Code-space recovery for sample-based quantum diagonalization beyond native symmetry constraints

Abstract

Sample-based quantum diagonalization (SQD) diagonalizes a Hamiltonian in a compact subspace built from quantum samples, and its performance often relies on recovery procedures that exploit native constraints such as particle-number symmetry. For a broad class of eigenvalue problems, however, no analogous constraint is guaranteed, limiting the applicability of SQD-type recovery. Here, we introduce code-space recovery, which engineers recoverable structure through encoding rather than assuming it in the target problem. Using a dual-rail representation, each logical qubit is mapped to a physical pair, $|0\rangle \to |01\rangle$ and $|1\rangle \to |10\rangle$, making code-space violations in noisy samples detectable and repairable. We combine this encoding with self-consistent recovery and benchmark it on transverse- and mixed-field Ising models with up to 36 spin sites. Despite increased circuit overhead, code-space recovery yields lower projected Ritz energies than unencoded sample-support diagonalization even at smaller projected-basis dimensions, suggesting that engineered recoverable structure can extend SQD beyond native constraints.