CHEMISTRY – Quantum Computing’s Killer App?
Chemistry is arguably the most important of sciences. We use chemistry to explain human evolution, how all living organisms work, how our brains function, how disease spreads, how food is produced and how our global environment behaves. We also use it to create the vast multitude of materials, fuels, nutrients, and drugs that society depends upon every day. Can quantum computing help advance our knowledge of chemistry and its applications? We explore the possibilities.
Quantum Computing’s Killer App?
It has been suggested that Chemistry is the Killer App for quantum computing:

It is the original application for which quantum computing was envisioned

Its potential business value is very high

It's not just speedup, it’s being able to do things impossible today

It has major potential benefits for humanity and the environment
Leading Management Consulting firms (e.g. McKinsey, BCG, Accenture) have highlighted the potential in terms of faster/better drug discovery, new materials design, new battery designs and better climate control through new catalysts for production of fuels and fertilisers. They estimate future value potential for the chemicals and pharmaceutical industries to be hundreds of $Billions in the next 10 to 15 years. But there are still a few hurdles to getting there, particularly the physical limitations of present NISQ quantum computers and the types of algorithms that can be run on them to achieve speedup of chemistry calculations AND deliver accurate answers.
Quantum Computational Chemistry
The terminology can get confusing, so let’s clarify the different methods:
1. Quantum Chemistry is the science of applying quantum mechanics (QM) to undertake theoretical calculations of the properties of atoms and molecules. It has been around since 1913 when Niels Bohr made the first calculation in quantum chemistry. He applied Planck’s quantum hypothesis to the hydrogen atom, modelled as a positively charged nucleus with a single electron orbiting it. His calculated electron orbitals and their energy levels accurately predicted the experimental spectra of hydrogen. Subsequently, as the full theory of QM was developed, Quantum Chemistry calculations were extended to more complex situations.
One problem soon became apparent in quantum chemistry – we cannot solve the equations! Although the hydrogen atom can be exactly solved, we cannot solve the next atom, helium. This is because helium has one nucleus and two electrons and, even today, we still have no mathematics to exactly solve the equations of motion of three or more interacting particles. This is the infamous “manybody problem”. We have to rely on various approximation methods that have been developed to solve the equations.
2. Computational Quantum Chemistry  the development of the (classical) computer gave a great boost to quantum chemistry since it enabled more of these approximate solutions to be made in greater detail, and for larger atoms/molecules. This field is now broadly described as “Computational Quantum Chemistry”. Dozens of different commercial and opensource software packages are now available, each suited to different types of problem. A list of these packages can be found here and their practical applications include drug design, catalyst evaluation, protein folding and new materials formulation.
However, there is still a problem  computing time. Calculations of chemical reaction processes require accurate calculations of molecular energies and our best (ab initio full configuration interaction) models for this require computation effort proportional to N^N
(N= no. of atoms). If a methane molecule calculation (5 atoms) takes 1 day on a (very fast) supercomputer, ethane (8 atoms) takes 15 years and propane (11 atoms) takes over 250,000 years on the same computer. For a complex, folded protein with hundreds of atoms, this computational task is impossible, even on today’s best supercomputers. This is the potential future opportunity for quantum computers.
3. Quantum Computational Chemistry – this is the application of quantum computers to the problems of computational chemistry. There are two fundamentally different types of computational chemistry problems:

Quantum Mechanical – where we aim to solve the detailed Shrödinger equation for the internal energy levels and electron densities of an atom or molecule to help us understand spectra, bond lengths, spatial configuration and reaction mechanisms/rates.

Molecular Dynamics – where we model molecules as “balls and springs” and use purely classical, statistical mechanics to determine how they interact with each other. This is a much more practical approach for the large, complex molecules typically studied in drug discovery.
Quantum computers have applications to both, but in different ways.
In Molecular Dynamics we use statistical methods and vast data libraries to help find a desired molecular structure. For example, thousands of hypothetical molecules can be screened for a specific disease application and just a small percentage progressed to real lab experiment. These methods have long been used in the pharmaceutical industry using software packages such as “Rosetta” run on supercomputers. In this area, machine learning (ML) has become a primary tool. It can rapidly sort and categorize patterns that are not intuitively obvious to humans. Several new ML/drug design startups such as Menten, ProteinQure, Qbit Pharmaceuticals and Qulab have emerged to promote the use of ML in drug discovery.
The linear algebra equations of ML/Deep Neural Nets are potentially quite suited for solution on quantum computers and there is now great focus on developing quantum computer algorithms for ML/DNN that will also support the efforts of Molecular Dynamics. Undoubtedly, NISQ computers will be able to speedup these calculations but, as yet, we have no evidence that they will be able to do better than classical supercomputers.
Solving the Schrödinger equation for atoms and molecules is a completely different problem than Molecular Dynamics and requires different computational techniques and skills. There are two parts to the solution:

Selecting the most suitable approximation method (remember we cannot exactly solve manybody problems!)

Selecting the most efficient quantum computer algorithm
The most suitable quantum mechanical approximation method is selected first on the basis of what we are trying to calculate (e.g. energy levels, electron densities) and then on the on the size of problem (no. of atoms/electrons), desired accuracy and available computer resources. Numerous different quantum mechanical approximations have been developed by chemists and physicists over the past 90 years and include methods such as HartreeFock, SelfConsistent Field, Density Function Theory, Coupled Cluster and Full Configuration Interaction (FCI). There will undoubtedly be more to come, perhaps new methods designed specifically for quantum computers.
FCI is the most accurate method. It describes a molecule in terms of a basis set of N functions (described as Gaussian or similar mathematical functions) which are typically the outermost electron orbitals of the N constituent atoms of the molecule. FCI requires calculation of the interactions between all ntuples of electrons (n= 1, N) and the problem size scales as N^N. It is very difficult for classical computers but can be done for small (3 to 4 atom molecules). Similar size calculations have also been performed on quantum computers.
The other (lesser) approximations only scale as around N^4 or N^5 so are much more feasible on classical computers but, even so, as N gets large, computational time/cost becomes prohibitive. So far, calculations have been done on supercomputers with N around 100 atoms, but accuracy can be variable. No quantum computing algorithms yet exist to handle a problem of this size on the quantum computers available today.
Quantum Computing Algorithms
Numerous different algorithms have been developed for the solution of quantum chemistry problems on quantum computers and more will be discovered in the next few years. Due to the present limitations of quantum computers, the algorithm must be tailored to the design of the machine (e.g. superconducting versus trapped ion) and also to the formulation of the chemical problem in terms of molecule size, approximation method used and mathematical form of the basis set functions. It is highly specific and requires collaboration between experts in both quantum computing and quantum chemistry. As yet, there are no onesizefits all general problem solvers. Ideally, we would like the quantum computer to perform a complete simulation of the Hamiltonian of our problem but this requires long complex circuits with 100,000’s of qubits and our machines are not there yet.
The most promising algorithm for present NISQ computers is the Variational Quantum Eigensolver (VQE). The VQE solves the specific problem of finding the ground state energy of an atom or molecule, which is an important and useful property in quantum chemistry. The VQE uses a combination of both quantum computer and classical computer to achieve its results and is based upon the Variational Principal of quantum chemistry. This states that the exact (true) wave function for a system always minimises the energy of the system. If we construct an approximate guess (“ansatz”) of the wave function in terms of some parameters x1, x2, x3 ….. and then calculate its energy, it will be higher than the ground state minimum. By varying the parameters “x” to minimise the calculated energy we can get closer and closer to the true wave function and the ground state energy.
Doing this on a classical computer can be very time consuming, so VQE splits the task. The ansatz is prepared on a quantum computer in terms of various parameters “x”, evolved according to the desired Hamiltonian and the energy measured (these are the hard parts for the classical computer). The measured energy and parameters are then fed into a classical computer optimisation routine and new values of “x” calculated which are then fed back into the quantum computer. The whole process is recycled many times until the minimum energy and associated “true” wave function are found.
VQE has been successfully demonstrated on several different types of quantum computer. However, it is a heuristic method with no proof that it will be faster than a classical computer. Also, it can take many iterations to get to the minimum energy and it is possible that the classical optimisation routine will settle on a local suboptimum minimum. Design of an efficient “ansatz” is also an art in its own right. All we can do is try it and see!
An excellent recent (technical) review of the present state of quantum computational chemistry was published by McArdle et al., 2020 (arxiv1808.10402) and they go into great details of the algorithms. Their focus was solely on Quantum Mechanical calculations and they did not discuss Molecular Dynamics problems. One of their key findings was as follows:
“ …. a small quantum computer, with around 100 perfect qubits, would be able to calculate the FCI energy of a system with around 100 spinorbitals in polynomial time. This would imply that these problems are among the best targets for quantum computers. It is important to note that being able to accurately predict the ground state energy of 100 spinorbital systems still leaves us far from our longterm goal of designing new medicines and materials with simulations.”
Conclusions
Quantum Computing theory suggests that quantum computers can simulate any known physical/chemical situation. However, we have no mathematical formalism to predict the success of quantum computers in ML/DNN methods for Molecular Dynamics, nor in the solution of the Schrödinger equation for moderately complex molecules. At present, it is all a matter of experiment – try it and see. Large corporations in the chemicals, materials, energy and pharmaceutical industries cannot afford to ignore the potential disruption of quantum computing and must invest in research programs in this technology. Some startups may discover the secrets first and become quite famous.
Success will depend upon the future ability to build large fullyerrorcorrected quantum computers and to assemble crossfunctional expert teams of quantum engineers, computer scientists and quantum chemists to codesign appropriate algorithms.
Finally, quantum chemistry and molecular dynamics are a great start to the discovery process, but there is much more to do in creating a successful new chemical product. For example. in drug discovery, the processes of pharmacokinetics, clinical trials, packaging, manufacturing etc. Quantum computing may also offer benefits in each of these processes.
We can expect steady progress over the next 10 years but should not be expecting to close down chemistry laboratories any time soon. The “Killer App” expectations may be somewhat overstated both in timing and $ value.