Basis Adaptive Algorithm for Quantum Many-Body Systems on Quantum Computers
We introduce a Basis Adaptive (BA) algorithm for hybrid quantum-classical simulation of correlated quantum many-body systems. Starting from a small set of physically motivated bitstrings, the algorithm iteratively applies a single-step first-order Trotterized circuit on a quantum processor, filters the sampled configurations by enforcing $U(1)$ spin conservation and lattice reflection symmetry, and classically diagonalizes the Hamiltonian in the resulting reduced Hilbert space. This design avoids the variational optimization overhead of VQE, the deep coherent circuits required by QPE, and the symmetry-violating subspaces that arise in SKQD. The ground-state energy error is bounded analytically by $\sqrt{8}\,\|H\|\left(1-\sqrt{\alpha_{D_T}}\right)^{1/2}$, where $\alpha_{D_T}$ is the probability weight captured by the $D_T$ sampled basis states. This bound connects algorithm performance directly to ground-state sparsity and explains the observed accuracy hierarchy across different phases. Benchmarked on the spin-$1/2$ Heisenberg XXZ chain (up to $N=62$ qubits on the IBM Heron processor), the algorithm achieves a $3.5\%$ energy error in the gapped Neel phase ($\Delta=2.0$) and below $0.5\%$ at the ferromagnetic boundary ($\Delta=-1.0$). The accuracy degrades to $28.7\%$ in the strongly quasi-long-range-ordered regime ($\Delta=0.5$). Spin-spin correlation functions are reproduced across all regimes, confirming that symmetry-filtered real-time sampling provides a practical and noise-resilient pathway to ground-state properties on near-term quantum hardware.