Tapering off qubits to simulate fermionic Hamiltonians

TL;DR

This paper introduces methods to reduce qubit requirements in simulating fermionic Hamiltonians by exploiting symmetries, proposing new encodings and improvements for quantum simulation efficiency.

Contribution

It presents novel encoding schemes based on first and second quantization, utilizing classical LDPC codes and graph structures to minimize qubit usage in quantum simulations.

Findings

Encoding with $Q=N ext{log}(M)$ qubits for particle number conservation.

Graph-based encodings can eliminate roughly $M/N$ qubits.

Symmetry exploitation can significantly reduce qubit count for molecular Hamiltonians.

Abstract

We discuss encodings of fermionic many-body systems by qubits in the presence of symmetries. Such encodings eliminate redundant degrees of freedom in a way that preserves a simple structure of the system Hamiltonian enabling quantum simulations with fewer qubits. First we consider $U(1)$ symmetry describing the particle number conservation. Using a previously known encoding based on the first quantization method a system of $M$ fermi modes with $N$ particles can be simulated on a quantum computer with $Q=N\log{(M)}$ qubits. We propose a new version of this encoding tailored to variational quantum algorithms. Also we show how to improve sparsity of the simulator Hamiltonian using orthogonal arrays. Next we consider encodings based on the second quantization method. It is shown that encodings with a given filling fraction $\nu=N/M$ and a qubit-per-mode ratio $\eta=Q/M<1$ can be constructed from efficiently decodable classical LDPC codes with the relative distance $2\nu$ and the encoding rate $1-\eta$. A family of codes based on high-girth bipartite graphs is discussed. Graph-based encodings eliminate roughly $M/N$ qubits. Finally we consider discrete symmetries, and show how to eliminate qubits using previously known encodings, illustrating the technique for simple molecular-type Hamiltonians.