Optimal Particle-Conserved Linear Encoding for Practical Fermionic Simulation

Abstract: Particle-conserved subspace encoding reduces resources for quantum simulations, but a scalable and resource-minimal protocol for $M$ modes and $N$ particles, $\mathcal{O}(N\log M)$ qubits and $\mathcal{O}(Poly(M))$ measurements bases, has remained unknown. We demonstrate optimal encoding with classical parity check code generated by the Randomized Linear Encoder 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.