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Number-conserved subspace encoding for fermionic Hamiltonians, whichexponentially reduces qubit cost, is necessary for quantum advantages invariational quantum eigensolver (VQE). However, optimizing the trade-offbetween qubit compression and increased measurement cost poses a challenge. Byemploying 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 theRandomized Linear Encoding (RLE) algorithm on VQE for $\text{H}_2$ and LiH inthe 6-31G* and STO-3G/6-31G* basis respectively. The resulting subspace circuitexpressivity and trainability are enhanced with less circuit depth and highernoise tolerance.

Unleashing Quantum Simulation Advantages: Hamiltonian Subspace Encoding for Resource Efficient Quantum Simulations