Basis Adaptive Algorithm for Quantum Many-Body Systems on Quantum Computers

A new basis adaptive algorithm for hybrid quantum-classical platforms is introduced to efficiently find the ground-state (gs) properties of quantum many-body systems. The method addresses limitations of many algorithms, such as Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (QPE) etc by using shallow Trotterized circuits for short real-time evolution on a quantum processor. The sampled basis is then symmetry-filtered by using various symmetries of the Hamiltonian which is then classically diagonalized in the reduced Hilbert space. We benchmark this approach on the spin-1/2 XXZ chain up to 24 qubits using the IBM Heron processor. The algorithm achieves sub-percent accuracy in ground-state energies across various anisotropy regimes. Crucially, it outperforms the Sampling Krylov Quantum Diagonalization (SKQD) method, demonstrating a substantially lower energy error for comparable reduced-space dimensions. This work validates symmetry-filtered, real-time sampling as a robust and efficient path for studying correlated quantum systems on current near-term hardware.

💡 Research Summary

The paper introduces a hybrid quantum‑classical “Basis‑Adaptive” (BA) algorithm designed to obtain high‑accuracy ground‑state properties of quantum many‑body systems on near‑term quantum hardware. Recognizing the limitations of existing approaches—Variational Quantum Eigensolver (VQE) suffers from barren‑plateau optimization problems for strongly correlated states, while Quantum Phase Estimation (QPE) requires deep, fault‑tolerant circuits, and Sampling Krylov Quantum Diagonalization (SKQD) can generate noisy, symmetry‑violating subspaces—the authors propose a method that combines shallow real‑time evolution, symmetry‑filtered sampling, and classical diagonalization in a reduced Hilbert space.

The algorithm proceeds in five steps. First, a small set of physically motivated computational basis states (bit‑strings) is prepared; for the spin‑½ XXZ chain these are the two Néel configurations. Second, each basis state is evolved independently on a quantum processor using a first‑order Trotter decomposition of the Hamiltonian with a short time step Δt≈0.25, implemented as a sequence of two‑qubit gates acting on neighboring spins. Third, the evolved states are measured in the computational basis, and the resulting bit‑strings are united across all initial states. A symmetry filter then discards any strings that violate conserved quantities: total Sᶻ, the U(1) particle‑number symmetry, and the chain reflection symmetry. The remaining strings, together with their symmetry‑related partners, define a symmetry‑adapted subspace. Fourth, the Hamiltonian matrix is constructed within this subspace on a classical computer and exactly diagonalized; the lowest eigenvalue and eigenvector provide an updated estimate of the ground‑state energy and wavefunction. Fifth, if the energy has converged, physical observables (spin‑spin correlations, fidelity, etc.) are computed; otherwise, the most probable basis states from the current eigenvector are selected as the seed for the next iteration.

The authors benchmark the BA algorithm on the antiferromagnetic spin‑½ XXZ chain for system sizes up to N=24 qubits, using IBM’s Heron processor. Starting from only two initial bit‑strings, they find that after just two iterations the ground‑state energy error drops below one percent when roughly 96 basis states (≈18 % of the full Hilbert space) are retained. Increasing the number of measurement shots (20 k vs. 40 k) reduces statistical fluctuations and further improves accuracy. Comparisons with exact diagonalization (ED) show sub‑percent energy errors, wavefunction fidelities exceeding 0.99, and spin‑spin correlation functions that match ED results across the entire anisotropy range (Δ < −1, −1 < Δ ≤ 1, Δ > 1). When contrasted with SKQD at comparable reduced‑space dimensions, the BA method achieves roughly a factor of two to three lower energy error, demonstrating superior efficiency of the symmetry‑filtered sampling.

Key insights include: (i) shallow Trotterized evolution suffices to generate the most relevant basis states for the low‑energy sector, avoiding deep circuits required by QPE; (ii) enforcing physical symmetries at the sampling stage dramatically stabilizes the reconstructed subspace and mitigates noise‑induced leakage; (iii) the hybrid workflow leverages the strengths of quantum hardware (state preparation and short‑time dynamics) while relying on classical exact diagonalization for precision, making it well‑suited to current noisy intermediate‑scale quantum (NISQ) devices.

The paper also discusses practical considerations: the choice of Δt and Trotter order affects the balance between circuit depth and basis‑state generation; the number of initial basis states (m₀) and measurement shots (Mₛ) control the growth rate of the reduced space; and while the method scales favorably for one‑dimensional models, extensions to higher dimensions or fermionic systems will require careful management of the subspace size. Future directions suggested include adaptive Δt schedules, higher‑order Trotter schemes, and application to more complex Hamiltonians such as Hubbard or frustrated spin models.

In summary, the Basis‑Adaptive algorithm provides a robust, hardware‑efficient pathway for studying strongly correlated quantum many‑body systems on near‑term quantum computers, achieving high accuracy with modest quantum resources by combining real‑time evolution, symmetry‑filtered sampling, and classical diagonalization.