Advancing Beyond Dreaming Or Giving Hopes About What Will Come 20-50 Years Later
All the problems in Physics, Chemistry, Biology boil down to finding just one thing – how does a given quantum system evolve over time? We could do it by simulating the entire process of quantum dynamics of multiple bodies involved in the system. Or we could use something like the Quantum Monte Carlo methods to estimate the ground state of the system without having to simulate the dynamics of the entire system.
If we try to do the first way, we will have to build Quantum Computer Hardware to simulate the dynamics. Because even with our supercomputers we cannot simulate Quantum Systems with more than 50-100 odd bodies.
The second way is to use Classical Quantum Monte Carlo methods to do the same thing.
There is actually a third method which nobody knows about, which has been invented at Automatski. Which is a breakthrough and is the topic of this article.
The Quantum Many Body Problem
Solving the many body problem for the electronic structure of molecules and solids has been one of the biggest challenges in quantum mechanics in the last century. Similar to the many body problem in classical mechanics, it cannot be solved analytically, but different iterative schemes exist, which in principle allow an exact numerical solution. They are based on variational calculus and on the knowledge of the full set of wave functions for a given system. Most of them require a huge amount of computational time and have a very unfavorable scaling with system size. This restricts such calculations to systems of roughly ten electrons. (This is only true for the most exact methods. Today it is possible to perform simulations based upon perturbation theory containing hundreds of electrons.)
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. Microscopic here implies that quantum mechanics has to be used to provide an accurate description of the system. A large number can be anywhere from three to infinity (in the case of a practically infinite, homogeneous or periodic system, such as a crystal), although three- and four-body systems can be treated by specific means (respectively the Faddeev and Faddeev–Yakubovsky equations) and are thus sometimes separately classified as few-body systems. In such a quantum system, the repeated interactions between particles create quantum correlations, or entanglement. As a consequence, the wave function of the system is a complicated object holding a large amount of information, which usually makes exact or analytical calculations impractical or even impossible. Thus, many-body theoretical physics most often relies on a set of approximations specific to the problem at hand, and ranks among the most computationally intensive fields of science.
Quantum Monte Carlo
Quantum Monte Carlo encompasses a large family of computational methods whose common aim is the study of complex quantum systems. One of the major goals of these approaches is to provide a reliable solution of the quantum many-body problem.
A Quantum Model of D dimensions maps to a Classical Model of D+1 dimensions using an imaginary temporal dimension. And hence can be sampled from reasonably easily.
QMC scales with O(N3) and the calculations can be distributed over a cluster very easily. But to get practical accuracies it requires exponential effort.
QMCPack is a distributed solver/tool for Quantum Monte Carlo Applications.
What are the largest problems mankind can solve today?
About ~1000 electrons using a lot of approximations which may or may not work.
These gentlemen below claim to have simulated a 30,000 particle system. We have not verified their claims due to lack of details.
Some Types of Monte Carlo Methods
- Sequential Monte Carlo
- Markov Chain Monte Carlo
- Gibbs Sampler
- Cluster Sampling
- Data Driven MCMC
- Langevin Monte Carlo
- Hamiltonian Monte Carlo
- Variational Monte Carlo
- Path Integral Monte Carlo
- Reptation Quantum Monte Carlo
- Self Healing Diffusion Monte Carlo
- Fermion Monte Carlo
Basically using these Monte Carlo Methods one can find the exact solutions for the Schrodinger’s Equations for the Ground States. Without simulating the Quantum Dynamics in its entirety.
Applications of Monte Carlo
Monte Carlo Methods have been used for many tasks such as…
- Simulating a system and its probability distribution
- Estimating a quantity through Monte Carlo Integration
- Optimizing a target function to find its modes (maxima or minima)
- Learn parameters from a dataset to optimize some loss function and learn the model
- Visualizing the energy landscape of a target function
Limitations of Quantum Monte Carlo
First of all QMC was created before Quantum Computers. And it is a 100% Classical Algorithm. It is not even a Hybrid Algorithm. It doesn’t involve using any Quantum Computer for any of its calculations.
QMC works only for Stoquastic Hamiltonians i.e. Hxy <= 0 for x!= y
Which means QMC only works if there are no negative terms in the Hamiltonian. If all terms are positive then we can use QMC by using Classical Sampling for our work. This is also called the [fermion] sign problem. If there are negative terms then when they interact with positive terms they cannot be integrated because they cancel each other out. They would require quantum entanglement to compute which cannot be done classically and would require a quantum computer. Note: Fermions have this sign problem. Hence QMC cannot be used for Fermions.
Examples of recent attempts to solve the Fermion Sign Problem using Time Reversal…
Where can I learn more?
The Infinity & Beyond Algorithm
It works for both Bosons and Fermions. It doesn’t have a problem with the sign problem. It computes ‘classically’ using entanglement and superposition. The two quantum principles used in Quantum Computers. Hence it can simulate arbitrary Hamiltonians.
Whatever be the underlying model, we use absolutely no Hamiltonian approximation’s, decomposition’s or reduction’s.
It is a Deterministic order O(N3) solution.
It is built over our Polynomial Time NP-Hard Global Optimization Solvers.
The demo’s are available today by invitation for Qualified Investors and Customers.
The Ground State calculation of a 100 Qubit/Particle Model, with full quantum state discovery plus the solution took ~21,810 seconds on an Asus Laptop with a quad core Intel i7 processor (3 year old machine)
The Infinity & Beyond Algorithm can be scaled to almost all systems of practical interest to us.
Time to light a cigar…
This is our website http://automatski.com