  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.
