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We discuss encodings of fermionic many-body systems by qubits in the presenceof symmetries. Such encodings eliminate redundant degrees of freedom in a waythat preserves a simple structure of the system Hamiltonian enabling quantumsimulations with fewer qubits. First we consider $U(1)$ symmetry describing theparticle number conservation. Using a previously known encoding based on thefirst quantization method a system of $M$ fermi modes with $N$ particles can besimulated on a quantum computer with $Q=N\log{(M)}$ qubits. We propose a newversion of this encoding tailored to variational quantum algorithms. Also weshow how to improve sparsity of the simulator Hamiltonian using orthogonalarrays. Next we consider encodings based on the second quantization method. Itis shown that encodings with a given filling fraction $\nu=N/M$ and aqubit-per-mode ratio $\eta=Q/M<1$ can be constructed from efficiently decodableclassical 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 considerdiscrete symmetries, and show how to eliminate qubits using previously knownencodings, illustrating the technique for simple molecular-type Hamiltonians.

Tapering off qubits to simulate fermionic Hamiltonians