Code-space recovery for sample-based quantum diagonalization beyond native symmetry constraints
Quantum algorithms that sample basis states and diagonalize a Hamiltonian in their span depend on repairing noisy measurement outcomes, using a constraint native to the target problem$\unicode{x2014}$typically particle-number symmetry. Many eigenvalue problems possess no such constraint. Here we show that the constraint can instead be engineered into the sampling representation. Using dual-rail encoding, $|0\rangle\to|01\rangle$ and $|1\rangle\to|10\rangle$, we implement sampling operations in encoded form, making code-space violations local recovery signals for self-consistent, reference-based repair. On transverse- and mixed-field Ising models of up to 36 spin sites, which lack a $U(1)$ symmetry usable for recovery, code-space recovery reached lower Ritz energies than diagonalization over the full observed support of matched unencoded samples while using 38-84% fewer basis states, despite a twofold qubit overhead. The margin was largest for the two-dimensional and 36-site systems and was already present in the first recovery iteration. Recoverable structure can be engineered rather than inherited.