  We present a method for encoding second-quantized fermionic systems in qubits
when the number of fermions is conserved, as in the electronic structure
problem. When the number $F$ of fermions is much smaller than the number $M$ of
modes, this symmetry reduces the number of information-theoretically required
qubits from $\Theta(M)$ to $O(F\log M)$. In this limit, our encoding requires
$O(F^2\log^4 M)$ qubits, while encoded fermionic creation and annihilation
operators have cost $O(F^2\log^5 M)$ in two-qubit gates. When incorporated into
randomized simulation methods, this permits simulating time-evolution with only
polylogarithmic explicit dependence on $M$. This is the first second-quantized
encoding of fermions in qubits whose costs in qubits and gates are both
polylogarithmic in $M$, which permits studying fermionic systems in the
high-accuracy regime of many modes.
