Hybrid Quantum-Classical Eigensolver with Real-Space Sampling and Symmetric Subspace Measurements

TL;DR

The paper addresses the challenge of simulating strongly correlated quantum many-body systems whose Hilbert space grows exponentially. It introduces a hybrid quantum-classical eigensolver that combines real-space sampling of tensor-network–bridged quantum circuits with symmetric-subspace measurements to compress the effective Hilbert space while preserving essential correlations. A key contribution is replacing deep quantum circuits with RDM-driven isometries (DMRG-inspired block truncation) and performing energy evaluations in a symmetry-reduced basis, enabling efficient, scalable optimization via METTS-like sampling. Benchmark results on the periodic $J_1$-$J_2$ Heisenberg model in 1D and 2D show absolute energy errors as low as $10^{-5}$ with bond dimensions as small as $D\le 6$, illustrating high accuracy and potential for scalable quantum simulations on near-term hardware.

Abstract

We propose a hybrid quantum-classical eigensolver to address the computational challenges of simulating strongly correlated quantum many-body systems, where the exponential growth of the Hilbert space and extensive entanglement render classical methods intractable. Our approach combines real-space sampling of tensor-network-bridged quantum circuits with symmetric subspace measurements, effectively constraining the wavefunction within a substaintially reduced Hilbert space for efficient and scalable simulations of versatile target states. The system is partitioned into equal-sized subsystems, where quantum circuits capture local entanglement and tensor networks reconnect them to recover global correlations, thereby overcoming partition-induced limitations. Symmetric subspace measurements exploit point-group symmetries through a many-to-one mapping that aggregates equivalent real-space configurations into a single symmetric state, effectively enhancing real-space bipartition entanglement while elimilating redundant degrees of freedom. The tensor network further extends this connectivity across circuits, restoring global entanglement and correlation, while simultaneously enabling generative sampling for efficient optimization. As a proof of concept, we apply the method to the periodic $J_1\!-\!J_2$ antiferromagnetic Heisenberg model in one and two dimensions, incorporating translation, reflection, and inversion symmetries. With a small matrix product state bond dimension of up to 6, the method achieves an absolute energy error of $10^{-5}$ for a 64-site periodic chain and a $6\times6$ torus after bond-dimension extrapolation. These results validate the accuracy and efficiency of the hybrid eigensolver and demonstrate its strong potential for scalable quantum simulations of strongly correlated systems.

Figures (4)

• Figure 1: Wavefunctions of the Hybrid Quantum-Classical Eigensolver. a, Tensor-network-bridged parallel quantum circuits efficiently generate real-space samples $a_{\rm real}$ according to the distribution $p(a_{\rm real})$, ensuring that the corresponding symmetric components $a_{\rm symm}$ follow $\widetilde{p}(a_{\rm symm}) = \sum_{g \in G} p(a_{\rm real}\,|\,a_{\rm repr}=g a_{\rm real})$. Here, $D$ denotes the MPS bond dimension, and $\chi$ represents the renormalized block degrees of freedom. b, An effective isometric transformation can be inserted in place of the quantum circuits, constructed from the dominant eigenvectors of the reduced density matrix obtained from an exactly solvable system.
• Figure 2: The METTS sampling algorithm for real-space configurations $a_{\rm real}$. a, Contraction of the RDM $\rho(s_{1}, s_{1}')$, whose diagonal elements determine the sampling probabilities $p(\uparrow_{1}) = \rho(\uparrow_1,\uparrow_1)/[\rho(\uparrow_1,\uparrow_1) + \rho(\downarrow_1,\downarrow_1)]$ and $p(\downarrow_1) = 1 - p(\uparrow_1)$. b, Contraction of the conditional RDM $\rho(s_{i}, s_{i}' \,|\, s_{i-1}, \cdots, s_{1})$, conditioned on the previously sampled spins. In both panels, the right-hand diagrams illustrate the simplification arising from the unitary contraction of the quantum circuits, $C_{i}^{\dagger} C_{i} = I_{\chi \times \chi}$.
• Figure 3: Absolute energy error for the one-dimensional $J_1\!-\!J_2$ periodic chain. The relative error is defined as $\epsilon(N) \equiv |[E(N) - E_{\rm DMRG}]/E_{\rm DMRG}|$, evaluated for chains of length $N = 32$ and 64 using MPS characterized by block size $b$, bond dimension $D$, and renormalized block basis dimension $\chi$. a, Singlet ground state with quantum numbers $(k,p,z) = (0,1,1)$ at $g = 0.2$. b, Lowest triplet excitation with quantum numbers $(k,p,z) = (\pi,-1,-1)$ at $g = 0.25$.
• Figure 4: Variational energy per site versus inverse bond dimension. Results of the hybrid quantum-classical framework for a one-dimensional $J_1\!-\!J_2$ chain ($N = 64$) and a two-dimensional $6 \times 6$ torus, obtained using MPS with block size $b$, bond dimension $D$, and renormalized block basis dimension $\chi$. Linear energy extrapolations (red dashed lines) use the three largest $D$ values, while black dotted lines indicate reference energies from ED and DMRG. a, Singlet state with quantum numbers $(k,p,z) = (0,1,1)$ at $g = 0.2$, using $b = 4$, $D = 2\!-\!6$, and $\chi = 11$. b, Triplet state with $(k,p,z) = (\pi,-1,-1)$ at $g = 0.25$, using $b = 4$, $D = 2\!-\!6$, and $\chi = 16$. c, Singlet state of a $6\times6$ lattice with $(k_x,k_y,p_x,p_y,\sigma_1,\sigma_2,z) = (0,0,1,1,1,1,1)$ at $g = 0.5$, using $b=2\times 2$, $D = 2,4,6$ and $\chi = 16$.