Excitation Spectra of the Frustrated Spin-1/2 J 1 - J 2 Model via Dual Spin-Sector Variational Quantum Eigensolvers

Finding excited states and excitation gaps in frustrated spin systems using Variational Quantum Eigensolvers (VQE) is complicated by state-ordering inversions. In the frustrated spin-1/2 J 1 - J 2 Heisenberg model, standard single-sector algorithms like Variational Quantum Deflation (VQD) in S z=0 fail when J 2/J 1 < 0.25. In this regime, the lowest-lying excitation in S z=0 is a Triplet (S=1), which lies below the Excited Singlet (S=0). Standard deflation projects out the ground singlet and converges onto the Triplet state, misidentifying it as the Excited Singlet. Here, this work presents a Dual Spin-Sector VQE framework that resolves these state-ordering inversions applied to frustrated J 1-J 2 models on 1D spin chains and 2D rectangular lattices. Firstly, by optimizing the ground Triplet state in the S z=1 sector (where singlet states cannot exist), mapping it into S z=0 via the total spin-lowering operator S- = sum i S i-, and evaluating a dual-sector penalty cost function, the Excited Singlet (E S) is isolated without variational ambiguity. Numerical state-vector simulations confirm exact convergence to machine precision with unit overlap fidelity. The method captures key physical features, including the gapless phase, the gapless-gapped critical point at J 2/J 1 about 0.2411, the Majumdar-Ghosh dimer point (J 2/J 1 = 0.50), and the 2D non-magnetic frustrated regime (J 2/J 1 about 0.40 - 0.60). Finite-shot sampling simulations and gate-scaling analytics up to N=100 qubits demonstrate the resource efficiency and shot resilience of the algorithm for near-term quantum execution.