Optimal Particle-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 $\mathcal{O}(N\log M)$ 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 encoder and decoder complexities appeared in error correction framework. We demonstrate optimal encoding with random classical parity check code and propose the Fermionic Expectation Decoder for scalable probability decoding in $\mathcal{O}(M^4)$ bases. The protocol is tested with variational quantum eigensolver on LiH in the STO-3G and 6-31G basis, and $\text{H}_2$ potential energy curve in the 6-311G* basis.