Efficient Implementation of the Fermionic Schrodinger Equation with Adiabatic Quantum Computation*

One obvious goal of quantum computing that dates back to its inception is to solve for the ground state of the Schrödinger equation.   The problem addressed here is the first quantization description of a position space multi-particle multi-dimensional Schrödinger equation on a periodic lattice where the particles are identical fermions.  

The objective is to maintain an exponential advantage over classical computing. In Chang, McElvain, Rrapaj, Wu, PRX Quantum 3(2), 020356,

the authors present a solution for distinguishable particles.  The work described in this talk is found in arXiv:2309.08101.

I associate a specific encoding for each anti-symmetric multi-particle state.  The encoding depends on an ordering of the particles according to their position in the multi-dimensional space.    Fermion exchange symmetry complicates the description of derivative operators like the Laplacian because the motion of a fermion to a neighboring lattice position in two or more dimensions can change the ordering the encoding depends on.  The key to success is that it is possible to bound the the circuit size required to restore the proper ordering. 

The key result is that the number of qubits and circuit size for the Laplacian scale logarithmically with the number of position states and linearly with the number of particles.