Second-Quantized Fermionic Operators with Polylogarithmic Qubit and Gate Complexity

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.

Automated summary

This paper presents a method for encoding second-quantized fermionic systems in qubits. The encoding reduces the number of required qubits significantly when the number of fermions is much smaller than the number of modes. By incorporating this encoding into randomized simulation methods, it enables polylogarithmic simulation of fermionic systems with low dependence on the number of modes.