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

TL;DR

Problem: accurately determining ground-state properties of quantum many-body systems is hindered by exponential Hilbert-space growth and circuit-depth limits on near-term quantum devices. Approach: the Basis Adaptive (BA) algorithm builds a symmetry-filtered, reduced basis via short-time real-time evolution on a quantum processor and resolves the ground state by classical diagonalization in this subspace. Contributions: demonstrated on the spin-$\tfrac{1}{2}$ XXZ chain up to $N=24$ qubits on IBM Heron, achieving sub-percent ground-state energy errors and high fidelity, with accurate spin-spin correlations, and outperforming SKQD at comparable reduced-space sizes. Significance: offers a practical, hardware-friendly route to studying strongly correlated quantum systems on current devices and can be extended to other models and symmetries.

Abstract

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.

Figures (3)

• Figure 2: Schematic workflow of the Basis Adaptive Algorithm. An initial set of $m_{0}$ bitstrings is evolved using the Trotterized time-evolution operator on a quantum device and measured to generate a new set of sampled bitstrings. These bitstrings are then expanded by adding symmetry-related configurations, forming a symmetrized subspace in which the Hamiltonian is classically diagonalized. The $m_{i}$ most probable bitstrings from the resulting eigenstate are selected and used to define the basis set for the next iteration.
• Figure 3: (a) Percentage error in the ground-state energy as a function of the maximum number of retained basis states ($m_1$), shown for different values of $M_s$. Increasing $m_1$ systematically improves the ground-state accuracy. (b) Percentage error as a function of $m_1$ for several values of $\Delta$, computed using $M_s = 40\,\text{K}$. The results highlight how enlarging the basis enhances the ground-state accuracy.
• Figure 4: For different $\Delta$ values correlation function is shown for 24 number of qubits with $M_s=40K$.