Unleashing Quantum Simulation Advantages: Hamiltonian Subspace Encoding for Resource Efficient Quantum Simulations
Number-conserved subspace encoding for fermionic Hamiltonians, which exponentially reduces qubit cost, is necessary for quantum advantages in variational quantum eigensolver (VQE). However, optimizing the trade-off between qubit compression and increased measurement cost poses a challenge. By employing the Gilbert-Varshamov bound on linear code, we optimize qubit scaling $\mathcal{O}(N\log_2M)$ and measurement cost $\mathcal{O}(M^4)$ for $M$ modes $N$ electrons chemistry problems. The compression is implemented with the Randomized Linear Encoding (RLE) algorithm on VQE for $\text{H}_2$ and LiH in the 6-31G* and STO-3G/6-31G* basis respectively. The resulting subspace circuit expressivity and trainability are enhanced with less circuit depth and higher noise tolerance.