d\vec{r}_{1}\chi_{i}^{*}\left(\vec{r}_{1}\right)\left(-\frac{1}{2}\nabla_{1}^{2}-\sum_{\sigma}\frac{Z_{\sigma}}{\left|\vec{r}_{1}-\vec{R}_{\sigma}\right|}\right)\chi_{j}\left(\vec{r}_{1}\right)$
(49)

$h_{ijkl}=\int
d\vec{r}_{1}d\vec{r}_{2}\chi_{i}^{*}\left(\vec{r}_{1}\right)\chi_{j}^{*}\left(\vec{r}_{2}\right)\frac{1}{r_{12}}\chi_{k}\left(\vec{r}_{2}\right)\chi_{l}\left(\vec{r}_{1}\right)$
(50)

This fermionic Hamiltonian can be mapped to Hamiltonian as the product of the
Pauli matrix using Bravyi-Kitaev transformation already discussed in 3.3. One
can also use parity basis and particle and spin conservation method to further
reduce the number of qubits needed.

In quantum computation, there are simulation methods which can be used to
simulate and calculate the energy states of the molecule using its
Hamiltonian. The phase estimation method using Trotterisation in which the
Eigen-energy values are encoded into the phase of the propagator is one
method. Also, another two ways for simulations are Direct implementation of
Hamiltonian using first-order [282] and second order Trotterisation [283]. Yet
another method is the Direct-measurement method. These algorithms are all
Phase Estimation Algorithm (PEA) type algorithms. The most useful method in
the NISQ era of quantum computers is the Variational Quantum Eigen Solver
[284]. The paper [285] shows that the VQE method requires the least number of
qubits for scaling. While PEA type of methods is shown to have higher accuracy
just by one measurement but they require a large number of qubits. This shows
that the VQE algorithm is best suited for NISQ era quantum computers while PEA
types are the ones best suited for long term quantum computers.

Number Operator

& T(θ)

Figure 8: Here $T(\theta)$ is the phase gate such that
$T(\theta)\ket{0}=\ket{0}$ and $T(\theta)\ket{0}=\exp(-\iota\theta)\ket{1}$

Number-excitation operator

p & M 2 2 M

q+1 [style=fill=red!30] 2 2 [style=fill=red!30]

q-1 2 2

r M [style=fill=red!30] R_z(θ) [style=fill=red!30] M

Figure 9: M gate is the combined set of $\\{H,Y\\}$ gates taken in order
($Y=R_{x}(-\frac{\pi}{2})$)

Double excitation operator

p & M_1 2 2 M_1^†

q M_2 [style=fill=red!30] 2 2 [style=fill=red!30] M_2^†

r M_3 2 2 M_3^†

s M_4 [style=fill=red!30] R_z(θ) [style=fill=red!30] M_4^†

Figure 10: In the circuit,
$(M_{1},M_{2},M_{3},M_{4})=\\{(H,H,H,H),(Y,Y,Y,Y),\newline
(H,Y,H,Y),(Y,H,Y,H),(Y,Y,H,H),(H,H,Y,Y),\newline (Y,H,Y,H),(H,Y,H,Y)\\}$

.

Excitation Operator

p & H 2 2 H Y 2 2 Y

q H [style=fill=red!30] R_z(θ) [style=fill=red!30] H Y [style=fill=red!30]
R_z(θ) [style=fill=red!30] Y

Figure 11: Y gate is nothing but $Y=R_{x}(-\frac{\pi}{2})$

Coulomb and exchange operators

p & G(θ) R_z(θ) 2 2

q R_z(θ) [style=fill=red!30] R_z(θ) [style=fill=red!30]

Figure 12: $G(\phi)$ is the global phase gate which is expressed as
$\exp{(-i\phi)}1$

 _Notation:_

& 3 3

[style=fill=red!30] [style=fill=red!30] = & 1 1
1 1
1 1

####  6.4.2 Molecular designing simulation

Being able to study the dynamics of the molecules and their time evolution
allows scientists to design and come up with new molecules which can be used
as products in the market or as a treatment for certain diseases. The last 2
years of the Covid-19 pandemic indicate the importance of speeding up these
processes of designing molecules. These can be solved using two methods. The
first approach is using the Born-Oppenheimer approximation. Alternatively, the
dynamics of the quantum molecular systems can be expressed as the simple
product of time-dependent electronic and nuclear wave functions [286].
Simulation using these methods requires a higher computational cost. Although,
the computational cost can be lowered by making certain approximations it
often increases the errors [287, 288].

Quantum dynamics has its relevance in the study of non-equilibrium processes
with potential energy surfaces, dynamics of molecular and solid state systems
with electron and nuclear dynamics and optimal quantum control theory. Quantum
optimal control theory is of high interest. It is nothing but the theory of
controlling the dynamics of quantum systems using external lasers.
Applications are immense and growing rapidly [289]. The theory has been
experimentally verified with bond dissociation experiments [290],
isomerisations [291] and molecular fragmentation [292].

This field has shown vast growth in recent years. Although, current QC
algorithms are often used for demonstration purposes only for which only
simple molecules are considered. This is because the current state-of-the-art
quantum computers are limited in terms of qubits. These confines the
algorithms to BO approximations and do not allow the inclusion of non-
adiabatic effects [293].

In this field of work, the most famous class of quantum algorithms that are
used is Variational Quantum Algorithms (VQAs). These algorithms use a hybrid
approach, that is, the simulation of the system is done on a parameterised
quantum circuit whose parameters are optimised classically using some cost
function. The authors of [294] use Variational Quantum Eigensolver to simulate
their molecular system. They specifically study the rearrangement dynamics of
the molecule. One reason why VQE is often used is that the current era of
quantum computers is noisy, but VQAs are adaptable to the noisy nature of QCs
if one uses Hardware efficient Ansatz [295]. In [296] a hybrid method for
calculating and designing Deuterated High-Efficiency Organic Light Emitting
Diodes was proposed. They use machine learning methods to calculate the Ising
model systems and then followed by the implementation of the VQE algorithm to
calculate the quantum efficiency of the molecular system to obtain the optimal
Deuterated molecule. Apart from using VQAs, one can also use the Digital
Quantum Simulation method for molecular dynamics. It has been used for laser-
driven systems. The work [297] describes the use of the theory of quantum
optimal control to simulate the dynamics of molecules. Although their approach
is also hybrid they do not use a variational approach. The steps involve
mapping to the qubits, and Hamiltonian simulation followed by the qubit
readout. To find the optimal control field, the readout states of the qubit
are used by the classical computer for optimisation. The optimisation function
can be decided based on the Quantum optimal control theory. This approach can
be used for controlled bond stretching, the orientation of dipole-dipole
coupled (OCS) rotors and preparing the state of the light-harvesting complex
in a controlled manner.

####  6.4.3 Spectral analysis

Spectral analyses refer to the study of spectral properties. These properties
include the spectrum of frequencies of vibrations and related quantities like
eigenvalues and energies. It is well known that matter can never be at rest.
At the quantum level, even the tightly bonded molecules execute oscillations.
More commonly these oscillations are called vibrations. One can study
vibrations in time independent and time dependent picture. The former allows
us to perform spectral calculations like Infrared and Raman spectroscopy [298]
and fluorescence [299] which have importance in determining solar cells
performance [300] and industrial dyes [301]. The dynamics of vibrations have
much more applications including the dynamics of reactions [302] and
electronic transport [303]. Also, these affect large frequency temporal
resolved laser experiments [58].

This is a field of great importance. There have been methods to accurately
simulate the systems but they are limited to a few particle systems. One such
method is Real-space, grid-based method. When the simulation of the molecular
systems is done on a classical computer, we are restricted to using a finite
basis for spaning the infinite dimensional Hilbert space. The full
configuration interaction or FCI method can provide accurate solutions for
electronic structures but scales up exponentially with the increase in system
size [304]. This field uses configurational methods discussed in 4.

Table 2: Quantum circuits for second quantisation operators. The circuits are presented above Operator Name |  Symbol |  Circuit
---|---|---
Number Operator |  $h_{pp}a^{\dagger}_{p}a_{p}$ |  circuit 1
Excitation Operator |  $h_{pq}(a^{\dagger}_{p}a_{q}+a^{\dagger}_{q}a_{p})$ |  circuit 2
Coulomb and exchange operators |  $h_{pqqp}a^{\dagger}_{p}a^{\dagger}_{q}+a_{q}a_{p})$ |  circuit 3
Number-excitation operator |  $h_{pqqr}(a^{\dagger}_{p}a^{\dagger}_{q}+a_{q}a_{r}+a^{\dagger}_{r}a^{\dagger}_{q}+a_{q}a_{p})$ |  circuit 4
Operator for double excitation |  $h_{pqrs}(a^{\dagger}_{p}a^{\dagger}_{q}a_{r}a_{s}+a^{\dagger}_{s}a^{\dagger}_{r}a_{q}a_{p})$ |  circuit 5

####  6.4.4 Chemical Reaction simulation

The analytical results in quantum chemistry are very important when it comes
to understanding a chemical reaction. They allow us to know the steps and
mechanisms involved in a chemical reaction [305]. Again, classical computers
pose the problem of fewer computational resources. Solving the Schrodinger
equation and simulating its time evolution requires an exponential increase in
the size of the system. Also, increasing the degree of freedom requires us to
have more size in the system.

Quantum Computers as already known, can easily simulate or can be used to
propagate the Schrödinger equation. They show a promise of completing the same
task in polynomial time [306] when compared to classical computers. Although
the limited number of qubits and noisy hardware makes things difficult for
simulating accurate results. Still, there are noise mitigation techniques
which can be used to reduce the noise. Most of the algorithms in Quantum
computation for chemical reactions are based on the approach of Digital
quantum simulations [307]. Specifically, there have been examples of reactions
which are controlled and driven by the external laser fields.

Figure 13: Figure shows the isomerisation of substituted malonaldehyde which
is non-symmetric. Authors of [308] performed the digital quantum simulation of
taking this molecule into consideration. A double well potential has been
considered during isomerisation.

The DQS based approach can be found in [309]. Authors of both have studied the
isomerisation reactions in a double welled potential and showed the time
evolution of the reactant and product states. The former implemented the
algorithm on an NMR based quantum computer, while the latter simulated the
same in the quantum simulator of IBM, called ibmq_qasm_simulator. The
theoretical approach remains the same in both the cases which are to find the
Hamiltonian in second quantisation and then to the qubit system. The former
uses a GRAPE technique [310] to implement the pulses on the NMR QC to
implement the unitary operations. This technique provides an efficient
implementation on the NMR quantum computers.

###  6.5 Bioinformatics

There has been a lot of development of algorithms and mathematical tools to
solve biological problems. Bioinformatics is one of the filed of utmost
importance as it provides solutions to a better lifestyle, helps fight against
widespread diseases and much more. This field explores complex areas like
human genome, modelling biomolecule’s behaviour in different environments,
calculating binding affinity of a ligand etc. These problems can widely be
categorised into three subsections. These are Drug Discovery, Genome
Sequencing and Proteomics.

The research in the field of finding better computational methods has been
tremendous. Some of these methods are used in alignment of sequence [311,
312], computational genetics [313], data processing for X-ray diffraction
[314]. These problems can be solved using computational methodologies. But the
current computational resources are not enough to simulate large bio-
molecules. Therefore, there has been a shift in attention from classical
computers to quantum computers as a computational platform. The following
subsections explain three major problems that can be found in the field of
Bio-informatics and also describes the emerging role of Quantum computation.

####  6.5.1 Drug Discovery

The Discovery of drugs were used to happen accidentally like penicillin. But
now with advancements in technology drug discovery has become not a random
process but a process involving steps and procedures. Various chemical
compounds are selected from the database and extensively searched if they can
be a potential drug. Also, their synthesisability is studied. Following this,
the compound is optimized to maximise the affinity and then it is gone through
trials including animals succeeded by human trials.

The development of drugs is a very long process and consists of the following
stages including target discovery then molecular design followed by pre-
clinical studies and lastly clinical trial. This makes the process of creating
a marketable drug expensive and time-consuming. Pharmaceutical research has
been using high-throughput screening technology Discovering drugs involves
searching through target-binding compounds. This is a long process and even
expensive. While the computational methods for molecular docking can help to
identify the molecules which bind with the target.

The computational accuracy of the docking search depends on the description of
the compound in the software’s library used for the process. This software is
often called the docking engine. The docking methods used by these engines can
vary from software to software. Some common examples of these methods are
Auto-Dock Vina [315], MedusaDock [316] and Glide [317]. These approaches
mostly try to properties of the compounds and receptors which can bind
together. The accurate results are given by the density functional theory. As
usual, the classical computational methods are limited to small sized
molecules and receptors [318].

The quantum algorithm for Quantum machine learning is a faster and cheaper
solution to the problem of classical computations. The most promising area of
quantum algorithms for this field is the Quantum Generative Adversarial
Network [319]. Quantum GAN with Hybrid Generator [320] is one of the QGAN
algorithms. It consists of a parameterised quantum circuit for finding the
feature vector and followed by a classical DNN to predict the bond matrix in
terms of graphs for the drug. In this method, there are various variations.
One is Patched-QGAN-HG [321]. Apart from QGAN based approaches, there are
Image search based methods. These methods involve the quantum convolution
neural network [322]. These are based on convolution neural networks. In QNN
the filters are replaced with the quantum circuits. The quantum variational
autoencoders have been tried to perform the drug simulations but have not yet
shown any better performance than classical VAE. VAE is a method which is
based on probabilistic search [318].

####  6.5.2 Genome sequencing

Genome sequencing is the process of figuring out the order of nucleotides of
the DNA. A DNA consists of an order of genetic letters As, Cs, Gs and Ts. The
human genome is made up of over 3 billion letters. To understand a genome, its
sequencing is very important. This will allow scientists to find genes much
more quickly as they contain information on where the genes are. The study of
genome sequence has great significance to scientists to understand how they
direct the growth and maintenance of a full organism.

Traditional computational methods use De Novo assembly [323] to construct an
original DNA from an unstructured set of reads without using any pre-requisite
knowledge like DNA sequence length, composition etc. The complexity depends on
the size of the genome. As an example, it takes nearly 50 hours for the human
genome on a supercomputer. This time might be acceptable for research tasks
but is not fit for the case of emergencies. These assembly tools are based on
Overlap Layout Consensus (OLC) algorithm [324]. It uses an OLC graph in which
the vertices are presented by a read while the overlap between any two reads
corresponds to the vertices of the graph. Then the Hamiltonian path is found
which is the path that consists of all the edges and each vertex is visited
only once giving the original genome.

Quantum computers specifically quantum annealing can solve this problem. Since
it is a graph problem, it can be formulated into a QUBO formulation. The OLC-
graphs are converted to the QUBO/Ising model [325] which is then embedded in
the quantum annealing system and then as an output one is given the
Hamiltonian path. Apart from QUBO formulation, the QAOA method is used for DNA
sequencing to accelerate the de-novo sequencing.

####  6.5.3 Proteomics

Proteomic is a field that has started merging with quantum technologies. This
field studies the electronic structure of the proteins in a given cellular
system of any organism. Proteomics is defined as a group of proteins present
in the organism. This allows scientists to study the properties of proteins
like energy levels, dipole moments, amino acid charges, their electric
potential and a plethora of many things. This field emerged after the human
genome project (HGP) was completed in 2001 [326]. HGP involved the
identification of more than 30,000 genes in humans, which gave way to the
study of the proteins expressed by the genes. The problems involved in this
field include the characterisation of protein structures, inter protein
interaction (interactome) and phosphorylation (phospho-proteomics). Protein
folding is one problem which comes up after the identification of proteins.
One needs to study the proteins to disclose the knowledge of how the proteins
are encoded in genes. Classical algorithms of protein folding can sample small
conformation space. Many quantum algorithms have been proposed to solve this
problem. The paper [327], proposes a hamiltonian and the variational quantum
algorithm for folding a polymer chain with N monomers on a lattice,
specifically for 10 amino acid Angiotensin worked out on 22 qubits. Gate based
algorithms have also been proposed as in paper [328]. This field has a great
potential for growth with quantum computers offering an exploration of large
conformational space for protein folding.

##  7 Error Mitigation

Near term quantum computers are not fault-tolerant. The two most significant
hurdles to scalable universal quantum computers are error sensitivity and
noise. Errors can arise in each quantum computation step, making it difficult
for efficient digital quantum simulations. The errors can be broadly
classified as (i) State preparation errors, (ii) Gate errors and (iii)
Measurement or Readout errors. The gate errors are further classified into
Coherent and Incoherent errors. The coherent errors preserve the purity of the
state. It typically occurs due to miscalibration in the control parameters.
Now, one can understand incoherent errors in two ways. Either it can be
considered coherent errors with randomly varying control parameters or an
operation that entangle the qubit with the environment. State Preparation and
Measurement errors are sometimes together addressed as SPAM errors. Compared
to gate errors, SPAM errors occur only at the beginning or end of the circuit
and do not accumulate with increasing circuit depth.

There are two proposals for achieving Fault-tolerant quantum computation. The
first method uses non-abelian quasiparticles called anyons in the topological
matter to perform error-free quantum computation (Topological Quantum
Computation) [329]. Another approach for fault-tolerant quantum computation is
using Quantum Error-Correcting (QEC) codes [330]. One can use these codes to
detect and remove gate errors during computation. While the first proposal is
still in its infancy, the second approach requires computational resources
unattainable in near-term devices. For instance, using Surface code, a
ubiquitous QEC code, one needs millions of physical qubits to perform the
fault-tolerant computation of Shor’s algorithm [331].

Readout errors make approximately $15\%$ of error in quantum computation
(Superconductor qubit based). Thus mitigating readout errors holds importance.
A straightforward approach is using the operator rescaling method for error
mitigation. It uses the documented readout errors to correct the results via
post-processing. But it cannot mitigate the correlated errors in the
computation. Another approach to minimize the readout errors is the
calibration matrix method.

_Calibration Matrix Method_ : In this method, before each time evolution
experiment, we perform a calibration experiment to characterize the device for
each of the $2^{N}$ basis states. We organize the results of each calibration
experiment in a $2^{N}\times 2^{N}$ matrix after the calibration experiment.
Each member of the matrix $Pij$ represents the probability of a system
preparing in-state $i$ and measuring in-state $j$. By applying the inverse of
this matrix to the noisy results, we would get results with mitigated
measurement errors. While applying this method, we need to make two crucial
assumptions; 1) The readout error is caused by classical noise, and 2) the
noise is independent of the quantum gates applied to the system. A recent work
[332] shows that readout errors in quantum systems based on superconducting
qubits can be effectively explained using simply classical noise models.
Further, to prevent the exponential scaling of the calibration matrix with
system size, we assume the error due to noise is local and correlates to a
subset of qubits [30, 333]. Then the error model is called tensored error
model, and the calibration matrix will be the tensor product of $2\times 2$
noisy matrice. Also, sometimes due to strong noise, the inverse of the
calibration matrix will not be well defined. In such a scenario, we have to
find the Moore-Penrose pseudo inverse of the calibration matrix. Tensored
error models do not address the errors due to cross-talk between qubits during
readout. Therefore, recently a measurement error mitigation scheme that
addresses cross-talk errors was proposed [334].

Next, let’s move on to the mitigation of gate errors. As mentioned earlier,
there are Coherent and Incoherent gate errors. Incoherent errors are usually
modelled as depolarizing noise. There exist methods to mitigate depolarizing
noise [335, 336]. Coherent errors are more damaging than incoherent ones. But
in [337] it was shown that one could convert coherent errors to incoherent
errors through randomized compiling. Thus in principle, the coherent errors
also can be mitigated. But there are other approaches to mitigating gate
errors, including the popular one called Zero Noise Extrapolation (ZNE) [338,
339]. We will discuss some of the schemes to reduce the gate errors in the
following section.

_Zero Noise Extrapolation_ : A ZNE consists of two steps Noise Scaling and
Extrapolation. In noise scaling, we would intentionally scale up the noise
level of the quantum circuit. Noise scaling can be done in two ways. The first
approach is called time scaling or pulse stretching. In this method, we
stretch the control pulses and thereby increase the noise in the circuit. The
second approach is called the unitary folding. Here we map the unitary
operation $U$ to $U(U^{\dagger}U)^{n}$, where n is an integer. Thus in the
quantum circuit, it would increase the circuit depth and thereby scale the
noise. The unitary folding can be applied globally (Circuit Folding) or
locally (Gate Folding) [340]. Using Noise scaling, we would calculate the
expectation value of the observable (that we want to measure) at different
noise levels. Once we have the regression between the expectation value (of
the observable) and noise, we can evaluate the expectation value at the zero-
noise limit through extrapolation. The model used for performing extrapolation
depends upon the noise model assumed. Polynomial extrapolation is generally
used in the weak noise limit if the number of data points ($m$) is equal to or
greater than $d+1$, where $d$ is the degree of the polynomial. Of the two
variants of polynomial extrapolation, linear extrapolation is used when $d=1$
and Richardson extrapolation when $d=m-1$ [338]. The polynomial extrapolation
is inefficient when there is a low error rate and a large number of noisy
gates. In such cases, we need to resort to other extrapolation methods such as
poly-exponential extrapolation [340] and exponential extrapolation [331].

_Probabilistic Error Correction_ : The Probabilistic Error Cancellation (PEC)
[338] works based on two ideas. The first idea is the quasi-probability
representation of the ideal gates. It essentially means that we should
represent an ideal gate as a linear superposition of noisy quantum gates
[341]. The real coefficients in such a representation form a quasi-probability
distribution, i.e., the sum of the coefficients will be normalized but differ
from the standard probabilities by taking negative values. Using quasi-
probability representation, any observable represented using an ideal gate set
can be translated into a noisy gate set representation. One could directly
find the expectation values of noisy gates from the hardware. Using it, we
could derive the ideal expectation value of any observable. Unfortunately,
this strategy demands the execution of a large number of circuits, which rises
exponentially with circuit depth and is often impractical. Thus we use the
second idea of probabilistically sampling from the quasi-probability
representation. The Monte Carlo average that follows would give the
approximate expectation value of the observables.

_Other methods for error mitigation_ : Methods like PEC and ZNE discussed
above require complete knowledge of the underlying noise model to be
efficient. In most cases, experimentalists only have imperfect knowledge of
the noise model. Therefore people are working on learning-based QEM techniques
using ZNE and PEC to repress errors via an automatic learning process.
Examples of such methods include _Clifford Data Regression_ [342], and
_Learning-based quasi-probability_ method [343]. In addition, there is also
another approach based on Gate Set Tomography for error mitigation without
being noise aware [331]. Apart from the popular ones, there are other methods
for error mitigation. Examples of such methods include _Dynamic Decoupling_
[344], _Quantum Subspace Expansion_ [36, 37], _Stochastic error mitigation_
[48] and so on. Most of the methods we discussed do not use ancilla qubits.
Another class of error mitigation methods also uses ancilla qubits for error
mitigation. One ubiquitous example is the _Stabilizer-based (Symmetry
verification) error mitigation_. It uses ancilla to perform measurements on
conserved quantities of the problem Hamiltonian, such as spin or parity. Any
error would change conserved quantities indicated upon ancilla measurement
(similar to stabilizers in QEC). This method is often used in variational
state preparation methods [345, 346]. Another error mitigation method that
utilizes ancilla qubit is the recently proposed _Virtual Distillation_ [347,
348]. It suppresses noise by simultaneously preparing numerous copies of a
noisy state, entangling and measuring them to virtually prepare a more pure
state than the original noisy one. Virtual distillation is quite promising
with its exponential suppression of error rate.

Another class of errors that we haven’t discussed yet is the Algorithmic
errors. Compared to others, these errors are not of physical origin. One
ubiquitous example is the trotterization errors arising in the Hamiltonian
simulation. The prevalent method for mitigating trotterization error is the
ZNE. We perform the noise scaling using small trotter steps and then apply
extrapolation. Recently, another approach that exploits symmetry of the system
to mitigate trotterization error was proposed [349]. It is a symmetry
protection technique that causes destructive interference of the errors
arising in each trotter step. An extensive introduction to algorithmic and
other error mitigation techniques is provided in [350].

##  8 Software tool sets

Quantum computing is a field which merges many disciplines. At its current
age, it is at a stage where it has evolved to enter the industry. Many start-
ups in quantum computing have come up in recent years including Xanadu, IonQ,
Zapata, PASQAL, just to name a few. These start-ups are now collaborating with
bigger organisations providing them quantum solutions to the existing
problems. Over the years cloud solutions have become very famous in this
field. Many companies have been building software which allows one to execute
their problems with real quantum hardware without the need to learn quantum
computing. This section provides the details of many software & online
platforms for quantum simulation encountered during the survey. Their details
have been given in the table 2.

Table 3: Resources for Numerical and Quantum Simulations. Software Package |  Domain |  Tasks |  Noteworthy attributes
---|---|---|---
QuASeR [351] |  Bioinformatics |  DNA Sequencing |  One can perform DNA sequencing using the de-novo assembly on gate based and quantum annealers. Uses TSP, QUBO, Hamiltonian methods and QAOA algorithms
InQuanto [352, 353] |  Chemistry |  Ground and Excited states,Spectroscopy, Molecular Geometry,Transition Pathways, Reaction Pathways, Ligand Protein Binding, Molecular Forces |  Mostly uses VQE methods and its variations like ADAPT-VQE, Penalty VQE, VQ Deflation, Quantum Subspace Expansion and Iterative Qubit-excitation Based VQE for computation of the tasks. The packages also comes with Error Mitigation techniques like PMSV and SPAM
MQS [353] |  Chemistry |  Computation of Solubility, Viscosity, Partition coefficient values, Phase equilibria calculations for vapour-liquid-, liquid-liquid and solid-liquid-equilibria. |  Maps the models of quantum chemistry models (DFT, PMx,COSMO-RS/SAC, GNFx-xTB) to Quantum computer hardware through cloud based methods. Allows submitting calculations which are accessed and pipelined to further steps. Has applications for process design, product design and Material Design
OpenFemion [354] |  Chemistry & Condensed Matter |  Computation of Trotter error operators, symbolic Fourier transformation, preparing fermionic Gaussian states, routines for generating Hamiltonians of the Hubbard model, the homogeneous electron gas (jellium), general plane wave discretizations, and d-wave models of superconductivity and wide range of data structures important in Quantum chemistry |  Everything from efficient data structures for encoding fermionic operators to fermionic circuit primitives for use on quantum devices is included in the package.
Fermionic.jl |  Chemistry & Condensed Matter |  Fermionic operators can be constructed both in the full Fock space or in the fixed particle number subspace, can be used to perform fermionic quantum computation. Compute average particle number, one body matrices entropy, partially traced systems , majorization relations, m-bodies density matrices, one body entropies and more. |  Julia tool kit for fermionic simulations and fermionic quantum computation
MISTIQS [141] |  Condensed Matter |  Translation of circuits into executable circuit objects for IBM, Rigetti, and Google quantum devices, domain-specific IBM and Rigetti compilers developed for TFIM simulations, support for user-defined time dependence functions for external fields, full XYZ model support in Hamiltonian constructions. |  A full-stack, cross-platform software for creating, constructing, and running quantum circuits for simulating quantum many-body dynamics of Heisenberg Hamiltonians-governed systems.
QuSpin [355, 356] |  Condensed Matter |  Can Implement Exact diagonalisation, Lanczos Algorithm, Floquet Hamiltonian simulation of a wide range of many-body system. Also have parallel computing capabilities |  An open-source Python package that supports the use of various (user-defined) symmetries in one and higher dimensions, as well as (imaginary) time evolution following a user-specified driving protocol, for exact diagonalization and quantum dynamics of arbitrary boson, fermion, and spin many-body systems.
Kwant [357] |  Condensed Matter |  Calculation of transport parameters (conductance, noise, scattering matrix), dispersion relations, modes, wave functions, different Green’s functions, and out-of-equilibrium local values, other computations involving tight-binding Hamiltonians |  Kwant is a Python package for computing numerical quantum transport. It provide a user-friendly, ubiquitous, and high-performance toolkit for simulating physical systems of any dimensionality and geometry that can be characterised by a tight-binding model.
ArQTIC [358] |  Condensed Matter |  Dynamic Simulation, QITE Simulation, can simulate materials that can be modeled by any Hamiltonian derived from a generic, one-dimensional, time-dependent Heisenberg Hamiltonain |  An open-source, full-stack software package built for the simulations of materials on quantum computers
Quantavo [359] |  Quantum Optics |  A framework which can declare, manipulate and characterize quantum states of light (finite number of modes, and finite dimensional), and implement linear optics circuits such as Beam Splitters (BS), Phase Shifters (PS), arbitrary unitary transformations of the modes etc. |  A Maple toolbox for linear optics and quantum information in Fock space
QuantumOptics.jl [360] |  Open quantum systems |  numerical simulation of the dynamics of OQS, finding the steady-state of OQS & time correlation functions, Optimizes processor usage and memory consumption |  A Julia framework for simulating open quantum systems
HOQST: Hamiltonian Open Quantum System Toolkit [361] |  Open quantum systems |  simulating the dynamics of OQS, supports various master equations, as well as stochastic Hamiltonians |  A Julia toolkit for simulating the open quantum system dynamics
Mitiq [362] |  Error Mitigation |  Zero-noise extrapolation, Probabilistic error cancellation, and Clifford data regression |  Python package for error mitigation on noisy quantum computers
QuaEC [363] |  Error Correction |  Support for maniuplating Pauli and Clifford operators, as well as binary symplectic representations and automated analysis of error-correcting protocols based on stabilizer codes |  Python library for working with quantum error correction and fault-tolerance
CHP: CNOT-Hadamard-Phase [364] |  Error Correction |  Construct quantum error-correction designs and debug them. Numerically study massive, highly entangled quantum systems. Generate random stabilizer quantum circuit, Shor 9-qubit quantum error-correcting code |  High-performance simulator of stabilizer circuits (Quantum Error Correction)
QuaSiMo [365] |  Hybrid quantum-classical algorithms |  Dynamicalsimulation, VQE, Symmetry reduction, Fermion qubit mapping, QITE, QAOA |  A composable design scheme for the development of hybrid quantum/classical algorithms and workflows for applications of quantum simulation
QuEST [366] |  - |  Many functions for simulating decoherence, Calculating density inner product, Hilbert Schmidt distance, Purity, Fidelity, many quantities from Density matrix |  simulator of quantum circuits, state-vectors and density matrices.
qsims |  - |  qsims represents the spatial wavefunction of a particle as a discretized wavefunction on a grid, Internal states of the particle can be represented using multiple grid, Potentials and couplings between the internal states can be specified, and potentials can be position- and state-dependent. |  A tool for studying quantum computing using optical lattices, General-purpose quantum simulation software package, capable of simulating the dynamics of systems with a wide range of Hamiltonians, qsims is not limited to optical lattices, and could be adapted for use in many other physical systems.

##  9 Concluding Remarks

In this paper, we have covered a handful of areas out of the vast versatile
potential domains which can show quantum advantage towards quantum simulation
in near future. Today, the real hardware implementation is limited to
elementary quantum systems and processes which is due to limitation in
realising decoherence free long circuit run time and inevitable gate errors.
But with every new day, we have seen new algorithms and techniques coming up
which have been enlightening the scientific community with optimised methods
to realise quantum simulation and this only narrates that, realising the
advantage of quantum simulation on quantum computers is a reality not very far
now. The realisation of quantum treatment with Hamiltonian simulation has also
expanded its branches to fundamental physics like- simulating gauge
theories[367][368], problems in high energy physics[369][370][371] and quantum
sensing. Its only a matter of time when quantum computers will be solving real
world problems addressing quantum simulations.

##  10 Acknowledgement

We are thankful to our colleagues and collaborators who contributed their
expertise, feedback, and valuable insights during the preparation of this
paper.

This work was made possible by the collective effort and unwavering commitment
of our team at Qulabs, and we are immensely grateful for their collaboration
and support.

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