Active Sampling Sample-based Quantum Diagonalization from Finite-Shot Measurements
This addresses the challenge of reliable quantum computation on near-term devices with limited measurements and noisy states, though it appears incremental as an enhancement to existing SQD methods.
The paper tackles the problem of estimating ground-state energies from finite-shot quantum measurements with imperfect state preparation by proposing AS-SQD, an active learning algorithm that selects basis states using perturbation theory. The method achieves substantially lower absolute energy errors than standard approaches on spin chains up to 16 qubits and demonstrates robustness on IBM Quantum hardware.
Near-term quantum devices provide only finite-shot measurements and prepare imperfect, contaminated states. This motivates algorithms that convert samples into reliable low-energy estimates without full tomography or exhaustive measurements. We propose Active Sampling Sample-based Quantum Diagonalization (AS-SQD), framing SQD as an active learning problem: given measured bitstrings, which additional basis states should be included to efficiently recover the ground-state energy? SQD restricts the Hamiltonian to a selected set of basis states and classically diagonalizes the restricted matrix. However, naive SQD using only sampled states suffers from bias under finite-shot sampling and excited-state contamination, while blind random expansion is inefficient as system size grows. We introduce a perturbation-theoretic acquisition function based on Epstein--Nesbet second-order energy corrections to rank candidate basis states connected to the current subspace. At each iteration, AS-SQD diagonalizes the restricted Hamiltonian, generates connected candidates, and adds the most valuable ones according to this score. We evaluate AS-SQD on disordered Heisenberg and Transverse-Field Ising (TFIM) spin chains up to 16 qubits under a preparation model mixing 80\% ground state and 20\% first excited state. Furthermore, we validate its robustness against real-world state preparation and measurement (SPAM) errors using physical samples from an IBM Quantum processor. Across simulated and hardware evaluations, AS-SQD consistently achieves substantially lower absolute energy errors than standard SQD and random expansion. Detailed ablation studies demonstrate that physics-guided basis acquisition effectively concentrates computation on energetically relevant directions, bypassing exponential combinatorial bottlenecks.