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We present a method for encoding second-quantized fermionic systems in qubitswhen the number of fermions is conserved, as in the electronic structureproblem. When the number $F$ of fermions is much smaller than the number $M$ ofmodes, this symmetry reduces the number of information-theoretically requiredqubits 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 annihilationoperators have cost $O(F^2\log^5 M)$ in two-qubit gates. When incorporated intorandomized simulation methods, this permits simulating time-evolution with onlypolylogarithmic explicit dependence on $M$. This is the first second-quantizedencoding of fermions in qubits whose costs in qubits and gates are bothpolylogarithmic in $M$, which permits studying fermionic systems in thehigh-accuracy regime of many modes.

Second-quantized fermionic operators with polylogarithmic qubit and gate complexity