Quantum Simulation

Quantum simulation can be applied to simulating the electronic states of the lithium hydride and beryllium hydride (BeH2) molecules on a quantum computer. Classical computers falter when simulating molecules with increasing numbers of size. We can calculate toy problems using variational quantum eigensolver (VQE). 

 

For example, in order to synthesize a novel and extremely strong polymer, researchers need to find what chemicals have historically made it strong and look up the reactions to produce strong materials, see who has tried it, know the environmental factors like temperature for the lab setup. So they need to combine the literature search and a little creativity to run an experimental reaction. If it does not succeed, they need to try something else. If yes, they have to take the notebook to a computational chemist to ask what chemical makes it so strong, or even stronger. That is the traditional approach.

 

Chemical reactions occur when it is energetically favored, the energy of the final products is lower than the energy of the chemical mixture and the energy barrier to get to the product is low enough to overcome in the reactive environment. To help tip the scale, adding catalysts can change the energy landscape. 

 

Question: Can we predict the properties of the final product? It can be computationally expensive, and long periods of time on supercomputers, mainly approximation. To certain, supercomputers cannot be as accurate as demanded. 

 

Now we can ask a quantum chemist to have a simulation before mixing the chemical and we predict the properties like toxic, pyrophoric, explosive. Running on a quantum computer can cut down the amount of literature search, trial and error, nebulous intuition, or serendipity to figure out a new molecule. 

 

The important impact is that we could model complex reactions like transition metal, the predictive potential for catalysts incorporating these metals. These catalysts can be important for the processes like nitrogen fixation, and they could reduce energy consumption by a few percentage points. 

The largest molecules for exact simulation are simple enough for classical computers, however, it is a test to push forward the boundary of the processing power of quantum simulation and further understand the requirements to enhance the accuracy and lay the foundation to explore molecular energy studies.

The best simulations of molecules running on classical computers use complex approximate methods to estimate the lowest energy of a molecular Hamiltonian (quantum mechanical energy operator to describe the interaction between all the electron orbitals and nuclei of the constituent atoms). The lowest energy can depict the molecular structure and how it will interact with other molecules, which is critical for chemists to design new molecules, reactions, and chemical processes for industrial applications. 

We can firmly know that quantum chemistry algorithms should be able to calculate the properties of different molecules accurately. 

 

 

By using Schrodinger's equation, we can describe the possible energy states of a molecule given the set of initial conditions of the system, with the help of hamiltonian. So ground state and excited state energies can be calculated.

 

Calculate the energy is more or equal to the exact energy (it should be the energy of the final product), adjust the parameters to get it, should be in quantum machine learning (the most recent research advocate that we can "simply" use analog quantum machine learning). We can get the state and then measure the expectation values based on the parameters, like energy, position, momentum, etc.

 

Rather than running faster than classical computers, most of the research goes into advancing the theory, software, hardware, some algorithms used to run the molecules on small, noisy devices i.e. variational quantum eigensolver (VQE) to estimate the minimum eigenvalue (ground state energy).

 

VQE Algorithm

To implement the VQE algorithm, preparation of the quantum state with a parametrized quantum circuit called ansatz is needed (an assumption about the form of an unknown function that is made in order to facilitate the solution of an equation or other problem). Parameter corresponds to the rotations of the qubits. Different parameters generate different wave functions. It is used to optimize the wavefunction by parameterizing the parameters, such that Hamiltonian is at its minimum. When the hamiltonian is converged to its minimum, the ground state of the molecular system can be found. It is to create dissociation curves, plots of how the ground state energy varies with the distance between atoms in the molecule. The simulated results should be close to the exact calculated values for both lithium hydride and beryllium hydride, which is the largest molecule simulated on a quantum computer to date.

Physically, IBM's superconducting quantum hardware can help to achieve the simulation by mapping the electronic structure of molecular orbitals onto a subset of 7-qubits quantum processor. Unintuivitvely, electrons exist in orbitals that are better visualized as a cloudy shell enveloping the nucleus. The density of the cloud describes the probability of finding an electron in that region, giving it its characteristic shape. This encoding form orbitals to qubits can simplify simulations of the even larger molecules when the quantum volume of the system has increased. The seven qubit quantum processor is not fully error-corrected and fault-tolerant, the coherence times of the individual qubits last about 50 us. Therefore, the efficiency of the algorithm in terms of the number of qubits used and the number of quantum operations performed is important.

There are more works that should be done on running faster with better code. Apart from VQE, quantum phase estimation offers a way to estimate the eigenvalues of a matrix and therefore could potentially find use in chemistry simulations. However, the very long quantum circuit is required for good accuracy, so it won't find a use for creating accurate simulations on today’s devices, as they are not powerful enough at this stage.

Anyway, VQE is a very good starting point for classical simulations, we need to think in a more quantum way to more powerful computing.

Why taking this approach?

The previous scheme focuses on adapting classical molecular simulation to quantum hardware, which is not effectively taking into account the limited overheads (running costs) of current realistic quantum devices. In contrast, we should ask: how can we extract the maximal quantum computational power out of the current processor?

To tackle this problem, a molecule's fermionic Hamiltonian is transformed into a qubit Hamiltonian, with a new efficient mapping that reduces the number of qubits required in the simulation. 

A new approach to find all possible states-Stochastic Process

A quantum device has been recently released which can simulate all different potential paths of the stochastic process.

The stochastic process describes each step from the previous one can produce different possible outcomes with a probability of occurrence, which makes stock market movement and diffusion gases as stochastic processes possible to be predicted. However, as you can imagine, the simulations become complex in a number of on-going runs. If we have two possibilities to choose from each minute, there will be 14 million possible states as future generated. For less than 24 hours, the number of futures will be more than we can count the atoms in the universe.

Quantum weirdness can acceleration the connectivity among all nodes simultaneously by quantum entanglement, the transfer of information from one qubit to another does or even all connecting nodes do not need to follow a classical displacement path or follow a single path. In the stochastic process, researchers are investigating how to preserve the details of all possible outcome in a coherent state, which keep the quantum information in superposition, while the outcome would be decohered at each time step. This idea was realized in an experiment using photons with information encoded sent to multiple paths at the same time in space-time locations, in which the probability of the occurrence can be weighted.

The researchers demonstrated this property in the principle by simulating a perturbed coin for 3 time-steps, it recreated up to 16 possible states simultaneously in a superposition. could measure the velocity of possible future would diverge with variance in the bias of the coin. Different biases could interfere with the superposition which in turn difference in the measurement result. This simulation technique could be implemented in predictive models for machine learning.

The above mentioned are only the tips of the iceberg, there are still many more applications to be discovered in quantum simulation. We are in the era of harvesting quantum computer advantage, just like what we were positioning in the 1960s. At that time, we could not imagine the computer science could elevate our quality of life to nowadays dimension.

Reference:

1. https://medium.com/qiskit/ive-simulated-a-molecule-with-a-quantum-computer-now-what-aa9e2dfd92c5
2. https://www.ibm.com/blogs/research/2017/09/quantum-molecule/
3. https://medium.com/qiskit/introducing-the-new-qiskit-chemistry-module-and-gradients-framework-for-next-level-quantum-ebaf2be4c1a
4. https://www.ibm.com/blogs/research/2020/10/qiskit-chemistry-module-gradients-framework/
5. https://qiskit.org/textbook/ch-algorithms/quantum-phase-estimation.html
6. https://www.quantumlah.org/about/highlight/2019-04-quantum-simulation-all-futures
7. https://www.quantumlah.org/about/highlight/2019-04-quantum-simulation-all-futures