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Materials modeling is very computationally intensive and first-principles simulation of a real system of reasonable size typically involves the use of classical supercomputers.

How can quantum computing accelerate materials modeling? Are there differences in how quantum computing can be used to accelerate wavefunction vs. density functional methods?

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Quantum computers provide the possibility of simulating systems that are so mathematically complex that classical computers cannot practically be used. The difficulty comes from the fact that the electrical properties of materials, and other chemical systems, are governed by the laws of many body quantum mechanics, which contain equations that are extremely hard to solve with classical computers. The calculations are essentially intractable, taking the age of the universe, or longer, to solve. A quantum computer doesn’t have this problem - by definition the qubits already know how to follow the laws of quantum physics - see the classic paper, Feynman, R., Simulating Physics with Computers, 1982, Int J. Phys..

In current quantum computers, aka noisy intermediate-scale quantum (NISQ) devices, the noise can actually be a feature when it comes to modeling chemical systems - see Noisy Quantum Computers Could Be Good for Chemistry Problems, Wired, April 2019. Some further research in this direction is given in the following papers.

Nam, Y. et al, Ground-state energy estimation of the water molecule on a trapped-ion quantum computer, Nature, npj Quantum Information, April 3, 2020, 6, 33

Reiher M. et al, Elucidating reaction mechanisms on quantum computers, PNAS, July 18, 2017, 114(29) 7555-7560

Rivera, N. et al, Ab initio calculation of phonon polaritons in silicon carbide and boron nitride, Aug 31, 2018

From a practical standpoint, we are still in the very early stages of developing quantum computers, however, and further research is needed to solve significant challenges, including creating systems with many more qubits, improving qubit performance, and developing software languages and frameworks for quantum computers. For more information see the NSF report Quantum Information and Computation for Chemistry, NSF Workshop Report, June 20, 2017

A recent report by McKinsey gives a higher level market overview of the field.

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Quantum computing has been extensively discussed as long as the first quantum computers have become a reality. As already pointed out, we are still at very early stages of such development. What we need to keep in mind is that a quantum computer needs a quantum computing algorithm. The latter can have few to no correlation with classical algorithms, such as the ones used in density functional theory for materials modeling.

The major advantage of using qubits (quantum bits) and quantum algorithms, against the classical ones, is that they are by essence fully parallelized since we are dealing with entangled states. Also, the sampling space grows with $2^n$, where $n$ is the number of qubits. That is the key: use the massive power of parallelism to perform tasks a classical (super)computer would suffer to do.

Since materials modeling relies on dealing with complex systems with lots of interactions, quantum computing can be thought of as a tool we should take care of. The main obstacles nowadays are the lack of specific algorithms for material modeling.

There are some efforts to adapt the classical algorithms, where the time-consuming parts such as the minimization problems are done by the quantum computer, and efforts to create new algorithms from scratch.

Qiskit from IBM is

an open-source quantum computing software development framework for leveraging today's quantum processors in research, education, and business.

One of its components is the Qiskit AQUA. It is a

package contains the core cross-domain algorithms and supporting logic to run these on a quantum backend, whether a real device or simulator.

Aqua includes the Chemistry package, a specific library for quantum chemistry in (IBM's) quantum computers.

The qiskit.chemistry package supports problems including ground state energy computations, excited states and dipole moments of molecule, both open and closed-shell. The code comprises chemistry drivers, which when provided with a molecular configuration will return one and two-body integrals as well as other data that is efficiently computed classically. This output data from a driver can then be used as input to the chemistry module that contains logic which is able to translate this into a form that is suitable for quantum algorithms. The conversion first creates a FermionicOperator which must then be mapped, e.g. by a Jordan Wigner mapping, to a qubit operator in readiness for the quantum computation.

Therefore, this is still an open question.

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Quantum computers potentially could provide a far more efficient way to solve the electronic structure problem: finding the ground state energy (or other properties) of electrons in a compound.

The most prominent quantum algorithms for this purpose are the Phase Estimation Algorithm (PEA) and the Variational Quantum Eigensolver (VQE). While PEA offers the potential for dramatic speedups over classical methods, it requires very high fidelity qubits. VQE in contrast is more tolerant of device error.

Recent advances in superconducting and ion-trap quantum computers have allowed for the experimental realization of both of these algorithms by Google, IBM, IonQ, and others using Noisy Intermediate-Scale Quantum (NISQ) devices. (However these experiments have been limited to simulations of small molecules and small active spaces.)

Note that both VQE and PEA are wavefunction approaches in the sense that the quantum computer is used to prepare a state that corresponds to the state of the electrons in the compound. For more details, I would recommend the review article Quantum Chemistry in the Age of Quantum Computing.

Additionally, there are potential applications for quantum computers in materials modeling besides electronic structure simulation. For example, some quantum algorithms may accelerate the computation of thermodynamic properties of classical lattice Hamiltonians.

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    $\begingroup$ +1. Nice first answer. I'll mention that VQE evaluates the energy and derivatives using qubits, then sends these to a classical computer, which does the optimization. For methods like coupled cluster and FCI, on classical computers we are limited by the amount of RAM we have available to store cluster amplitudes or CI determinants, so I'm not very optimistic about quantum computers being able to use VQE on cases which require more of these than classical computers can handle. But it is also too early to be too pessimistic. $\endgroup$ – Nike Dattani May 14 at 3:47

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