appealing options
for edge applications. Even if using sub-byte networks can help speeding up
computation in neural networks, to the best of our knowledge, there are
currently no other vector processors that can efficiently store, load, and
operate on sub-byte operands.

This paper makes the following contributions: first, we modified Ara [5], a
RISC-V V compliant vector processor to support operations on sub-byte
operands. To achieve this, we added custom instructions to accelerate low-
bitwidth operations. Indeed, the majority of quantized DNN inference is
performed only in the integer domain, hence, we removed the floating-point
unit from Ara. This resulted in each lane of Quark being 2$\times$ smaller and
1.9$\times$ more power efficient than the ones in Ara. On the other hand, in
quantized DNNs, to recover accuracy, there is a re-scaling step after each
convolution or linear layer that requires floating-point operations. In Quark,
higher precision operations can be performed in the CVA6 [6] scalar processor.

Second, we developed a set of custom instructions, specifically designed for
running sub-byte DNNs. Then, using these custom instructions, we developed a
small vector DNN runtime that provides common DNN kernels operating on sub-
byte data. We show that compared to Int8 and FP32 kernels, the quantized
kernels achieve higher performance. Finally, we have implemented Quark in
GlobalFoundries’ 22FDX Fully-Depleted Silicon-on-Insolator (FD-SOI) technology
and provide its detailed power and area usage report.

Figure 1:  This figure shows the content of 8 vector elements of v1 and v2
when vbitpack instructions are executed consecutively on the same input and
output vector registers. In this example, we are packing the content of v1
into v2 when 2 bit precision is used.  and  show different bit slices.

The rest of the paper is organized as follows. In Section II, we review the
related works. In Section III, we describe the architectural changes made to
Ara to make it more efficient for running quantized neural networks. In
Section IV, we provide performance comparisons between Quark and Ara, and an
analysis of Quark’s implementation results. Finally, Section V concludes this
work.

##  II Background and Related Works

Quantized neural networks are a well-studied research topic [7]. However,
supporting sub-byte precision has not been as popular. In the following, we
have divided the related works for these types of quantized neural networks
into three different groups depending on how they are implemented: on custom
accelerators, commodity CPUs and custom RISC-V Microprocessors.

###  II-A Custom Accelerators

Since the inception of quantized neural networks, many custom accelerators
based on ASIC and FPGA were proposed. Loom [8] supports sub-byte computation
through bit-serial computation. Loom accelerates the inherent computation cost
of bit-serial by using a large number of parallel bit-serial units that
perform a 1-bit by 1-bit multiplication. Loom expects the input data to these
parallel units to be in a bit-stream format. Because of that, Loom transposes
data into a bit-stream format at each output unit, with a significant amount
of overhead in area and power consumption. In [9] a DNN accelerator based on
Loom was proposed and implemented in 65 nm technology, achieving 805 GOPS when
operating on 8-bit precision operands. BitFusion [10] and BitBlade [11] use a
set of parallel 2-bit by 2-bit multipliers followed by a shift accumulation
unit to support sub-byte computation. To further optimize for area and power,
BitBlade [11] replaces the shift accumulation units with bitwise summation.
RaPiD [12] is an ultra low precision accelerator for AI training and
inference. RaPiD supports 16 and 8-bit floating-point and 4 and 2-bit fixed-
point and was implemented in 7 nm technology. It delivers a peak performance
of 3.5 TOPS/W in its 8-bit floating-point mode and 16.5 TOPS/W in its 4-bit
fixed-point mode.

There are examples of FPGA-based quantized neural network accelerators as
well. FINN [13] is a framework that was originally designed for the
acceleration of binary neural networks. It has now evolved to a data-flow
style framework with support for low precision operations for DNN
acceleration. FINN can process quantized neural networks in ONNX format. Then,
FINN’s front-end compiler optimizes the computational graph for the FINN’s
back-end compiler, which uses an HLS hardware library to generate hardware
supporting input quantized models. However, FINN requires the entire model to
be implemented all at once on the FPGA, which limits its flexibility. BARVINN
[14] is an FPGA based neural network accelerator that was designed to support
arbitrary precision operations. It consists of a barrel RISC-V processor [15]
used as a controller and an array of Matrix Vector Units (MVU) [16]. Each MVU
has a set of configuration registers that can be programmed by the RISC-V
processor through RISC-V’s control status registers. This allows the MVU to
perform a set of neural network specific operations in a pipelined fashion
with any precision. The MVUs are connected to a crossbar that allows them to
send the results of one MVU to another. Furthermore, BARVINN has a Python code
generator library that can consume a quantized DNN in ONNX format and
generates RISC-V code.

###  II-B Sub-byte Precision on Commodity CPUs

In addition to custom hardware accelerators, there have been efforts to
support low-precision inference on commodity CPUs. In [17], the authors added
support for bit-serial convolution in TVM to generate code for Arm devices.
They exploit the vector instructions of Arm devices to overcome the
computationally expensive task of formatting tensors in bit-stream format.
They show that on a Cortex-A72 CPU in a Raspberry Pi 4B device, the relative
speedup over 32-bit floating-point kernels for 1-bit and 2-bit precision is
6.2$\times$ and 1.7$\times$, respectively. With ULPPACK [18], the authors
propose a novel packing method that allows a commodity CPU to operate on sub-
byte precision. In this packing method, the quantized inputs are packed into
CPU’s registers with enough space to allow for overflow. In [18], the authors
show that on a Cortex-A72 in the Raspberry Pi 4B CPU, compared to [17],
ULPPACK has a better performance for 3-bit or higher precision inputs.

###  II-C Custom RISC-V MPUs

Dustin [19] is a 16-core RISC-V processor based on OpenHW Group’s CV32E40P
[20, 21] cores. Each core has a special dot product unit that can be
programmed through RISC-V’s control status registers (CSRs) to perform a dot
product operation with different precision (2, 4, 8 or 16-bit operations are
supported). Dustin was taped-out in 65 nm CMOS technology and achieves a peak
performance of 58 GOPS and a peak efficiency of 1.15 TOPS/W. Darkside [22] is
a heterogeneous RISC-V compute cluster specifically designed for DNN inference
at the edge. It has 8 RISC-V cores enhanced with 2-bit to 32-bit mixed-
precision integer arithmetic. To improve performance, Darkside is equipped
with a specialized datamover, a 16 bit floating point Tensor Product Engine
and an accelerator for computing depthwise convolution. Darkside was
implemented in 65 nm CMOS technology and achieves a peak integer performance
of 65 GOPS on 2-bit integer kernels.

Compared to the existing works, Quark is a vector processor that extends the
RISC-V ISA by adding new instructions to accelerate packing methods for bit-
serial computation. As we will discuss in Section III, compared to Ara, Quark
is a much slimmer processor since it does not require any vectorized floating-
point operations. Compared to ASIC or FPGA based accelerators and due to the
nature of general processors, Quark is very flexible and more likely to
support any type of neural network that might come up in the literature in the
future. Finally, none of the previous works used a vector RISC-V to provide
sub-byte support for quantized DNNs.

##  III Architecture

Quark’s architecture is based on Ara, a modular open-source vector processor
recently updated to the RISC-V V 1.0 vector ISA that is able to reach almost
its theoretical peak performance when running matrix multiplication between
“large enough” matrices [23]. The Ara system is a decoupled vector
architecture composed of a CVA6, the scalar RV64GC core, and Ara, its vector
engine. Each unit has independent access to the upper level of the memory
hierarchy via a shared AXI bus.

CVA6, being the only unit capable of accessing the instruction side of the
memory, fetches both the scalar and vector instructions from its private L1
instruction cache, executes the former ones, and dispatches the latter ones to
Ara. Since CVA6 is an in-order issue/commit processor that can still execute
instructions out-of-order by means of a scoreboard, the dispatch of vector
instructions to Ara happens non-speculatively as soon as the instruction
reaches the top of the scoreboard. Then, CVA6 waits for Ara’s acknowledgment
and answer, before committing the instruction. This is a fire-and-forget
dispatch since Ara, if there is no result to return, acknowledges CVA6
immediately after checking that no exceptions can occur. Ara’s architecture
matured through time following the RISC-V V specifications, from RVV0.5 to
RVV1.0.

Ara has been designed to accelerate a wide range of computations by means of
vectorization. However, it only supports standard data types. In quantized
neural networks, we try to avoid floating-point operations since they are
computationally costly. Therefore, our design focuses on maximizing the
throughput of integer computations, paying only a negligible degradation in
accuracy. This allowed us to remove Ara’s floating-point units and to save
precious area to make room for our custom bit-serial computation instructions.
In the following sub-sections, we briefly discuss these modifications.

###  III-A Bit-Serial Computation

Figure 2 illustrates a typical computation flow for quantized neural network
inference. One way to accelerate the inference of neural networks is to
accelerate convolutions or matrix multiplications by using low precision
operations. However, as it can be seen, the rest of the computation has to
happen in full precision. Since the computation complexity of convolution or
matrix multiplication kernels is much higher than the rest of the operations
in Figure 2, it makes sense to accelerate them first. As discussed in Section
II, there are many ways to support low precision and particularly sub-byte
operations. With Quark, this is achieved by exploiting its vector processing
capacity.

Figure 2: Typical computation graph in a forward path of a quantized neural
network. Here, we show the computation flow for 2 bit inference.

As mentioned earlier, Quark extended the RISC-V vector ISA with custom
instructions to speed up bit-serial computation and packing. To understand how
these bit-serial computations operate, let us consider a vector-vector
multiplication:

$w\cdot a=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}2^{n+m}\text{popcnt}(w_{m}\land
a_{n})$ (1)

Equation (1) describes a vector-vector multiplication in bit-serial format
between two input vectors $w$ and $a$, where $m$ and $n$ are the precision
used for $w$ and $a$, respectively. To perform this operation in a vector
processor, we need logical AND, popcount, and shift-and-accumulate operators.
Moreover, Equation (1) expects the data to be in bit-stream format, and since
this data transposition has to happen for each input of a convolution or
linear layer, this data transformation should be fast to avoid making it a
bottleneck to the whole computation.

Figure 3: Per layer relative speed up when running Resnet18 on CIFAR100 with
batch size of 1, on Quark with Int1 and Int2 over Ara with Int8. Figure 4:
Roofline plot for Quark with 8 lanes and Ara with 4 lanes on Conv2d with a
$3\times 3$ kernel. AraAn Ara LaneQuarkA Quark Lane Figure 5: Ara and Quark
system placed-and-routed designs.  is vector register file related ,  is
operand queue,  is the vector floating-point unit,  is the vector register
file and  is the vector ALU.

Although the base RISC-V vector ISA includes vectorized AND, it does not
support the other operators needed. To accelerate the bit-serial computation,
we added three custom instructions as follows: vpopcnt, vshacc and vbitpack.
Our popcount operator requires an instruction able to count the number of bits
at 1 for every element of the input vector; however, the base RISC-V vector
ISA can only count the global number of 1s over the entire vector length. Our
special vpopcnt instruction performs popcount on each vector element. We also
developed vshacc to fuse the shift and accumulation operations in a single
instruction. Finally, we developed a custom instruction to accelerate bit-
packing operations. vbitpack slices the elements of a vector register into
bits and then packs these bits into an output register. To accumulate the bit
slices of previous elements, each call to vbitpack shifts the target register
to the left and then performs the packing operations. Figure 1 illustrates
this operation in detail.

TABLE I: Quantization of Resnet18 Using LSQ [3] Quantization method. Dataset | Model | Precision (W/A) | Accuracy | Size (MB)
---|---|---|---|---
CIFAR 100 | Resnet18 | LSQ(1/1) | 57.32 | 1.45
LSQ(2/2) | 76.81 | 2.89
LSQ(8/8) | 78.45 | 10.87
FP32 | 76.82 | 42.80

##  IV Results and Performance Analysis

###  IV-A Performance Analysis

To demonstrate the benefits of adding our custom instructions, we benchmarked
Ara and Quark with different kernel sizes and shapes. To do this, we developed
customized bit-serial programs for conv2d, matrix multiplication, and other
common kernels needed to run typical neural networks on Quark.

We benchmarked the system performance by running Resnet18 (with CIFAR100
dataset) on both Ara and Quark. To simulate, we benchmarked both Ara and Quark
with Questasim 10.7c. For each experiment, we used CVA6’s cycle CSRs to know
exactly how many clock cycles each kernel takes. To preserve the accuracy of
the model, we used full precision data type for input and output layers.
Figure 3 illustrates the per-layer performance from running Resnet18 on Quark
and Ara under different precisions. We ran Resnet18 on Ara with Int8 and FP32
data types and on Quark with Int1, Int2 data types. To show the importance of
using a specialized packing operation, we ran the Int2 experiment once with
RISC-V vector instructions and then with the custom vbitpack instruction. As
Figure 3 shows, running Resnet18 with Int1 precision on Quark outperforms Int8
on Ara in each layer. However, as Table I shows, quantization of Resnet18 with
1-bit precision comes with a significant accuracy degradation. On the other
hand, with 2-bit quantization, the accuracy drop is negligible, but the
performance boost over Int8 is significant. For 2-bit quantization, we
measured the performance of each layer with and without vbitpack instruction.
Finally, although Int2 without the vbitpack instruction performs better than
Int8 on average, the improvement is not significant. On the other hand, Figure
3 shows that Int2 with vbitpack instruction outperforms Int8 by an average of
$5.67\times$.

###  IV-B Physical Implementation Results

We implemented Quark and Ara with GlobalFoundries’ 22FDX FD-SOI technology.
For both Quark and Ara, we used 4 lanes, a vector length of 4096 bits (i.e.,
16 KiB of Vector Register File), and an equally-configured CVA6. We used
Synopsys Design Compiler 21.06 for synthesis and Cadence Innovus 21.14 for
physical implementation. In Table II, we report Innovus’s static power
analysis for both Quark and Ara, in typical operating conditions. Figure 5
illustrates the placed and routed design for both Ara and Quark. Based on the
results in Figure 5 and Table II, each Quark lane is about $2.3\times$ smaller
and consumes $1.9\times$ less power than Ara. Nevertheless, both cores can
operate at the same 1 GHz clock frequency. Finally, Figure 4 illustrates the
roofline plot for Quark and Ara on conv2d with a $3\times 3$ kernel. In this
graph, considering the same power and area budget for Ara and Quark as shown
in Table II, Quark outperforms Ara in all the input tensor sizes.

TABLE II: Physical Implementation of Ara and Quark | Ara | Quark
---|---|---
Number of Lanes | 4 | 4 | 8
VRF Size [KiB] | 16 | 16 | 32
Lane Cell Area [mm2] | 0.120 | 0.051 | 0.046
Die Area [mm2] | 1.09 | 0.69 | 1.09
TT Frequency [GHz] | 1.05 | 1.05 | 1.00
Core power per lane [mW] | 229 | 119 | 97

##  V Conclusion

This paper introduced Quark, an integer RISC-V vector processor based on Ara.
It extends the RISC-V vector ISA by adding custom instructions for sub-byte
computations in DNNs. We modified Ara by removing the floating-point unit and
adding custom instructions to accelerate bit-serial computations. Our
simulation shows that on Resnet18 with 1-bit and 2-bit precision, Quark is
5.7x and 3.5x faster than Ara with 8-bit precision on average for each
kerenel. Finally, our implementation results of Quark in GlobalFoundries’ 22
nm technology show that Quark’s lanes are 2.3x smaller than the corresponding
ones in Ara.

##  VI Acknowledgements

The authors thank CMC Microsystems and Global Foundries for access to design
tools and technologies. This research was funded by CMC Microsystems, Mitacs
and the OpenHW Group.

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