  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.
