Accuracy vs Assumption in the Quantum Age - Where Chemistry Meets Machine

Everyone seems to agree that the absolute understanding and control of elementary and composite particles, as well as atoms and molecules, is the pot of gold at the end of the rainbow - and I believe we can get there. 

Unfortunately, after many years and many more millions of dollars invested, the quantum industry at large still seems to only see a shimmer in the distance, with no solid proof of gold, some clouds on the horizon and fears of quantum winter growing.  

The cause for this malaise, in my opinion, is in the use of an approximation by the broader scientific community for nearly a century, the Born-Oppenheimer (BO) Approximation. This method served a real practical purpose in 1926, in a world with slide rules and pencils, yet somehow, its use has persisted in the modern age of computing and supercomputing. Even in the quantum computing era of today, it remains the preferred method as the status quo in today’s quantum computing and computational chemistry industries.  

It may seem self-evident that one must account for the nuclei in order to accurately understand complex chemistry. However, almost all of quantum chemical modeling stems from the use of BO methods - a mass approximation - radically inhibiting our ability to describe fundamental physical phenomenon.

As debates heat up regarding the attributes, benefits, and challenges of the three main qubit types: photonic, superconducting, and ion traps, I remain an ardent believer that all three systems are fundamentally handicapped by theory. The only real way to see inside the mystery box before us, regardless of the chemical system being used as a qubit, is to start thinking explicitly about all particle interactions from the beginning, especially electronic interactions with nuclei. I am not the first to make this argument. The idea of performing a calculation with explicit particle correlation was alien technology mid-20th century due to the massive scale of the calculation.  Today, it is within reach. I have spent my career understanding and working with explicitly correlated function spaces and am excited to see the science implemented at scale.

Regarding quantum applied chemistry, again, the major hurdle that needs to be overcome in order to realize the accurate implementation of quantum chemistry is the same BO approximation. To elaborate: the BO world is one where electrons “see” or “experience” nuclei as having an infinite mass, a sort of black hole without complicated math - completely ignoring the nuclei’s effect on all other particle interactions. Furthermore, this approximation assumes that the motions of the nuclei and the motions of the electrons are separable and that the nuclei do not behave quantum mechanically, i.e., nuclei do not exhibit wave-like behavior.

This mass approximation has major consequences when trying to understand particle correlations. The effect not being accounted for is called mass polarization.

Consider wave-particle duality and temperature. Temperature, in the most basic chemical definition, is the average value of the motions of all of the particles in the system or said another way, the average value of the kinetic energy of all of the particles in a system. The kinetic energy of each particle is defined in terms of the mass of that particle. This, to me, seems like an acceptable definition of temperature where temperature and mass have some relationship (or connectivity). Cool, right? It seems that mass and kinetic energy are fundamental to our understanding of how heat is “distributed” in any system through wave-like behavior. 

To apply chemical insight to my point: “In electrically conductive solids, the Wiedemann-Franz law requires the electronic contribution to thermal conductivity to be proportional to electrical conductivity. Violations of the Wiedemann-Franz law are typically an indication of unconventional quasiparticle dynamics, such as inelastic scattering, or hydrodynamic collective motion of charge carriers, typically pronounced only at cryogenic temperatures. We report an order-of-magnitude breakdown of the Wiedemann-Franz law at high temperatures ranging from 240 to 340 kelvin in metallic vanadium dioxide in the vicinity of its metal-insulator transition. Different from previously established mechanisms, the unusually low electronic thermal conductivity is a signature of the absence of quasiparticles in a strongly correlated electron fluid where heat and charge diffuse independently.” 

This excerpt implies that heat is nuclear motion, not electronic motion, at both temperature extremes. I find this result to be particularly exciting because the effects that are being leveraged at cryogenic temperatures are also being seen at high temperatures as well - near the metal-insulator transition. My hope is that these effects exist at room temperature for some materials without creating unnecessary electronic heat - this would be especially exciting news for our computing industry.  Perhaps there is a way to tap into these nuclear motion states at room temperature by not assuming the BO approximation.

Nearly all quantum algorithms hinge on one equation, where the mass of all particles is assumed to be equal to 1, the mass of the electron, and that nuclei are massless point particles (only occupying a single point in 3D-space), they then solve an electronic Hamiltonian to obtain eigenvalues of the ground and a few excited states, then they shift the nuclei slightly, and resolve the electronic Hamiltonian again creating a potential energy surface - a reiterative process called Quantum Monte Carlo Simulation. This method is now being married with neural net type artificial intelligence algorithms.

 The reiterative method of “walking” nuclei does not produce the wave-like properties of nuclei, and in turn, does not represent quantum heat motion within a system, even if we are describing the lowest energy state of the system.

Even the lowest energy state of a system still has some inherent heat compared to the entire universe of all the particles that exist. The fact is simple: electrons couple to the motions of the nuclei, and we are not going to cool (or freeze) our way out of nuclear motions - no matter how hard we try. What we need to do is learn how to control for nuclei (composite particles), and the electrons will follow. Let’s stop boxing electrons in an infinite potential well and instead, explore the world in which they really live, transiting nuclei as they commingle in an optimized current.

The BO approximation is actively being implemented in OpenFermion, a computational chemistry package that Google is developing, as well as IBM’s Qiskit Aqua, and Microsoft’s & Pacific Northwest National Lab’s NWChem implementation on Azure with QSharp, just to name a few. Google’s main goal in OpenFermion is “minimizing the amount of domain expertise required to enter the field” as stated in their abstract of an article reported on ArXiV. The BO approximation is presented in Google’s equation number 1. 

While the BO method has been an innovative and useful approach to understanding and predicting electron structure in molecules over the last century, it leaves no explanation of what the nuclei are actually doing today. I say it’s high time to retire this overused shortcut. 

As a thought experiment, let’s formulate the problem that falls from the BO approximation in terms of the uncertainty principle, one of the postulates; if I know exactly where a quantum particle is I know nothing about its momentum and therefore, know nothing about its kinetic energy. How can we then, in the Born-Oppenheimer approximation, even define temperature as the average kinetic energy if we do not even know how to compute that energy for nuclei? Nuclear heat is not currently represented in a molecule and there will always be heat present as long as there exist different particle types in a system, i.e., particles with different masses and a different set of quantum numbers will, in fact, have a minimum heat experience.

We can draw similar conclusions in the human experience: if I observe the action of another (an event at some point in space and time), yet have no way of exactly knowing or measuring the intention behind the action (the momentum), do I really know the truth about my observed reality? This might be a bit of an extrapolation of quantum mechanics on the human condition but I think it works well especially if we are interested in uncovering the truth about our existence in this realm. As it stands now, the Postulates of Quantum Mechanics is just that: a set of rules assumed to be fact-based on conventional reasons or beliefs. If we continue to modify our understanding of the postulates and how these rules apply to all particle types including atoms and molecules with a very large number of particles (with heavier mass, and with complex components), beyond the nanoscale, then we have a better shot at being able to create our existence with real matter in an artificial device that could be built on the cellular level that interacts with our chemical pathways affecting our health, emotional wellbeing, and beyond. Let alone the benefits with regard to material design.

There exist methods for allowing the nuclei to act quantum mechanically and have the electrons correlate their motions accordingly which yields optimal efficiency of superconductivity for computing that is not currently being utilized by industry. I am most excited about predicting states of matter, material design, and drug discovery. I am disappointed, however, that the industry at large still refuses to acknowledge the meaningfulness of the mass of the nucleus when dealing with particle physics.

The mass of the particles in a system becomes increasingly more important when there are spin-orbit interactions and relativistic effects to consider, especially if we want to use excited states with non-zero orbital angular momentum. The BO approximation inhibits our use of relativistic corrections which will come into play for superconductivity and magnetically trapped atomic and molecular ions. 

I propose using a quantum applied chemistry algorithm with the following features: 

1. Explicit correlation between all particles - entanglement is included. I am working on ideas of how to overcome the factorial dependence. 
2. All the masses of nuclei are included and therefore treats nuclei quantum mechanically through the kinetic energy operator. Without including the mass of nuclei they cannot act mechanically through kinetic energy.
3. Use Young tableaux to build the symmetry requirements for all particle sets: fermions and bosons. Think of this as how to build properly behaving or symmetrized particles - qubits.
4. Include non-zero angular momentum states in addition to the ground state.

If we are able to map all the motions and interactions of the molecule (both nuclear and electronic on equal footing) to the qubit of a particular choice, then we should be able to image any atom or molecule using a quantum computer.

To learn more about pursuing explicitly correlated methods, visit www.odestar.com.