# Clifford Circuit Initialisation for Variational Quantum Algorithms

M. H. Cheng<EMAIL_ADDRESS>QOLS, Blackett Laboratory, Imperial College
London, London, SW7 2AZ, UK
Fraunhofer Institute for Industrial Mathematics, Fraunhofer-Platz 1, 67663
Kaiserslautern, Germany
K. E. Khosla<EMAIL_ADDRESS>
These two authors contributed equally to this work QOLS, Blackett Laboratory,
Imperial College London, London, SW7 2AZ, UK
C. N. Self  Quantinuum, Partnership House, Carlisle Place, London, SW1P 1BX,
United Kingdom
QOLS, Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK
M. Lin  QOLS, Blackett Laboratory, Imperial College London, London, SW7 2AZ,
UK
B. X. Li  QOLS, Blackett Laboratory, Imperial College London, London, SW7 2AZ,
UK
A. C. Medina  Fraunhofer Institute for Industrial Mathematics, Fraunhofer-
Platz 1, 67663 Kaiserslautern, Germany
M. S. Kim  QOLS, Blackett Laboratory, Imperial College London, London, SW7
2AZ, UK

###### Abstract

We present an initialisation method for variational quantum algorithms
applicable to intermediate scale quantum computers. The method uses simulated
annealing of the efficiently simulable Clifford parameter points as a pre-
optimisation to find a low energy initial condition. We numerically
demonstrate the effectiveness of the technique, and how it depends on
Hamiltonian structure, number of qubits and circuit depth. While a range of
different problems are considered, we note that the method is particularly
useful for quantum chemistry problems. This presented method could help
achieve a quantum advantage in noisy or fault-tolerant intermediate scale
devices, even though we prove in general that the method is not arbitrarily
scalable.

Quantum computing, variational algorithms, quantum information

††preprint: APS/123-QED

##  I Introduction

Intermediate Scale quantum devices are rapidly improving and are potential
candidates for demonstrating quantum advantage. While algorithms such as the
phase estimation [1], factorisation [2], or Deutsch-Jozsa [3] are too resource
intensive to demonstrate quantum advantage on current devices, intermediate
scale devices have inspired an important class of hybrid classical-quantum
algorithms based on classical optimisation – Variational Quantum Algorithms
(VQA)[4, 5, 6]. Device noise is a significant barrier to achieving supremacy
with VQAs on noisy Intermediate Scale Quantum (noisy ISQ) devices [7];
however, due to their ability to efficiently estimate classically intractable
cost functions, VQAs remain a possible candidate for demonstrating quantum
advantage on small, or intermediate scale error corrected devices. This class
of algorithms divides an eigenvalue problem into a classical optimisation
task, using a quantum computer to estimate the cost function. The calculation
of the cost function requires only a relatively short quantum circuit,
replacing the requirement for deep circuits and ancillary qubits with a run-
time overhead. The Variational Quantum Eigensolver (VQE), is a specific
variational algorithm that aims to minimize the expectation value of a given
Hamiltonian. The VQE is sufficiently flexible to solve many optimization
problems, including MAXCUT [8, 9], portfolio optimisation [10], financial
transaction settlement [11], and finding the ground state of solid-state
Hamiltonians [12] or interacting fermions [13].

The VQE has had some initial success in demonstrating simulations of quantum
chemistry energy levels within the chemical accuracy [14], exotic phases of
matter such as time crystals [15], and high energy physics [16]. However,
scalability remains a concern as the numbers of qubits and parameters increase
[17], notwithstanding the issue with noise [7]. The main obstacles to
scalability are the onset of barren plateaus [18, 19] and the trade-off
between expressibility and trainability [20] of a parameterized quantum
circuit.

To tackle these problems, we propose a classical initialization strategy for
the VQE: the Clifford Circuit Pre-Optimisation, which classically searches for
a good initial condition before running the VQE on device. Clifford circuits
are quantum operations that map Pauli strings to Pauli strings, and when
applied to an initial stabilizer state (i.e. an eigenstate of a Pauli string),
the resulting measurement outcomes are known to be efficiently simulable [21].
This pre-optimisation can help avoid barren plateaus and remove the influence
of device noise during the classical optimisation. Given an ansatz circuit
consisting of parameterized $R_{z}$, $R_{x}$, and $R_{y}$ rotations as well as
CNOTs, there exists a finite set of parameters (Clifford points) such that the
final quantum state is a stabilizer state and Pauli measurement outcomes are
classically simulable. Since stabilizer states are uniformly distributed in
the Hilbert space, one can envisage a search algorithm over these discrete
points to seek a good initialization. In this work, we consider classical
annealing with threshold resetting as the search method, however, in writing
the current manuscript, we became aware of a complimentary work using Bayesian
optimisation with encouraging results [22].

A good initialization needs not just to minimize the cost function; ideally we
would like an initialization with both low cost, and high gradient. This
gradient can be efficiently evaluated using the parameter-shift rule [23],
which itself also only requires Clifford circuits for a suitable choice of
parameters.

In this paper, we initially study the advantages of the Clifford pre-
optimization by considering the initial condition with low cost before running
the continuous parameter variational optimisation. We numerically show how
Clifford pre-optimisation works with different Hamiltonian models, and that
starting at a low-cost initial condition drastically reduces the gradient
descent times. In section II we overview the Clifford pre-optimisation method,
comparing and contrasting it to standard continuous parameter VQE. We then
present numerical results in section III for quantum chemistry, Transverse
Field Ising Model (TFIM), and Binary optimisation problems using the Quantum
Approximate Optimisation Algorithm (QAOA). We also investigate the
relationship between ansatz types, circuit depth and numerical scalability.
Finally, in section IV, we use a counting argument to show the method is not
arbitrarily scalable, but is still relevant for pushing true quantum advantage
for ISQ circuit sizes of several tens of qubits.

##  II Clifford pre-optimisation

Figure 1: (a) Schematic representation of Clifford searching for the VQE, with
crosses depicting the stabilizer states in the Hilbert space. A circuit
typically parameterizes a subset (darker crosses) of all stabilizer states —
generally including the logical Z-basis states (blue crosses). A cost function
can be efficiently classically simulated at these points, and a pre-
optimisation method can be used to find a state close to the VQE target state
(green dot). (b) A hardware efficient ansatz can become a Clifford circuit by
restricting ($\theta_{i}$)-parameterised rotations to a finite set, shown here
as single direction Pauli rotations with angles being multiples of $\pi/2$.

The VQE algorithm requires a parameterised ansatz circuit $U(\theta)$ to
construct the trial wave function $\ket{\psi(\theta)}$, combined with a
classical optimisation that minimizes the expectation value
$\left\langle\psi(\theta)\right|H\left|\psi(\theta)\right\rangle$ for some
operator $H$ (which is decomposed into a weighted sum of Pauli strings).
Selecting the structure and initial parameters for this ansatz circuit is an
active area of VQE research [24, 25, 26, 19]. For typical ansatz circuits,
there is a finite set of parameter points that implement a Clifford circuit,
with the resulting states being the so-called stabilizer states, Fig. 1 (a).
In the following we make use of the Gottesman-Knill theorem [27] to
efficiently classically sample the cost function (over these Clifford circuit-
parameterized stabilizer states) before any quantum hardware is needed. A
hardware-efficient Clifford circuit can be constructed by restricting the
parameter angles in single-direction rotation gates, Fig. 1 (b).
Alternatively, one could consider a regular circuit of random single qubit
Clifford gates between entangling C-NOT (or other two-qubit Clifford) layers,
and later decide which possible angles to relax for a continuous variable
circuit.

For some problems the Clifford points of the relevant ansatz may not be the
physically interesting part of the variational parameter space, e.g. small
perturbations using the Universal Coupled Cluster ansatz [28]. For the present
work, we restrict our analysis to the problem agnostic ansätze, where the
physically relevant part of the variational parameter space is unknown. Before
presenting numerical results, we first consider in which case this Clifford
pre-optimisation search may be useful and identify potential limitations.

Implementations of continuous parameter VQE are known to have a trade-off
between expressibility and trainability [20]. The same problem is expected in
the stabilizer pre-optimisation: (i) the ansatz may not be sufficiently
expressive to include the best stabilizer state (i.e. the stabilizer state
that minimises the energy), or (ii) the circuit can express too many
stabilizer states and the pre-optimisation itself becomes difficult. If there
are too many parameters, the finite-size ansätze begin to approximate a
2-design [29], where barren plateaus and uniform cost function conditions
apply [18]. Thus, the right number of parameters and circuit design will be
the key to a successful Clifford pre-optimisation.

For quantum chemistry tasks, a Fermion-to-qubit transform, e.g. Jordan-Wigner
[30], Parity [31], or Bravyi-Kitaev [32], is required to generate a suitable
qubit Hamiltonian. This transform has a direct impact on “where” (in Hilbert
space) the ground state is located, in contrast to parametrically accessible
stabilizer states which are fixed in the Hilbert space. Hence different
Fermion-to-qubit transforms may be more or less suitable for pre-optimisation
given a particular hardware-efficient ansatz. Unfortunately, it is not
possible to _a priori_ know the best transform, but running the stabilizer
search over each transform is a linear overhead and the best transform for the
quantum device can therefore be selected in a scalable way.

Figure 2: Comparison of different ansatz types (rows), ansatz depths
(columns), and Fermion transforms (line colours) for a linear chain of eight
hydrogen atoms (see inset). The energy curve is obtained from ten thousand
iterations of Clifford pre-optimisation, with no quantum VQE. Grey dashed
lines show ground and first excited states obtained via exact diagonalization.
The real-valued ansatz of depth two, and Trotter ansatz of depth five perform
equally well at approximating the ground state energy for low bond lengths.
The Trotter ansatz at depth two is insufficiently expressive, while the real-
valued ansatz at depth five is over parameterized. The plots highlight how the
pre-optimisation is not a panacea to all problems, and the ansatz structure,
depth and Fermion-to-qubit transform each play an interesting role. Further
on-device VQE is necessary to find better energy solutions (e.g. up to
chemical accuracy). We note that these curves are found classically and can
therefore be used to select the best ansatz for on-device optimisation.

Finally, we note that a given Pauli string has far more $0$ expectation value
stabilizer states than $\pm 1$ eigenvalue stabilizers, and therefore the
expectation value of a Pauli string with a random stabilizer state is likely
zero. These zero expectation value states give no information to the optimiser
resulting in a barren plateau-like flattening of the optimisation landscape.
Higher numbers of non-commuting Pauli strings will help combat this, hence the
more Pauli strings, the more likely a given stabilizer state will give you
useful information. These arguments are expanded in detail in section IV, but
for now we simply postulate that Hamiltonians with more (non-Commuting) Pauli
strings may be better suited to Clifford pre-optimisation. Hence the Clifford
pre-optimisation cannot be expected to be useful for all problems at every
scale. Since quantum chemistry problems typically require many non-commuting
Pauli strings, we might expect that the stabilizer search is more useful for
chemistry problems compared to binary optimisation (e.g. QAOA), or quadratic
condensed matter interactions.

##  III Numerical results

Here we consider several example problems to test the pre-optimisation method.
Different Hamiltonians are chosen to span local (MAXCUT and Ising model) and
global (quantum chemistry examples) cost functions. Here the distinction
between local and global loosely refers to how many non-identity Pauli
operators are in each of the Pauli strings in the decomposition of the
Hamiltonian [33]. Furthermore, we investigate the effect of ansatz circuit
type, circuit depth, and Fermion-to-qubit transformations (for quantum
chemistry problems) on the pre-optimisation. In each case, simulated annealing
with a threshold resetting method is used to optimize each Clifford circuit.
Our annealing method begins with random Clifford parameters and changes two
Clifford gates each iteration, keeping the new gates if a lower energy state
is found, and probabilistically keeping the new gate if a higher energy state
is found (see appendix A for details).

First, as a typical benchmark, we consider ground state problems in quantum
chemistry. We use the qiskit_nature library to compute the molecular orbitals
of a second quantized Fermionic Hamiltonian with frozen core electrons (if
applicable). The same library is used to generate a qubit Hamiltonian via the
parity [31], Jordan-Wigner [30], or Bravyi-Kitaev [32] transforms. These
quantum chemistry Hamiltonians are expressed as a weighted sum of Pauli
strings, with many groups of non-commuting terms.

Following on from quantum chemistry problems, we then analyze Clifford pre-
optimization as it applies to the MAXCUT and Transverse Field Ising Model
(TFIM) Hamiltonians. Both of these Hamiltonians can be expressed as,

$\displaystyle H=\sum_{ij}w_{ij}Z_{i}Z_{j}+\sum_{i}(g_{x}X_{i}+g_{z}Z_{i}).$
(1)

For MAXCUT, $w_{ij}$ is the graph adjacency matrix and $g_{x}=g_{z}=0$, while
for the TFIM $w_{ij}=J\delta_{i,i+1}$ with coupling constant $J$ (and
Kronecker-delta $\delta_{ij}$), and $g_{x}$ ($g_{z}$) the transverse
(longitudinal) field. Other constrained and unconstrained quadratic binary
optimisation problems can be written in this Ising-like form [34, 11]. While
the TFIM Hamiltonian is a local cost function Hamiltonian, it is known to have
a phase transition where the ground state exhibits volume law entanglement
[35], allowing the potential for a non-trivial stabilizer state to be found
via the Clifford search.

As well as different types of Hamiltonians and transforms, we compare the
performance of different types of ansatz: (i) an ansatz that parameterizes
only real-valued stabilizer states, consisting of only controlled $Y$
rotations and CNOT gates, (ii) an ansatz inspired by the Trotterized unitary
for the TFIM Hamiltonian that allows complex-valued stabilizer states, and
(iii) for the MAXCUT task we consider the standard QAOA ansatz. The real-
valued and QAOA ansätze are more constrained than the trotterized ansatz and
parameterise fewer stabilizer states for a given depth, but may be more suited
to problems with known real-valued ground state eigenvectors, such as some
quantum chemistry problems and binary optimisation. It turns out however, that
this is not necessarily the case, as we shall later show.

Figure 3: Ground state energy surface of H2O. (left) Clifford pre-optimisation
with Parity mapping (mid) Clifford pre-optimisation with Jordan-Wigner
mapping, and (right) exact diagonalization. There is a noticeable bias towards
shorter distances/large bond angles where the pre-optimisation gets closer to
the ground state. The patches in the Jordan-Wigner plot suggest the
optimisation landscape under this transform is more difficult for the
simulated annealing. The numerical results show that the energy obtained from
Clifford Pre-optimisation can find a good initialization for the ground state
energy.

###  III.1 Quantum chemistry

Fig. 2 shows the effect of ansatz type, ansatz depth, and Fermion transform
for a chain of eight hydrogen atoms. The H8 chain is parameterized by the
dimerisation length (see inset in Fig. 2), and the plots show the best
expectation value found over $10^{4}$ iterations of simulated annealing of the
Clifford circuits (see appendix A). The real ansatz outperforms the
Trotterized ansatz at depth $D=2$, indicating the Trotterized ansatz struggles
to parameterize Clifford points close to the ground state despite having the
same entanglement structure. In contrast, the Trotterized ansatz performs
better at depth $D=5$ as it allows more flexible (complex valued) stabilizer
states. Interestingly, although the depth $D=5$ real-valued ansatz can
parameterise a super-set of the $D=2$ stabilizer states, it performs
significantly worse. We attribute this to ansatz over-parameterization as the
circuit expressibility and parameter dimension are both significantly
increased. Correspondingly the $D=2$ Trotterized ansatz is underparameterized,
as the more complex (and more difficult to optimize) $D=5$ circuit seems to
find a good solution. There is also a noticeable difference between the
Fermion transforms: each transform results gives a different set of weights
and Pauli strings, and therefore the ground state has different coefficients
in the logical basis. This places the ground state at different parts of the
Hilbert space, and therefore has different projections onto the set of
circuit-accessible stabilizer states. While here it looks like the Bravyi-
Kitaev slightly underperforms compared to the Parity and Jordan-Wigner maps,
we have no current understanding as to if this is expected in general.

Next, we consider the ten qubits H2O Hamiltonian with varying O-H bond length
and H-O-H bond angle, Fig. 3. Again we note the parity transform outperforms
the Jordan-Wigner transform (Fig. 3). In both transformations, there is a
noticeable bias in the minimum towards shorter bond lengths, and larger bond
angles. That is, Clifford pre-optimisation finds a more accurate energy
solution near shorter bond lengths and larger bond angles. Since quantum
chemistry Hamiltonians have far more Pauli strings (typically
$\mathcal{O}(poly(n))$) compared to solid-state ($\mathcal{O}(n)$) or binary
optimisation ($\mathcal{O}(n^{2})$), we expect them to benefit the most from
circuit-agnostic Clifford pre-optimisation, due to the higher probability of
finding non-zero expectation value stabilizer states.

###  III.2 TFIM

Figure 4: Eight-qubit transverse field Ising model [Eq. (1)] energy as a
function of dimensionless transverse ($g_{x}$) and longitudinal ($g_{z}$)
fields with $J=1$. Plots show the lowest energy seen over the first $10^{4}$
annealed stabilizer states (left), lowest energy after two hundred further
rounds of SPSA starting from the best initial point (middle), and solution
from exact diagonalization (right). The Clifford results (left) show that as
$g_{x}$ becomes larger than $g_{z}$, the ansatz seems to find the same energy
(independent of $g_{z}$). This suggests that the stabilizer state is
preferentially finding $X_{i}$ eigenstates over $Z_{i}$, or $Z_{i}Z_{i+1}$
eigenstates (a similar effect is seen for large $g_{z}$ and $g_{x}\leq 0.5$).

Here we benchmark the Clifford pre-optimisation with an eight-qubit TFIM
Hamiltonian, with varying $g_{x}/J$ and $g_{z}/J$ fields (see Eq. (1) with
$w_{ij}=J\delta_{i,i+1}$). Fig. 4 shows the Clifford search finds low energy
states for low fields, but begins struggling for higher fields. We also plot
the solution after two hundred rounds of continuous parameter SPSA descent to
resolve the effects of pre-optimisation and circuit expressibility. For
$g_{x}=g_{z}\approx 0.9$, the ground state exhibits long-range entanglement;
however, since the continuous parameter gradient decent (using Simultaneous
Perturbation Stochastic Approximation — SPSA) quickly converges within two
hundred iterations, this suggests the ansatz is sufficiently expressive for
the required entanglement, but a non-trivial stabilizer solution isn’t found.

For $g_{x}\geq g_{z}$, the Clifford ansatz finds the $X_{i}$ stabilizer states
(i.e. the eigenstate of each of the Pauli $X_{i}$ operators), independent of
the $g_{z}$ field. This can be understood by noting that $Z_{i}$ generates
$Z_{i}Z_{i+1}$ terms in the stabilizer group (i.e. all $Z_{i}Z_{i+1}$ are
products of the single $Z_{i}$ operators), thereby giving correlated
eigenvalues as they all commute. These correlations preclude a low energy
solution: the low energy (i.e. -1 eigenstates) states of $Z_{i}$ and
$Z_{i+1}$, result in a positive energy contribution for $Z_{i}Z_{i+1}$ (i.e.
is a +1 eigenstate), while no such restriction applies for $X_{i}$ and
$X_{i+1}$ stabilizers. Likewise, all low energy (i.e. odd parity) stabilizer
states of $Z_{i}Z_{i+1}$ will give identically zero energy for $X_{i}$,
$X_{i+1}$ terms, and zero energy for $Z_{i}+Z_{i+1}$ terms111A low energy
$\left\langle Z_{i}Z_{j}\right\rangle$ stabilizer state is either an odd
parity logical state (in $i,j$) in which case $\left\langle
Z_{i}\right\rangle=\pm 1=-\left\langle Z_{j}\right\rangle$, or a Bell-like
state (in $i,j$) in which case $\left\langle Z_{i}\right\rangle=\left\langle
Z_{j}\right\rangle=0$ . For large $g_{x}$, the Clifford pre-optimisation
simply finds the ground state of the $J=g_{z}=0$ Hamiltonian, and there
appears to be little benefit in running it. This emphasises that Clifford pre-
optimisation can be parameter-specific and not just problem-specific.

Following this logic, we could expect that when the Hamiltonian contains many
non-commuting equally weighted Pauli strings, the lowest energy Clifford state
may not give a good ground state estimate. Nonetheless, in some cases (e.g.
$g_{z}\approx g_{x}$) it can give a substantially better initial condition,
thereby allowing a continuous parameter VQE optimisation to convergence
faster.

###  III.3 MAXCUT Problem

Finally, we consider Clifford pre-optimisation for the MAXCUT problem. The
MAXCUT task is qualitatively different from quantum chemistry and TFIM tasks,
as all of the $\mathcal{O}(n^{2})$ Pauli Strings commute with each other.
Furthermore, it is an example of a fundamentally classical problem of
unconstrained quadratic binary optimisation.

When the ansatz circuit is not expressive enough to include the ideal
stabilizer state (i.e. the solution), one might expect the Clifford search to
be unsuccessful in finding a low expectation value initialisation.
Interestingly, this is also the case for the problem-specific (QAOA) ansatz
circuits. The QAOA ansatz can be readily tuned to a Clifford circuit (by
setting all parameters to be multiples of $\pi/2$). However, due to the
constrained nature of the circuit, the stabilizer search is not a useful
initialisation strategy. For binary optimisation problems the target state is
a logical basis state, and while there are guarantees that the optimal circuit
gets close to the target basis states [8], the circuit cannot exactly
parameterize a good stabilizer state. We note that since the target state is a
logical basis state, the optimal Clifford ansatz is constructed only from a
single layer of single-qubit X-gates — in which case the Clifford search
trivially becomes identical to an implementation of classical simulated
annealing.

##  IV Scaling

Figure 5: Minimum pre-optimised energy for the TFIM problem
($g_{z}=g_{x}=0.85J$), as the number of qubits increases, but keeping the
Clifford annealing time fixed to $10^{4}$ iterations. The dashed line shows
the (linearly extrapolated) ground state. The relationship between the depth
(different colours) and number of qubits suggest the method is not arbitrarily
salable for the TFIM, but for shallower circuits and several tens of qubits,
Clifford pre-optimisation can find a good initial condition.

Here we analytically argue that a Clifford based initialisation is not
arbitrarily scalable, but due to practical considerations, may still be useful
for VQE tasks involving several tens of qubits. In agreement with the
numerical results above, we show that the method is particularly attractive
for quantum chemistry problems, compared to binary optimisation or solid-state
problems.

The ansatz circuit evaluated at the Clifford parameter points results in a
stabilizer state. The utility of the pre-optimisation depends on how many
stabilizer states give a non-zero expectation value of the cost function
operator $H$. For an $n$ qubit cost function, this can be estimated given the
Pauli string decomposition, $H=\sum_{i}c_{i}P_{i}$, for Pauli strings $P_{i}$
and coefficients $c_{i}$. Stabilizer states ($\ket{s}$) either return zero
expectation value $\left\langle s\right|P\left|s\right\rangle=0$ for a Pauli
string $P$, or are eigenstates $P\ket{s}=\pm\ket{s}$, in which case $P$ is
said to stabilize $\ket{s}$.

Following Ref. [21], the number $N_{\mathrm{P}}$ of stabilizer states which
can be stabilized by a single $n$ qubit Pauli string is
$N_{\mathrm{P}}=S(n-1)$, with

$\displaystyle S(n)=2^{n}\prod^{n}_{k=1}(2^{k}+1)$ (2)

while the total number of stabilizer states is given by $N_{\mathrm{T}}=S(n)$.
$N_{\mathrm{P}}$ is exponentially smaller than $N_{\mathrm{T}}$. Hence, for a
Hamiltonian which is decomposed into $poly(n)$ Pauli strings, the probability
of getting a non-zero expectation value becomes vanishing small
asymptotically. Therefore in the most general case, to keep up the scaling of
the stabilizer state space, the number of Pauli Strings in the Hamiltonian
must also grow exponentially. Furthermore, anti-commuting Pauli strings do not
share any stabilizers, and two commuting strings share a set of
$N_{\mathrm{C}}=S(n-2)$ common stabilizers. Hence, finding a single stabilizer
state that is stabilized by two commuting Pauli strings also becomes vanishing
small asymptotically.

Combing the above counting arguments, the number of stabilizer states that
give a non-zero expectation value to the Hamiltonian ($N_{\mathrm{H}}$) has an
upper bound, $N_{\mathrm{H}}<N_{\mathrm{P}}M$, where $M$ is the number of
terms in the Pauli decomposition of $H$. If $M$ scales as $poly(n)$, then the
Hamiltonian cannot escape the fate of having an exponentially shrinking size
of stabilizer states, giving an effective barren plateau for Clifford points.
This exponentially growing zero-eigenvalue stabilizer state will become the
main obstacle to scaling up the Clifford pre-optimisation, and is equivalent
to the barren plateau in the continuous parameter case. Nevertheless,
stabilizer state pre-optimisation is useful for device bench-marking, and
possible (non-scalable) small-scale demonstration of quantum advantage, and
selecting useful ansatz circuits.

For less-than-linear depth circuits, the number of accessible stabilizer
states is far smaller than the (exponential in $n^{2}$) number of all
stabilizer states, $N_{T}$ (this is also the case for any depth circuits with
a restricted number of parameterized gates). This smaller set of circuit-
constrained set of states could allow a higher-than-random chance of a given
state being stabilized by Pauli strings in the Hamiltonian. This has the
potential for limited scalability of classical preoptimisatoin, perhaps into
the quantum advantage regime, see Fig. 5. Moreover, classically finding a good
initial state, or suitable ansatz circuit will allow increased throughput of
quantum devices.

Section III showed the quantum chemistry problems (with their many groups of
non-commuting Pauli strings), benefited more from the Clifford search, when
compared to the TFIM (with only two groups of non-commuting Pauli strings), or
MAXCUT (where all Pauli strings commute). Scaling arguments notwithstanding,
the more non-commuting Pauli strings in the Hamiltonian, the more useful a
stabilizer state search. We reiterate that quantum chemistry problems fit this
paradigm, suggesting that quantum chemistry problems, in contrast to binary
optimisation and solid-state models, are particularly well suited to
stabilizer pre-optimisation in the ISQ regime.

##  V Conclusion

In this paper, we presented Clifford pre-optimisation; a classical tool that
reduces the load on the quantum part for the VQE. This algorithm optimizes for
stabilizer states accessible from a given quantum circuit (with the aid of
Clifford gates) to minimize a cost function. Many stabilizer states are
accessible by a parameterized circuit, and these uniformly distributed,
problem independent states, make them ideal for comparing the cost functions
before a quantum device is used for full continuous parameter optimization.
The total number of stabilizers scales faster-than-exponentially with the
number of qubits (exponential with $n^{2}$), but exponentially with the number
of ansatz parameters, thereby highlighting the importance of the ansatz
circuit. Furthermore, the number of stabilizer states that are Pauli string
eigenvalues also scales exponentially in $n^{2}$ (although it is exponentially
small compared to the total numbers of stabilizer states), but a given
restricted depth ansatz circuit does not uniformly sample all states,
potentially allowing more useful states to be found.

We presented numerical studies using Clifford circuit annealing as the pre-
optimisation method for quantum chemistry, TFIM, and QAOA problems. We
unraveled the effects of circuit choice, Hamiltonian structure (global, local,
and the number of commuting/non-commuting Pauli terms), and Fermion transform
on the pre-optimisations performance. We identified that the Clifford search
finds a single stabilizer of a (commuting) set of Pauli strings in the cost
function, and this is not necessarily the maximally commuting set, Fig. 4.

Our results show that Clifford pre-optimisation provides a powerful new method
to lower the cost of required quantum processing time but pushing the
classical limit of the VQE. The method is useful for both device bench
marking, and likely (albeit not provably) useful for ISQ quantum advantage, in
particular for quantum chemistry problems, Fig. 2 and 3. However, we noted
that for many qubits and deep circuits, the counting arguments of Sec. IV,
show the eventual onset of a barren plateau, and limit the arbitrary
scalability of the method, even when individual Clifford circuits themselves
are efficiently evaluated.

##  VI Acknowledgements

We thank Tobias Hung for his helpful insights during this work, and David
Amaro for constructive comments on the manuscript. KEK and MSK, acknowledge
funding from the Samsung GRC grant, KIST grant and EPSRC Quantum computing and
simulation hub grant. MHC and ACM acknowledge funding from the Fraunhofer
Institute. Source code available on request.

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  * Note [1] A low energy $\left\langle Z_{i}Z_{j}\right\rangle$ stabilizer state is either an odd parity logical state (in $i,j$) in which case $\left\langle Z_{i}\right\rangle=\pm 1=-\left\langle Z_{j}\right\rangle$, or a Bell-like state (in $i,j$) in which case $\left\langle Z_{i}\right\rangle=\left\langle Z_{j}\right\rangle=0$.
  * Note [2] Here local minima is referring to the $\theta$ vector locality, not Hilbert-space locality as change in a gate angle can have a large change of the parameterised state in the Hilbert space.

##  Appendix A Clifford circuit annealing

Here we provide a detailed description of the Clifford pre-optimisation
algorithm. The ansatz circuits we consider are (i) Complex valued ansatz that
has the same structure as a Trotterized dynamics of an Ising model, and (ii)
Real valued ansatz consisting of $CX$ and parameterised $R_{y}$ only, see Fig.
6. Note that for a given depth $D$ and qubit count $n$ the Trotter ansatz as
$D(3n-1)$ parameters while the Real ansatz has $Dn$. Each parameter is an
$R_{x},~{}R_{z}$ or $CZ$ (controlled Z) rotation. The algorithm considers the
classically simulable sector of an ansatz quantum circuit, i.e. restricting
parameterised rotations in the parameter vector
$\theta=(\theta_{1},\theta_{2},...\theta_{N})$ to each be multiples of
$\pi/2$.

Figure 6: Ansatz circuits used for TFIM and quantum chemistry problems.
Trotterised inspired ansatz (with independent parameters for each control-Z
and single qubit rotations in each layer) allows complex valued wave
functions. The real-valued ansatz has only Y-rotations and control not gates
and ensures the final wave function is real valued.

To minimize the state over this discrete set we consider classical annealing
in the Clifford angles. This method randomly changes exactly two components
(chosen heuristically) of the parameter vector $\theta$ — keeping the changed
parameters in $\\{0,\pi/2,\pi,3\pi/2\\}$ — to give a new $\theta^{\prime}$. If
the new cost
$\left\langle\psi(\theta^{\prime})\right|H\left|\psi(\theta^{\prime})\right\rangle$
is lower than the previous cost
$\left\langle\psi(\theta)\right|H\left|\psi(\theta)\right\rangle$ , the
updated $\theta^{\prime}$ vector is accepted with probability one, otherwise
the new $\theta^{\prime}$ is accepted with probability $\exp(-\beta\Delta E)$
with $\Delta
E=\left\langle\psi(\theta^{\prime})\right|H\left|\psi(\theta^{\prime})\right\rangle-\left\langle\psi(\theta)\right|H\left|\psi(\theta)\right\rangle$.
Here $\beta$ is the effective inverse temperature and is set to be on the
order of the (estimated) characteristic energy gap between low energy states.
If this gap is unknown then beta may be chosen to be arbitrarily large
corresponding to zero temperature annealing.

To stop the annealing algorithm getting stuck in a local minima 222Here local
minima is referring to the $\theta$ vector locality, not Hilbert-space
locality as change in a gate angle can have a large change of the
parameterised state in the Hilbert space we randomly reset the full $\theta$
vector if the annealing hasn’t found a better $\theta^{\prime}$ point over the
last $k$ iterations. For all plots we heuristically choose $k$ in the range of
400-1000. We note other annealing methods may be applicable, e.g. parallel
tempering, or different (perhaps iteration-dependent) rules for selecting
which individual $\theta_{i}$ parameters to update. Or other discrete
optimisation algorithms, such as Bayesian optimisation [22].

The annealing trajectory is local in the parameter space (vector space of
$\theta$), but can be non-local in terms of Hilbert space jumps of the
resulting stabilizer state. There trajectory can therefore be dependent on the
initial choice of the Clifford point. That can lead to initial condition-
dependent local minima in the classical optimisation process. We note that our
resetting after $k$ sequential non-improving evaluations helps prevent this
issue. An example annealing iteration series in Fig. 7 shows the utility of
threshold-dependent resetting in finding a good initial solution.

Figure 7: Plots shows ansatz circuit selection trajectories for three
different quantum chemistry problems: (left) H2 (middle) H4 and (right) H6,
corresponding to four, six and eight qubits respectively (not including
symmetry reduction). Rows correspond to different depth ansatz circuits, (top)
depth one, (middle) depth three and (bottom) depth five, with the colors
corresponding to the Trotter ansatz (blue) and real ansatz (orange). Line
styles correspond to different dimer distances with a fixed inter dymer
distance. Each curve is efficiently classically calculated and the set can be
used to select an appropriate circuit for the on-device problem. Note the
depth one real ansatz seems to find as good a solution as the depth five
Trotter ansatz, but due to the few parameter, tends to find a good solution
faster, and more consistently.

##  Appendix B Clifford solution vs true ground states

Other than observing the exact ground state energy, we can compare the
stabilizer state and the ground state solution through looking at their
respective contribution from individual Pauli strings in the Hamiltonian. The
set of expectation values also tells us about the amount of magic in the exact
ground state solution, since a state with high magic will have expectation
values that deviate from $1,0,-1$. If we observe a low magic ground state, it
means that there is a stabilizer state that is close to the exact solution,
which is good for the gradient descent. In the case of a high magic ground
states (see Fig. 8), we can still find a stabilizer state with relatively low
expectation value compared to random parameterisation. However, since the
ground state solution is far from a stabilzier state, the stabilizer state
found is not guaranteed to be close to ground state, and still may be
difficult to optimise..

Figure 8: Expectation values of each individual Pauli string in a nine qubit
TFIM Hamiltonian $H=\sum_{i}(Z_{i}Z_{i+1})+2Z_{i}+2X_{i}$, comparing the exact
expectation values (orange circles) and stabilizer solution (blue crosses).
The stabilizer solution (constrained to be $\pm 1$ or zero) gets the general
pattern of low/high correct, but with an emphasis on the $X_{i}$ stabilizer
states as discussed in the main text.


compat=1.1.0

# Meromorphic CFTs have central charges c = 8$\mathbb{N}$: a proof by the MLDE
approach

Arpit Das<EMAIL_ADDRESS>School of Mathematics, University of Edinburgh,
EH9 3FD, U.K.
Higgs Centre for Theoretical Physics, University of Edinburgh, Edinburgh, EH8
9YL, U.K.

###### Abstract

In this short note, we present a simple and elementary proof that meromorphic
conformal field theories (CFTs) have central charges of the form: $c=8N$ with
$N\in\mathbb{N}$ (the set of natural numbers) using the modular linear
differential equations (MLDEs) approach. We first set up the 1-character MLDE
for arbitrary value of the Wronskian index: $\ell$. From this we get the
general form of the meromorphic CFT’s character. We then study its modular
transformations and the asymptotic value of it’s Fourier coefficients to
conclude that odd values of $\ell$ make the character in-admissible implying
that the central charge for admissible character has to be a multiple of 8.

###### Contents

  1. I Introduction
  2. II Basics of RCFTs
    1. II.1 Modular transformation of RCFT characters
    2. II.2 MLDEs and RCFTs
    3. II.3 Solutions $\to$ admissible solutions
  3. III $(1,\ell)$ MLDE
    1. III.1 Elementary proof
  4. IV Conclusion

##  I Introduction

A rational conformal field theory (RCFT) is a 2D CFT, which has a finite set
of holomorphic characters $\chi_{i}(\tau)$ and a torus-partition function of
the form Anderson:1987ge ; Moore:1988qv ; Mathur:1988na :

$\mathcal{Z}(\tau,\bar{\tau})=\sum_{i,j=0}^{n-1}M_{ij}\,\bar{\chi}_{i}(\bar{\tau})\chi_{j}(\tau)$
(1)

where $\tau\in\mathbb{H}=\\{z\in\mathbb{C}\,|\,\textrm{Im}\,(z)>0\\}$ is the
modular parameter on the torus, $n$ is the total number of linearly
independent characters of some – typically extended – chiral algebra
$\hat{\mathfrak{g}}$ and $M_{ij}$ denotes the multiplicity matrix where the
multiplicity captures the number of primaries sharing the same character. Let
us denote the number of primaries by $p$. Usually, we have $p\geq n$ 111This
often happens when the chiral algebra includes a Kac-Moody algebra and there
is a symmetry in the Dynkin diagram of the corresponding finite-dimensional
Lie algebra (see Das:2023qns ; DiFrancesco:1997nk  for more details)..

A meromorphic CFT is a RCFT which has a single primary field and hence a
single-character: the identity character. This means all other fields in the
theory are descendants over the identity. In the mathematics literature,
meromorphic CFTs are also known as holomorphic vertex operator algebras (VOAs)
(see, for example, definition B.50 in Rayhaun:2023pgc ). It is well known that
the central charge of a meromorphic CFT is a multiple of 8 (see, for instance,
Lin:2021bcp ; Dolan:1994st ; Lin:2019hks ) however a simple proof of this
result is difficult to find in the RCFT literature. The main purpose of this
note is to present a simple proof of this ubiquitous fact from the perspective
of modular linear differential equations.

This note is organised as follows. In section II we review some basic aspects
of RCFTs: modular transformations of its characters and their relations to
vector-valued modular forms and MLDEs. Readers familiar with the MLDE approach
to RCFTs can skip this section and directly move on to section III. In section
III we present the main result. First we give the single-character MLDE and
from it we get the form of the single-character solution in terms of the
Klein-j, $j(q)$, modular invariant function. We then study its modular
transformation properties and use Rademacher sums to analyse the asymptotic
behaviour of its Fourier coefficients. This analysis enables us to conclude
the even-ness of the Wronskian index, $\ell$, for single-character RCFTs
implying that their central charges come in multiples of 8. We conclude this
note by a short concluding section IV.

##  II Basics of RCFTs

In the section we provide a brief overview of the basic aspects of RCFTs. Let
us start by reviewing modular transformations of an RCFT’s characters.

###  II.1 Modular transformation of RCFT characters

Since its torus-partition function is of the form given in (1) and it is
modular invariant:

$\displaystyle{{\mathcal{Z}}}(\gamma\tau,\gamma\bar{\tau})={{\mathcal{Z}}}(\tau,\bar{\tau}),\
\ \forall\,\gamma\in\text{SL}(2,\mathbb{Z})$ (2)

we get that the holomorphic characters transform into linear combinations
among themselves as (see Zhu:1996  for a rigorous proof and Harvey:2018rdc ;
Rayhaun:2023pgc  for more details),

$\displaystyle\chi_{i}(\tau)=\sum\limits_{j=0}^{n-1}\rho_{ij}(\gamma)\chi_{j}(\tau),$
(3)

where $\rho$ is a finite dimensional unitary representation of the modular
group SL$(2,\mathbb{Z})$. Characters obeying (3) are called vector-valued
modular forms (VVMFs) of weight 0. Comparing (1) and (3) we see that for the
partition function to be modular invariant we should have,

$\displaystyle\rho^{\dagger}M\rho=M,$ (4)

We know that SL$(2,\mathbb{Z})$ is generated by the $S=\begin{pmatrix}0&-1\\\
1&~{}0\\\ \end{pmatrix}$ and $T=\begin{pmatrix}1&1\\\ 0&1\\\ \end{pmatrix}$
matrices and hence one way to determine $\rho$ is to look at its value on
these generators, $S:\tau\to-\frac{1}{\tau}$ and $T:\tau\to\tau+1$, subjected
to the following modularity constraints (see Harvey:2018rdc ; murty:2016 ),

$\displaystyle(\rho(S))^{2}=1,\qquad(\rho(S)\rho(T))^{3}=C,$ (5)

where $C$ is the charge-conjugation matrix satisfying $C^{2}=\mathds{1}$.

###  II.2 MLDEs and RCFTs

It is known that the characters of an RCFT satisfy a $n^{th}$-order modular
linear differential equation (MLDE) of the form Mathur:1988na ; Eguchi:1988wh
,

$\displaystyle\left({{\mathcal{D}}}^{n}+\sum\limits_{s=1}^{n-1}\mu_{2s}\phi_{2s}(\tau){{\mathcal{D}}}^{n-s}\right)\chi(\tau)=0,$
(6)

where the Serre-Ramanujan derivative, in $\tau$ (or, in
$q\equiv\text{e}(\tau)\equiv e^{2\pi\text{i}\tau}$), denoted
${{\mathcal{D}}}$, is defined as follows. Let

$\displaystyle{{\mathcal{D}}}_{w}\equiv\frac{1}{2\pi\text{i}}\frac{\partial}{\partial\tau}-\frac{w}{12}E_{2}(\tau)=q\frac{\partial}{\partial
q}-\frac{w}{12}E_{2}(q),$ (7)

be the derivative acting on a modular form of weight $w$. Since
${{\mathcal{D}}}_{w}$ raises the weight of the form from $w$ to $w+2$, we
define:

$\displaystyle{{\mathcal{D}}}^{n}\equiv{{\mathcal{D}}}_{w+2n-2}\circ{{\mathcal{D}}}_{w+2n-4}\circ\ldots\circ{{\mathcal{D}}}_{w+2}\circ{{\mathcal{D}}}_{w}.$
(8)

In (6), the above definition applies with $w=0$ since characters are weight 0
VVMFs. $\mu_{2s}\in\mathbb{C}$ are arbitrary parameters and $\phi_{2s}$ are
meromorphic modular functions of weight $2s$ 222here by meromorphic modular
functions we mean modular forms of SL$(2,\mathbb{Z})$ but which are now
allowed to have poles in the upper-half plane and hence the term meromorphic.
An example would be $\phi_{2}=\frac{E_{6}}{E_{4}}$ where $E_{4}$ and $E_{6}$
are the normalised weight 4 and weight 6 Eisenstein series respectively, and
$\phi_{2}$ is a meromorphic modular form of weight 2.. whose poles are
governed by the zeroes of the Wronskian, and whose overall normalisations are
specified so that their leading term is unity. Explicitly we have
Mathur:1988na ; Mathur:1988gt :

$\mu_{2s}\,\phi_{2s}=(-1)^{s}\frac{W_{n-s}}{W_{n}}$ (9)

where:

$W_{s}(\tau)\equiv\left|\begin{matrix}\chi_{0}(\tau)&\chi_{1}(\tau)&\cdots&\chi_{n-1}(\tau)\\\
\vdots&\vdots&\vdots&\vdots\\\
{{\mathcal{D}}}_{\tau}^{s-1}\chi_{0}(\tau)&{{\mathcal{D}}}_{\tau}^{s-1}\chi_{1}(\tau)&\cdots&{{\mathcal{D}}}_{\tau}^{s-1}\chi_{n-1}(\tau)\\\
{{\mathcal{D}}}_{\tau}^{s+1}\chi_{0}(\tau)&{{\mathcal{D}}}_{\tau}^{s+1}\chi_{1}(\tau)&\cdots&{{\mathcal{D}}}_{\tau}^{s+1}\chi_{n-1}(\tau)\\\
\vdots&\vdots&\vdots&\vdots\\\
{{\mathcal{D}}}_{\tau}^{n}\chi_{0}(\tau)&{{\mathcal{D}}}_{\tau}^{n}\chi_{1}(\tau)&\cdots&{{\mathcal{D}}}_{\tau}^{n}\chi_{n-1}(\tau)\end{matrix}\right|,$
(10)

where $W\equiv W_{n}$ is the usual Wronskian. It is easy to see from the above
definition (10) that $W_{n-1}={{\mathcal{D}}}W_{n}$. From (9) we find the
following useful relation:

$\mu_{2}\phi_{2}=-\frac{W_{n-1}}{W_{n}}=-{{\mathcal{D}}}\log W_{n}$ (11)

Note that the modular $T$-transformation on the characters imply the following
Fourier expansion, in the variable $q$, also called the $q$-expansion,

$\displaystyle\chi_{i}(q)=q^{\alpha_{i}}\sum\limits_{j\in\mathbb{N}\cup\\{0\\}}a_{i,j}\,q^{j},$
(12)

where the exponents $\alpha_{i}\equiv-\frac{c}{24}+h_{i}$ with $c$ being the
central charge of the RCFT and $h_{i}$s being the conformal dimensions of the
characters. Also, $h_{0}=0$ denoting the conformal dimension of the identity
character. Now since the solutions of the MLDE – in the above Frobenius form –
should furnish irreducible representations $\rho$ of the modular group, we
have by definition (see Harvey:2018rdc ; Das:2023qns  for details),

$\displaystyle\rho(T)=\exp{\Big{[}2\pi\text{i}\,\,{\rm
diag}\left(-\frac{c}{24},-\frac{c}{24}+h_{1},\cdots,-\frac{c}{24}+h_{n-1}\right)\Big{]}},$
(13)

The order of the above MLDE corresponds to the number of linearly independent
characters of the RCFT, denoted by $n$. The pole structure of the MLDE is
captured by the poles of the coefficient functions $\phi_{2s}$ or the zeros of
the Wronskian (see (9)). The zeros of the Wronskian is captured by an integer
called the Wronskian index, denoted by $\ell$. It is defined to be an integer
such that $\frac{\ell}{6}$ is the total number of zeroes of $W_{n}$ (see
Das:2023qns  for more details).

Note that, $W_{n}$ is a modular form of weight $n(n-1)$ and at $q\to 0$ its
leading behaviour is, $W\sim q^{\sum\limits_{i=0}^{n-1}\alpha_{i}}$, since
$\chi_{i}(q)\sim q^{\alpha_{i}}$, applying the valence formula of
SL$(2,\mathbb{Z})$ to $W_{n}$, we get the following all important relation
Mathur:1988na ; Harvey:2018rdc ; Das:2020wsi ,

$\displaystyle\sum\limits_{i=0}^{n-1}\alpha_{i}=\frac{n(n-1)}{12}-\frac{\ell}{6},$
(14)

which is also called the Riemann-Roch relation in RCFTs.

The classification programme of RCFTs based on the above MLDE approach is
dubbed the Mathur-Mukhi-Sen (MMS) classification and it revolves around two
parameters of the modular differential equation – its order and its pole
structure, namely: $(n,\ell)$. This programme has been very successful in
classifying RCFTs for small values of $(n,\ell)$ and has been pursued by both
mathematicians and physicists in more recent times Naculich:1988xv ;
Kiritsis:1988kq ; Bantay:2005vk ; Mason:2007 ; Mason:2008 ; Bantay:2007zz ;
Tuite:2008pt ; Bantay:2010uy ; Marks:2011 ; Gannon:2013jua ; Kawasetsu:2014 ;
Hampapura:2015cea ; Franc:2016 ; Gaberdiel:2016zke ; Hampapura:2016mmz ;
Arike:2016ana ; Tener:2016lcn ; Mason:2018 ; Harvey:2018rdc ; Chandra:2018pjq
; Chandra:2018ezv ; Bae:2018qfh ; Bae:2018qym ; Mason:c8c16 ;
franc2020classification ; Bae:2020xzl ; Mukhi:2020gnj ; Kaidi:2020ecu ;
Das:2020wsi ; Kaidi:2021ent ; Das:2021uvd ; Bae:2021mej ; Duan:2022ltz ;
Das:2022slz ; Das:2022uoe ; Das:2023qns ; Duan:2022ltz ; Mukhi:2022bte ;
Rayhaun:2023pgc

###  II.3 Solutions $\to$ admissible solutions

Once we have set up the MLDE as in (6), we seek Frobenius type of solutions to
it as given in (12). After obtaining these solutions, one has to impose the
following constraints,

  1. 1.

Uniqueness of the vacuum – Corresponding to the identity character’s Fourier
expansion, we should have $a_{0,0}=1$ which measures the degeneracy of the
vacuum

  2. 2.

Integrality and positivity of the degeneracies of states – The Fourier
coefficients $a_{0,j}$s, which correspond to the identity character, should be
non-negative integers as they capture the state-degeneracies in a Verma
module. For the non-identity characters, the Fourier coefficients are allowed
to be positive rationals, since in this case the first term in the
$q$-expansion need not begin with unity, as long as these rationals do not
grow out of bound and there exists a common denominator – large but finite –
which when multiplied as an overall factor makes the coefficients $a_{i\neq
0,j}$s integral. In connection to RCFT, this common denominator corresponds to
the dimension of the highest weight integral representation (HWIR) – with
conformal dimension $h_{i}$ – of the fully extended chiral algebra of the
theory (see DiFrancesco:1997nk ; Hampapura:2015cea ; Hampapura:2016mmz ;
Das:2020wsi  for more details).

  3. 3.

Multiplicity of the trivial primary – The trivial primary correpsonds to the
identity character and hence in the torus-partition function in (1) we should
have $M_{00}=1$.

  4. 4.

Positivity and integrality of multiplicities – Since $M_{ij}$ correspond to
the number of primaries that share the same character, we should have
$M_{ij}$s as non-negative integers.

After the solutions to the MLDE satisfy all of the above constraints then they
are termed as admissible.

Note that having admissible solutions doesn’t mean having a genuine RCFT. For
instance, note that, the following solution to a $(1,6)$ MLDE at central
charge 24,

$\displaystyle
j(q)+{{\mathcal{N}}}=q^{-1}(1+(744+{{\mathcal{N}}})q+196884q^{2}+{{\mathcal{O}}}(q^{3}))$
(15)

is admissible for ${{\mathcal{N}}}\geq-744$ but only 71 of them correspond to
genuine RCFTs Schellekens:1992db ; vanEkeren:2017scl ; van2020dimension ;
vanEkeren:2020rnz ; Hohn:2020xfe ; Betsumiya:2022avv ; Moller:2019tlx ;
Moller:2021clp . In the above, $j(q)$ is known as the Klein-j function which
is a modular invariant function.

However, we are not interested in classifying RCFTs in this note. So, we won’t
discuss the various ways one needs to follow to go from constructing genuine
RCFTs from admissible solutions of MLDE. Interested readers can refer to the
following articles for more details Gaberdiel:2016zke ; Harvey:2018rdc ;
Das:2022slz ; Das:2022uoe ; Mukhi:2022bte ; Rayhaun:2023pgc ; Duan:2022ltz .
We will only need the admissibilty criterion described above to get to our
main result.

##  III $(1,\ell)$ MLDE

In this section we shall apply the aforementioned constructions to the case of
$(1,\ell)$ MLDE. For single-character theory note that, the multiplicity
matrix is just, $M=1$ and hence (4) becomes $\rho^{\dagger}\rho=1$ which
implies that the character is allowed to be modular invariant upto an
arbitrary phase $\rho=e^{\text{i}\theta}$ with $\theta\in[0,2\pi)$.

Next let us note that for 1-character theory, we have from (14), that its
central charge is related to the Wronskian index as,

$\displaystyle c=4\ell,$ (16)

which immediately implies that the central charge of meromorphic theory is a
non-negative integer since $\ell\in\mathbb{N}$.

Notice, it is clear from above that, if we can show that meromorphic theories
have only even Wronskian indices then it readily follows that their central
charges have to be multiples of 8. This is what we intend to show below. To be
precise, we will show that odd Wronskian indices lead to in-admissible single-
character solutions. Hence, these cannot be realised as characters of a
meromorphic RCFT.

Let us start by considering the generic $(1,\ell)$ MLDE,

$\displaystyle\left({{\mathcal{D}}}+\mu_{2}\phi_{2}\right)\chi=0,$ (17)

where $\phi_{2}$ is given as (see Eq.(2.13) of Das:2023qns ),

$\begin{split}\ell=6r:\quad\mu_{2}\,\phi_{2}&=E_{4}^{2}E_{6}\sum_{I=1}^{r}\frac{1}{E_{4}^{3}-p_{I}\Delta}\\\
\ell=6r+1:\quad\mu_{2}\,\phi_{2}&=\frac{2E_{6}}{3E_{4}}+\frac{E_{4}^{2}}{2E_{6}}+E_{4}^{2}E_{6}\sum_{I=1}^{r-1}\frac{1}{E_{4}^{3}-p_{I}\Delta}\\\
\ell=6r+2:\quad\mu_{2}\,\phi_{2}&=\frac{E_{6}}{3E_{4}}+E_{4}^{2}E_{6}\sum_{I=1}^{r}\frac{1}{E_{4}^{3}-p_{I}\Delta}\\\
\ell=6r+3:\quad\mu_{2}\,\phi_{2}&=\frac{E_{4}^{2}}{2E_{6}}+E_{4}^{2}E_{6}\sum_{I=1}^{r}\frac{1}{E_{4}^{3}-p_{I}\Delta}\\\
\ell=6r+4:\quad\mu_{2}\,\phi_{2}&=\frac{2E_{6}}{3E_{4}}+E_{4}^{2}E_{6}\sum_{I=1}^{r}\frac{1}{E_{4}^{3}-p_{I}\Delta}\\\
\ell=6r+5:\quad\mu_{2}\,\phi_{2}&=\frac{E_{6}}{3E_{4}}+\frac{E_{4}^{2}}{2E_{6}}+E_{4}^{2}E_{6}\sum_{I=1}^{r}\frac{1}{E_{4}^{3}-p_{I}\Delta}\end{split},$
(18)

Note that, $(n,1)$ MLDE is not allowed by construction 333One quick way to see
this is to note that the lowest order zero that the Wronskian can have is of
order $\frac{1}{3}$ and not below.. Thus, we disregard single-character
solutions with $\ell=1$ which implies there are no $c=4$ meromorphic theories.

Now let us consider the MLDE for $\ell=6r+1$ with $r=1$ and obtain its
solution. It is easier to work in the $j$-space and so we will transform the
$\tau$-space MLDE to the $j$-space MLDE using the following identities,

$\begin{split}&j=\frac{E_{4}^{3}}{\Delta},\qquad(j-1728)=\frac{E_{6}^{2}}{\Delta},\\\
&{{\mathcal{D}}}\chi=-j\frac{E_{6}}{E_{4}}\partial_{j}\chi,\end{split}$ (19)

where $\Delta$ is the modular discriminant and is a cusp form of weight 12 for
the modular group. Sunc transformations of MLDEs between the $\tau$-space and
the $j$-space have been recently studied in Das:2023qns . We get,

$\displaystyle\partial_{j}\chi-\left(\frac{2}{3j}+\frac{1}{2(j-1728)}+\sum\limits_{I=1}^{r-1}\frac{1}{j-p_{I}}\right)\chi=0,$
(20)

The solution to the above equation can be readily given as,

$\displaystyle\chi(j)=j^{\frac{2}{3}}(j-1728)^{\frac{1}{2}}\prod\limits_{I=1}^{r-1}(j-p_{I}),$
(21)

where the integration constant is fixed by demanding that the leading term in
the $q$-expansion of $\chi$ is unity. Similarly, one can immediately write
down the solutions for the other two remaining odd cases. For $\ell=6r+3$ we
get,

$\displaystyle\chi(j)=(j-1728)^{\frac{1}{2}}\prod\limits_{I=1}^{r}(j-p_{I}),$
(22)

and for $\ell=6r+5$ we get,

$\displaystyle\chi(j)=j^{\frac{1}{3}}(j-1728)^{\frac{1}{2}}\prod\limits_{I=1}^{r}(j-p_{I}).$
(23)

Now let us consider all the individual multiplying factors listed in the above
solutions. Note, $j(q)$ is admissible – as it happens to be the character of
the meromorphic theory $E_{8,1}^{3}$ – and hence $j(q)^{\frac{1}{3}}$ and
$j(q)^{\frac{2}{3}}$ are also admissible – which are the characters of the
meromorphic theories $E_{8,1}$ and $E_{8,1}^{2}$ or,
$\mathcal{E}_{1}[D_{16,1}]$ respectively444where the notation
$\mathcal{E}_{1}[\hat{\mathfrak{g}}]$ denotes the 1-character extension of the
$n$-character meromorphic theory with chiral algebra $\hat{\mathfrak{g}}$ (see
Mukhi:2022bte ; Das:2022uoe ; Rayhaun:2023pgc  for more details). (see
Rademacher:1939fou ; Schellekens:1992db ; Das:2022uoe ; Mukhi:2022bte ;
Rayhaun:2023pgc  for more details). However, the factor
$(j-1728)^{\frac{1}{2}}$ is in-admissible and has all negative Fourier
coefficients except the first term which is unity Chandra:2018pjq . This rules
out any admissible solutions at $\ell=3$ or $c=12$.

In spite of this, it might still happen that considering enough $(j-p_{I})$
factors might turn the overall in-admissible solution into an admissible
solution. So, at higher central charges, odd $\ell$ values still remains a
possibility.

###  III.1 Elementary proof

To remedy this, we need to analyse the asymptotic behaviour of the above
solutions. For this let us employ the Rademacher sums analysis as outlined in
appendix A of Chandra:2018pjq  (see also Cheng:2012qc  for more details).
Using Eq.(A.4) of Chandra:2018pjq  for single-character solution we get,

$\displaystyle
a_{0}(j)=\rho(S)_{00}^{-1}\,a_{0}(0)\exp\left\\{4\pi\sqrt{\frac{\ell}{6}\left(j-\frac{\ell}{6}\right)}\right\\}+\text{(sub-
leading)},$ $\displaystyle
a_{0}(j)=\rho(S)_{00}^{-1}\,\exp\left\\{4\pi\sqrt{\frac{\ell}{6}\left(j-\frac{\ell}{6}\right)}\right\\}+\text{(sub-
leading)},$ (24)

where $j$ denotes the order in the $q$-expansion in (12). The above holds in
the limit of $j\to\infty$. In above, going to the second-line we have used
$a_{0}(0)=1$. So, $\text{sgn}(\rho(S)_{00}^{-1})=\text{sgn}(a_{0}(j))$ for
$j\to\infty$. Let us determine $\text{sgn}(\rho(S)_{00}^{-1})$ next.

Since $c=4\ell$, we have
$\rho(T)=\text{e}(-c/24)=e^{-\text{i}\pi\frac{\ell}{3}}$. From the first
modularity constraint in (5) we have, $\rho(S)^{2}=C=e^{\text{i}m\pi}$. This
implies,

$\rho(S)=e^{m\text{i}\pi/2}.$ (25)

The second modularity constraint in (5) we have
$(\rho(S)\rho(T))^{3}=C=e^{\text{i}m\pi}$. This implies,

$\displaystyle\exp\left[\text{i}\pi\left(\frac{m}{2}-\ell\right)\right]=1=\text{e}(k),$
(26)

where $m,k$ are integers. Thus, we get from above:

$\begin{split}&\frac{m}{2}-\ell=2k,\\\
\text{implying,}\,\,&\ell=\frac{m}{2}-2k,\\\
\text{implying,}\,\,&m>4k\qquad\text{ and, }m\in 2\mathbb{N}\end{split},$ (27)

since $\ell\in\mathbb{N}$.

Now consider odd $\ell$ which implies $m=4v+2$ (from (27)), with $v$ being an
integer. Then, we get: $\rho(S)=e^{\text{i}\pi}=-1$ which implies
$\text{sgn}(\rho(S_{00})^{-1})=-1=\text{sgn}(a_{0}(j))$ implying that odd
$\ell$ results in in-admissible characters solutions to the MLDE. Note that
this in-admissibility essentially arises from the fact that
$(j-1728)^{\frac{1}{2}}$ has all Fourier coefficients negative except the
first term and this spoils the admissibility as we go to higher and higher
orders in the $q$-expansion or, as $j\to\infty$ in (12).

For the sake of completeness, let us consider even $\ell$, which implies
$m=4v$ (from (27)), with $v\in\mathbb{Z}$. Then, we get:
$\rho(S)=\text{e}(v)=1$. Thus, $a_{0}(j)>0$ for $j\to\infty$ in (12).

Thus, we see from above that single-character solutions to $(1,\ell)$ MLDE
with odd $\ell$ lead to in-admissible solutions implying that for admissible
solutions $\ell$ must be even. So, the space of genuine RCFTs must have even
$\ell$ implying that their charges will be $c=4\ell=8N$ with $N\in\mathbb{N}$.
With this we can write down the ansatz for the most general single-character
admissible solution as,

$\displaystyle\chi(q)=j^{\frac{N}{3}-s}(j^{s}+a_{1}\,j^{s-1}+a_{2}\,j^{s-2}+\ldots\ldots+a_{s}),$
(28)

where $s:=\left\lfloor\frac{N}{3}\right\rfloor$. This ansatz is also given in
Eq.(2.2) of Das:2022slz .

##  IV Conclusion

In this short note, we presented an elementary proof of the fact that central
charges of meromorphic CFTs are multiples of 8. The approach we took in
proving this was studying the MLDEs that single-character theories’ satisfy
and by observing their solution space. We then analysed the asymptotic
behaviour of the Fourier coefficients of the single-character solutions using
Rademacher sums which led to the conclusion that odd $\ell$ solutions are in-
admissible. Thus, admissible solutions and hence genuine meromorphic CFTs will
have even $\ell$ implying that their central charges will be multiples of 8.

Such analysis of the values of the Wronskian index was done using MLDEs for
the two-character case in Das:2023qns  where it led to the conclusion that the
Wronskian index should be even for all two-character admissible solutions too.
There is an older result on the even-ness of $\ell$ for the two-character case
in Naculich:1988xv  where this fact was proven using monodromy arguments of
the modular $S$ and $T$ matrices. In an ongoing work, we are trying to extend
this analysis to three or more character-solutions and MLDEs. We are hopeful
that by studying the relevant MLDEs and their solution spaces – across all the
orbifold points in the fundamental domain of SL$(2,\mathbb{Z})$ – we should be
able to re-derive established results like: $\ell=3\mathbb{N}$ for the three-
character case, $\ell=2\mathbb{N}$ for the four-character case,
$\ell=\mathbb{N}\setminus\\{1\\}$ for the five-character case, etc., as given
in Kaidi:2021ent ; and also extend upon these results.

## Acknowledgements

We would like to thank Chethan N. Gowdigere, Sunil Mukhi, Jagannath Santara,
Jishu Das and Naveen Balaji Umasankar for useful discussions on modular forms
and MLDEs. We are also immensely grateful to Sigma Samhita and Shaymin for
their constant support and love.

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# Mapping the South African health landscape in response to COVID-19

Nompumelelo Mtsweni<EMAIL_ADDRESS>Independent
South Africa ,  Herkulaas MvE Combrink  0000-0001-7741-3418 University of the
Free State
Dept. Computer Science, University of Pretoria
South Africa<EMAIL_ADDRESS>and  Vukosi Marivate  0000-0002-6731-6267
Dept. Computer Science, University of Pretoria
Council for Scientific and Industrial Research
South Africa<EMAIL_ADDRESS>
###### Abstract.

When the COVID-19 disease pandemic infiltrated the world, there was an
immediate need for accurate information. As with any outbreak, the outbreak
follows a clear trajectory, and subsequently, the supporting information for
that outbreak needs to address the needs associated with that stage of the
outbreak. At first, there was a need to inform the public of the information
related to the initial situation related to the “who” of the COVID-19 disease.
However, as time continued, the “where”, “when” and “how to” related questions
started to emerge in relation to the public healthcare system themselves.
Questions surrounding the health facilities including COVID-19 hospital bed
capacity, locations of designated COVID-19 facilities, and general information
related to these facilities were not easily accessible to the general public.
Furthermore, the available information was found to be outdated, fragmented
across several platforms, and still had gaps in the data related to these
facilities. To rectify this problem, a group of volunteers working on the
covid19za project stepped in to assist. Each member leading a part of the
project chose to focus on one of four problems related to the challenges
associated with the Hospital information including: data quality, data
completeness, data source validation and data visualisation capacity. As the
project developed, so did the sophistication of the data, visualisation and
core function of the project. The future prospects of this project relate to a
Progressive Web Application that will avail this information for the public as
well as healthcare workers through comprehensive mapping and data quality.

COVID19,Public Health, Health Facilities, Mapping

††copyright: none††conference: KDD 2020 Workshop on Humanitarian Mapping; ;
††booktitle: KDD 2020 Workshop on Humanitarian Mapping, San Diego, USA††price:
none Figure 1. Proof of concept Health facility map of South Africa.
https://dsfsi.github.io/healthfacilitymap/

##  1\. Introduction

In the early phases of the COVID-19 disease in South Africa, with the number
of infectious and infected people on the incline. During the early phases of
the COVID-19 disease in South Africa, the Data Science for Social Impact
(DSFSI) research group at the University of Pretoria and collaborators,
experienced challenges related to information on a facility level related to
the COVID-19 disease readiness (Marivate and Combrink, 2020).

As we move to the next phase of South Africa’s experience of COVID-19 and its
responses, we continue to look back at the progress of the Coronavirus
COVID-19 (2019-nCoV) Data Repository for South Africa project (Marivate et
al., 2020)111https://github.com/dsfsi/covid19za, from hereon referred to as
the _covid19za project_. The focus for this case study is on understanding the
health system capacity in South Africa. A challenge that was identified early
on in the project was the lack of updated health system data (data about
hospitals, clinics and health resources). This challenge expanded when there
was a realisation of the magnitude of work that would need to be done to
collect and harmonise public health systems data for South Africa. This work
builds on our initial work on collecting and collating COVID-19 data for South
Africa (Marivate and Combrink, 2020). In this case study, we discuss these
challenges and present what has been achieved so far.

##  2\. The beginning

Initially, our goal was to collect information on the public healthcare system
related to the South African COVID-19 response. We had anticipated that with
the growth of cases, better health system information would be needed for more
localised decision making. We collected the health system data for four
primary reasons. We wanted to be able to identify:

  * •

Where the COVID-19 hospitals were going to be in the country;

  * •

What the current situation in the country is related to the COVID-19 response;

  * •

When the shortages were going to arise within the public healthcare system
related to the COVID-19 response;

  * •

and how to assist the public in understanding which healthcare facilities were
used for what specific response amidst the COVID-19 pandemic.

At the beginning of this task, these quetions where in their infance and
fairly descriptive in nature. Initially, a few members in the _covid19za
project_ volunteer group started asking questions about health system
information (through posting issues on GitHub). These questions led to the
intial directions of what would be tackled in building data towards mapping
health systems information, capability and capacity in South Africa. We next
describe our data collection and collation.

##  3\. Data

###  3.1. Sourcing the data

The initial data contributed to the repository for health systems was a
pivoted dataset, in a CSV format, containing the geo-coordinates of the
locations of South African hospitals in the public sector and some of the
hospitals in the private sector. This dataset is a subset of a much larger
dataset from the South African National Department of Health (NDoH)’s Data
Dictionary222https://dd.dhmis.org/. The Data Dictionary provides a reference
point for selected health information standards to support healthcare
activities within South Africa. Unfortunately, there are differences in some
of the variables in the data dictionary, lacking some of the vital information
required for the COVID-19 response that have not been updated since 2018.

Although the data from the NDoH Data Dictionary listed a large number of
facilities; it did not represent the current realities of some hospitals.
There were some parts of the data that were outdated such as the hospital
names(some hospital names have changed). The dataset also had a large number
of missing data points including physical addresses and contact details. The
facilities listed did not include attributes such as capacity (number of beds,
number of Intensive Care Unit (ICU) beds), services available, and
infrastructure (ambulances, ventilators, etc). Despite the fact that the
dataset contains great details about the administrative division within which
a facility resorts, it lacks information about health districts and clusters.
More information about the health district approach in South Africa can be
found at the Health Systems Trust (Nicholson,
2001)333https://www.hst.org.za/publications/. The Gauteng Province Cluster
Policy defines the concept of health clusters (Health, 2019) as _’A collection
of health facilities at different levels of healthcare provision (primary,
secondary, tertiary and quatenary) which are grouped together in a designated
demographic area.’_

The Gauteng Cluster Policy did not only have details about health facilities
in each cluster in Gauteng but also provided some details about the number of
hospital beds in each cluster. It should be noted that some hospitals were
duplicated (meaning one hospital could be in two different clusters) making
the accounting of hospital beds per cluster questionable. This could lead to
challenges when tallying capacity later.

Given that one of the _covid19za project_ contribtors, Anelda van der Walt, is
involved in the Afrimapr444https://afrimapr.github.io/afrimapr.website/ the
health systems task received further input and resources to use to collect
further data. A prior study (Dell, 2016) had looked at the geographical mal-
distribution of surgical resources in South Africa. We reached out to the
author and obtained the underlying data. The data is now available through a
permissive creative commons license(Dell, 2020) to allow and encourage re-use
upon our request. The dataset has data about the number of beds (classified
into which units of hospitals they belong to) in each hospital per province.
This dataset has been the most useful in filling the gaps from the NDoH data
dictionary. We next had to work on data cleaning and munging.

###  3.2. Data cleaning

To maintain the integrity of the data, we had to work on maintaining the
provenance of all of the data that is being collated in the dataset. Since the
data is being collected from a wide variety of sources, and the data is
available under a variety of licenses (or without licenses), we had to
document data sources and ensure that we attribute people/organisations who
have collected the data and are willing to share it. This route is chosen not
just for attribution’s sake, but to encourage others to also open up their
data to assist in such situations.

Combining various health facility lists to create a single, complete view of a
country’s health system, is not a trivial task. For example, lists cannot be
merged simply by using facility names. As mentioned before, facility names
change over time and lists are often not updated. Furthermore, facilities may
be listed under their full names, short names, and/or common names. Typos,
capitalisation, misspelling, and varying naming conventions all contribute to
the challenge of merging health facility lists by facility name. To understand
some of the real-world challenges experienced in the analysis and merging of
health facility lists and identify steps that can be streamlined for others
wanting to do the same, we set out to assist with collating the South African
data. Some aspects of the datasets and challenges experienced are discussed in
work by Afrimapr555http://www.afrimapr.org/blog/2020/merging-health-facility-
lists-part1/ and come with supporting R Markdown
file666https://github.com/anelda/merge_open_hospital_data/blob/master/reports/merge_open_hospital_data_part1.Rmd.
The full report discussing these challenges is also available (van der Walt,
2020).

The work on the _covid19za project_ is also feeding into Afrimapr. Afrimapr is
a Wellcome Open Research Fund project hosted at the Liverpool School of
Tropical Medicine. The overarching aim of Afrimapr is to develop reusable
open-source R building blocks to make open African health data more
accessible. One of the focus areas at this stage is the
afrihealthsites(Afrimapr, 2020)777https://github.com/afrimapr/afrihealthsites
package that currently provides easy access to two continental open health
facility datasets - one developed by the KEMRI—Wellcome Trust Research
Programme and hosted by WHO and a second collated by healthsites.io. Ongoing
work includes adding functionality to include self-defined health facility
lists (such as the ones described in this
blog888https://dsfsi.github.io/blog/mapping-healthsystem/ post) for
visualisation and analysis.

###  3.3. Gaps in the data

We have made progress with getting data but it is still not enough. Currently,
there are still useful bits of information about hospitals that are still
missing like clear hospital descriptions, their operational functions and most
importantly the statuses on whether they are COVID-19 ready or not. Data about
contact details of hospitals and contact personnel per hospital is also
currently not complete. Another crucial missing bit of data in our healthcare
facilities dataset is the estimate of population in some local municipalities
and districts. Current bed numbers in hospitals are not verified as yet, the
locations of the recently developed field hospitals are not available as yet
and also a clear indication of which hospitals (both in the public and the
private sector ) are being upgraded has not yet been made available.

##  4\. Inspiration from across the world

During the COVID-19 pandemic, there have been many projects covering data
analysis, modelling and mapping. Here we share a few mapping projects that
have served as inspiration for some of the work that is being done on mapping
the South African health system. One such project worked to map the
availability of hospital resources in German hospitals. The project visualised
all hospitals and provides easy to understand triage of the health system
capacity for each hospital. An example is shown in Figure 2

Figure 2. German Health System Capacity Map

We were inspired by a COVID-19 testing facility
map999https://www.ineff.ch/cov19testmap/ that visualised testing facilities in
South Africa, Switzerland, Slovenia, Poland and the Czech Republic. One of the
authors of this case study contributed to the translation of the map into
isiZulu and also connected the map to the South African data in our project.
The map is shown in Figure 3.

Figure 3. Country testing map from https://www.ineff.ch/cov19testmap/

##  5\. Ongoing and future work

Our ongoing work focuses on obtaining more sources of data to enhance the data
that we already have. As the COVID-19 pandemic progresses in South Africa, we
believe there will be more need for coordinating the health care system
response. As such next discuss two sets of future work that we are planning.

###  5.1. Current state of mapping

We created a mapping visualisation of the health systems data (as shown in 1.
Our next goal is to complete missing data and update the dataset with the
current developments made by the South African governments. These developments
include upgrading of public hospitals, additions of beds in some hospitals and
development of field hospitals for triage. Populating the dataset with data
related to these current developments will assist in preparing for the surge
of COVID-19 that is anticipated to peak in a few months (Modelling et al.,
2020).

###  5.2. Tools for healthcare workers

We are planning to develop a Progressive Web Application that will make
accessibility of this data (amongst other data in the repository) easier. This
will make it possible to get the information even to parts of the population
that lives in remote low-bandwidth areas where internet quality is not so
strong or fast enough. This will be both for health care workers as well as
the public. This will then open up the next phase of the project to develop
tools for better understanding of individual health facility planning for the
pandemic akin to the work on hospital projections (Zhang et al., 2020).

##  6\. Conclusion

The work described in this case study has been focused on how we have worked
to collect, collate and clean health systems data for South Africa in response
to the COVID-19 pandemic. This project emerged from the necessity to assess
the healthcare system from a public point of view and inform not only the
public, but the healthcare workers on the current situation within their
facilities. Through this project, we have created our own maps of the health
system (public and private) and also contributed to other mapping projects
through making data available in an open manner. There is a large need for the
NDoH to keep their health systems data up to date and easy to use and
integrate with other systems both in public and private domains. This data
becomes critical at a local management level if one needs to understand how
well the health system is coping during a pandemic (we showed an example of
the German ICU map that has a simple triage of each hospital). Furthermore,
other innovations can be developed on top of the data (we showed an example of
a testing facility map) that now includes South African (publicly) testing
facilities that were announced by the Minister of Health, South Africa.

###### Acknowledgements.

We would like to acknowledge Anelda van der Walt who has assisted in guding
this work from the beginning and had numerous inputs into the process. We
would like to acknowledge every person from the public that dedicated their
time, effort and energy to assist during this pandemic. For the _covid19za
project_ we would like to acknowledge all contributors. As this is an open
contribution project, the updated list of contributors is available on the
github repo101010https://github.com/dsfsi/covid19za/graphs/contributors. We
would like to thank the DSFSI research group at the University of Pretoria for
all their expertise, patience and hard work during this time. We also would
like to thank all the employees of the NICD, DoH and WHO who assisted with
data during this time. We would like to acknowledge ABSA for sponsoring the
industry chair and it’s related activities to the project.

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