Second-quantized fermionic operators with polylogarithmic qubit and gate complexity

TL;DR

This paper introduces a novel encoding method for fermionic systems on qubits that significantly reduces resource requirements, enabling efficient high-accuracy simulations of large fermionic systems with polylogarithmic complexity.

Contribution

The authors develop the first second-quantized fermionic encoding with polylogarithmic qubit and gate complexity in the number of modes, leveraging fermion number conservation.

Findings

Reduces qubit count from Θ(M) to O(F log M) for F fermions in M modes.

Achieves O(F^2 log^4 M) qubits and O(F^2 log^5 M) gates for fermionic operators.

Enables time-evolution simulation with polylogarithmic dependence on M.

Abstract

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.