Symmetry-Adapted Eigensolver from Real-Space Sampling of Tensor-Network-Bridged Quantum Circuits

The gist: We propose a hybrid quantum-classical eigensolver that combines real-space sampling of tensor-network-bridged quantum circuits with symmetric subspace measurements to efficiently simulate strongly correlated quantum many-body systems.

How it works

1. The system is partitioned into equal-sized subsystems, where quantum circuits capture local entanglement and tensor networks reconnect them to recover global correlations (Page 1).

2. Real-space sampling exploits the generative capability of tensor-backed parallel quantum circuits enabling efficient exploration of spatial configurations (Page 2).

3. Symmetric subspace measurements project the quantum state onto a drastically reduced symmetric Hilbert space, effectively filtering out irrelevant degrees of freedom while establishing long-range entanglement between distant real-space components (Page 2).

4. The core idea behind this hybrid approach is a many-to-one mapping defined as pe(asymm) = P g∈G p(areal arepr = gareal), which aggregates the probabilities of all symmetry-equivalent real-space configurations, assigning their collective contribution to the corresponding symmetric basis state asymm (Page 2).

5. This construction links the real-space distribution function ϕ(areal) into a symmetry-constrained wave function ψ(asymm) via ln ψ(asymm) = 1/2 ln X areal∈garepr ϕ(areal)2, which means the amplitude of asymm⟩ equals the square root of the total probability weight aggregated from all configurations in the class (Page 4).

Key Components and Techniques

- Wavefunction Construction: The real-space distribution function p(areal) is generated by a set of parallel quantum circuits Ci and a MPS backbone Bi (Page 3). The final wave-function amplitude is given by a complex expression involving these components, where ϕ(areal) = X...B [a¯nb] ·... B γnb µnb−1Cγnb (Page 3). This structure is used to define the target symmetric subspace based on symmetry-constrained superpositions of Ising configurations (Page 4). The amplitude of asymm⟩ equals the square root of the total probability weight aggregated from all configurations in the class, while its phase is inherited from the representative configuration (Page 4). The relative phases between distinct representative states are encoded by the tensor-backed quantum circuits (Page 2). This intrinsic correspondence means that relative phases among symmetry-related configurations become redundant, as they contribute only through their combined amplitudes (Page 4). The method is initialized from an MPS optimized within a traditional real-space algorithm for a good starting point (Page 4). The relevant subspace is defined by the χ dominant eigenvectors of the reduced density matrix ρb = Tr¯b ψED⟩⟨ψED, which alleviates the need for deep and highly parameterized quantum circuits (Page 7). This substitution embeds entanglement into a compact, low-rank representation, effectively reproducing DMRG’s block-wise truncation mechanism (Page 7). The relevant subspace associated with ρb remains nearly invariant under shifts of the block position along a chain of length N due to translational symmetry (Page 7). The number of variational parameters equals 2χD + (N/b − 2)χD 2 (Page 9). This method inherits DMRG’s efficiency in identifying the most relevant subspace while maintaining compatibility with shallow-depth quantum circuit architectures (Page 7). The energy and its gradient are evaluated within the symmetric subspace, as shown in Eq. (10), which highlights that sampling is performed in real space, while energy and its gradient are evaluated within the symmetric subspace (Page 6). The condition for a valid sample is that each sampled areal must satisfy the condition that its corresponding symmetric state asymm⟩ has a nonzero norm Narepr!= 0; otherwise, the sample is discarded (Page 6). The energy expectation value is expressed as = P asymm ψ(asymm)2 Eloc(asymm) P asymm ψ(asymm)2 (Page 6). The energy gradient can be written in terms of the real-space and symmetric basis components, simplified to ∂E/∂θ

Validation and Results

The method was applied to the periodic J1−J2 antiferromagnetic Heisenberg model in one and two dimensions, incorporating translation, reflection, inversion, and particle number conservation symmetries (Page 3). The results for a 64-site periodic chain and a 6×6 torus showed an absolute energy error of 10−5 for a 64-site periodic chain and a 6×6 torus after bond-dimension extrapolation (Page 1). For the one-dimensional J1−J2 model, precisions of 3 × 10−4 and 5 × 10−5 are achieved with block sizes b = 4 (308 parameters) and b = 8 (252 parameters) for N=32, and 2 × 10−3 and 4 × 10−4 are obtained with b = 4 (660 parameters) and b = 8 (588 parameters) for N=64 in the singlet sector (Page 9). For the triplet sector at N=32, precisions of 1.8×10−3 and 1× 10−3 are achieved with block sizes b = 4 (448 parameters) and b = 8 (408 parameters) (Page 9). Linear extrapolations of the variational energy with respect to the inverse bond dimension yield remarkable energy precision on the order of 10−5 for the singlet ground state and 10−4 for the triplet sector in one dimension (Page 10). These results demonstrate that the hybrid scheme can faithfully capture intricate entanglement structures and long-range correlations across different dimensionalities (Page 10).

Conclusion

This work introduces a hybrid quantum-classical eigensolver that bridges real-space sampling with symmetric-subspace measurement, offering an efficient and scalable approach for simulating strongly correlated quantum many-body systems (Page 11). The framework successfully captures local entanglement directly on quantum subcircuits while restoring global correlations through classical tensor reconnection (Page 11). The method’s efficiency, accuracy, and adaptability make it a promising framework for both high-fidelity classical simulations and scalable implementations on near-term quantum hardware (Page 4). Benchmark calculations validate the power of this approach, achieving DMRG-level precision even with extremely small bond dimensions (D ≤ 6) (Page 10). The method establishes a coherent framework in which real-space quantum sampling, symmetry-constrained measurement, and tensornetwork reconstruction operate synergistically (Page 11). These results highlight the robustness and scalability of the method, achieving DMRG-level precision in large-scale simulations with dramatically reduced computational overhead (Page 10). The hybrid real-space-symmetric-space architectures can overcome the scaling bottlenecks of both classical tensor-network and variational quantum approaches (Page 11). The method’s efficiency, accuracy, and adaptability make it a promising framework for both high-fidelity classical simulations and scalable implementations on near-term quantum hardware (Page 4). The method establishes a coherent framework in which real-space quantum sampling, symmetry-constrained measurement, and tensornetwork reconstruction operate synergistically (Page 11). These results highlight the robustness and scalability of the method, achieving DMRG-level precision in large-scale simulations with dramatically reduced computational overhead (Page 10). The hybrid real-space symmetric space architectures can overcome the scaling bottlenecks of both classical tensor-network and variational quantum approaches (Page 11). The method’s efficiency, accuracy, and adaptability make it a promising framework for both high-fidelity classical simulations and scalable implementations on near-term quantum hardware (Page 4).

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Hybrid Quantum-Classical Eigensolver with Real-Space Sampling and Symmetric Subspace Measurements The gist: We propose a hybrid quantum-classical eigensolver that combines real-space sampling of tensor-network-bridged quantum circuits with symmetric subspace measurements to efficiently simulate strongly correlated quantum many-body systems.

The system is partitioned into equal-sized subsystems, where quantum circuits capture local entanglement and tensor networks reconnect them to recover global correlations (Page 1).

Real-space sampling exploits the generative capability of tensor-backed parallel quantum circuits enabling efficient exploration of spatial configurations (Page 2).

Symmetric subspace measurements project the quantum state onto a drastically reduced symmetric Hilbert space, effectively filtering out irrelevant degrees of freedom while establishing long-range entanglement between distant real-space components (Page 2).

The core idea behind this hybrid approach is a many-to-one mapping defined as pe(asymm) = P g∈G p(areal arepr = gareal), which aggregates the probabilities of all symmetry-equivalent real-space configurations, assigning their collective contribution to the corresponding symmetric basis state asymm (Page 2).

This construction links the real-space distribution function ϕ(areal) into a symmetry-constrained wave function ψ(asymm) via ln ψ(asymm) = 1/2 ln X areal∈garepr ϕ(areal)2, which means the amplitude of asymm⟩ equals the square root of the total probability weight aggregated from all configurations in the class (Page 4).

1. Real-Space Sampling for Generative Exploration of Configuration Space: real-space sampling enables efficient exploration of spatial configurations. This allows for a direct mapping from a real-space distribution function to symmetric basis components, obviating the need for Markov chain updates.

2. Symmetric Subspace Measurement for Hilbert Space Reduction: The method exploits symmetry through a many-to-one mapping between realspace and symmetric bases, which projects the quantum state onto a drastically reduced symmetric Hilbert space, effectively filtering out irrelevant degrees of freedom.

3. Tensor Network Coordination for Global Correlation Restoration: Quantum circuits capture local entanglement while the MPS backbone reconnects them to recover global correlations, ensuring that the resulting wavefunction is a faithful representation of strongly correlated many-body features.

4. RDM-based Isometries for Efficient Ansatz Design: The framework utilizes isometric maps derived from reduced density matrices (RDM) to identify and retain the most relevant subspaces of the Hilbert space, which alleviate the need for deep and highly parameterized quantum circuits.

5. Scalable Benchmarking with Reduced Parameters: The method demonstrates that even with extremely small bond dimensions (D ≤ 6), linear extrapolation of the variational energy yields absolute errors as low as 10−5—comparable to conventional density matrix renormalization group (DMRG) calculations employing bond dimensions in the thousands.

We propose a symmetry-adapted eigensolver for strongly correlated systems that separates where the wave function is sampled from where it is defined. The lattice is divided into equal blocks: parallel quantum circuits absorb the entanglement within each block, while a matrix product state (MPS) backbone bridges the blocks and restores global correlations. The resulting real-space amplitude ϕ(a real) acts only as a sample generator---configurations are drawn from ϕ(a real) squared directly, block by block, without any Markov chain. The variational state itself is defined in a symmetry-adapted basis by two separate rules: the modulus of ψ(a symm) is the square root of the total weight accumulated over the entire equivalence class, (a symm)= sum a real ing a reprϕ(a real) squared, whereas its phase is inherited from the single representative amplitude ϕ(a repr). The modulus rule is what makes the scheme both exact and cheap: the distribution (a symm) sampled in the symmetric basis is exactly the one induced by the mapping from the real-space distribution p(a real), so every generated configuration is an unbiased sample of ψ(a symm) squared; the phase rule then fixes the only freedom that remains. Because amplitudes within a class are combined in probability rather than summed with symmetry characters, ψ is not a group projection of the circuit-MPS state; the class-wide aggregation instead supplies long-range, symmetry-enforced entanglement that the MPS never has to carry itself.