Quantum Many Body Hamiltonians stored symbolically in polynomial memory

The quantum many particle Hamiltonian contains many undiscovered new physics, however remains intractable due to its large size. Traditional methods of storing these Hamiltonians in memory will consume exponential amount of memory. If we want to simulate N particles, where each particle can only occupy two states, then the numerical size of the Hamiltonian stored in traditional sparse matrix form will consume O(2^N) number of bits. The largest computer we can build is limited by the number of atoms in the observable universe. If every atom in the observable universe is assigned one bit of memory storage, then the maximum number of particles that can be simulated is 270. A typical unit cell simulated in materials science can contain from 100 to 1000 electrons. This illustrates the need for a better method for quantum many particle simulations. We will present a numerical method of storing the quantum many particle Hamiltonian in polynomial memory and still retain useful diagonalization tools for polynomial computation time.