Optimal number-conserved linear encoding for practical fermionic simulation

Number-conserved subspace encoding reduces resources needed for quantum simulations, but scalable complexity trade-off bounds for 𝑀 modes and 𝑁 particles with 𝒪⁡(𝑁⁢log⁡𝑀) qubits have remained unknown. We study qubit-gate-measurement trade-offs through the lens of classical/quantum error correction complexity and develop a framework of fermionic gate and measurement complexity based on classical encoder/decoder appearing in the error correction framework. We demonstrate optimal encoding with random classical parity check code and propose the Fermionic Expectation Decoder for scalable probability decoding in 𝒪⁡(𝑀4) bases. The protocol is tested with variational quantum eigensolver on LiH in the STO-3G and 6-31G bases, and H2 potential energy curve in the 6-311G* basis.