A Hybrid Algorithm Combining Matrix-Product States and Quantum Monte Carlo for Efficiently Simulating Correlated Quantum Matter in Two Dimensions

Core Concepts

This paper introduces a novel hybrid algorithm combining matrix-product states (MPS) and auxiliary-field quantum Monte Carlo (AFQMC) to overcome the limitations of each method in simulating two-dimensional correlated quantum matter, demonstrating its potential to efficiently address previously intractable problems.

Abstract

• Bibliographic Information: Bollmark, G., Mardazad, S., Hofmann, J. S., & Kantian, A. (2024). Fusing matrix-product states with quantum Monte Carlo: reducing entanglement and sign problem at the same time. arXiv preprint arXiv:2411.00480v1.
• Research Objective: This paper presents a new hybrid algorithm that combines the strengths of MPS and AFQMC methods to overcome their individual limitations in simulating two-dimensional correlated quantum matter. The objective is to develop an algorithm that can efficiently simulate systems that are challenging for traditional methods due to entanglement growth (MPS) or the sign problem (AFQMC).
• Methodology: The researchers develop a hybrid algorithm that utilizes MPS to represent one-dimensional slices of a two-dimensional system and AFQMC to treat the coupling between these slices. This approach leverages the efficiency of MPS in one dimension and the ability of AFQMC to handle higher-dimensional systems. The algorithm is tested on two model systems: a repulsive Hubbard model and a model with an effective pair-hopping interaction.
• Key Findings: The hybrid algorithm successfully simulates both model systems, demonstrating its ability to address the sign problem and entanglement growth. The researchers find that the algorithm outperforms standard AFQMC for the repulsive Hubbard model, exhibiting no sign problem for the studied parameter range. While a sign problem is observed for the effective hopping interaction model, the hybrid algorithm still provides valuable insights into the system's behavior.
• Main Conclusions: The hybrid MPS-AFQMC algorithm presents a promising approach for simulating a wide range of correlated quantum matter systems that are difficult to address with traditional methods. The algorithm's ability to mitigate the sign problem and entanglement growth opens up new possibilities for studying complex quantum phenomena in two dimensions.
• Significance: This research significantly contributes to the field of computational physics by providing a new tool for simulating complex quantum systems. The hybrid algorithm has the potential to advance our understanding of strongly correlated materials, quantum phase transitions, and other challenging problems in condensed matter physics.
• Limitations and Future Research: The current implementation of the algorithm is limited by the requirement of weak inter-chain coupling. Future research could focus on developing more efficient update schemes within the algorithm to overcome this limitation and extend its applicability to a broader range of systems. Further investigation into the interplay between the MPS and AFQMC components could lead to a deeper understanding of the sign problem and its mitigation in this hybrid approach.

Stats

The time step size used in the simulations is dτ = 0.125.
The MPS+MC routine maintains a discarded weight of ϵψ = 10−8, resulting in an error of ϵMPS ∼10−4.
The on-site interaction strength for the repulsive Hubbard model is U = 8t.
The on-site interaction strength for the effective hopping interaction model is U = −10t.
The density for the mass imbalance phase analysis is n = 0.890625.

Quotes

Key Insights Distilled From

Fusing matrix-product states with quantum Monte Carlo: reducing entanglement and sign problem at the same time

by Gunnar Bollm... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00480.pdf

Deeper Inquiries

How does the computational cost of the hybrid algorithm scale with system size compared to pure MPS and AFQMC approaches?

The hybrid MPS-AFQMC algorithm presents a trade-off in computational cost compared to pure MPS and AFQMC approaches, excelling in certain aspects while being potentially limited in others:
Scaling Advantages over pure MPS:

Reduced Entanglement: By treating the inter-chain coupling (in the y-direction) with AFQMC, the MPS component only deals with 1D chains. This drastically reduces the bipartite entanglement the MPS needs to capture compared to a full 2D MPS simulation. Consequently, the required bond dimension for the MPS scales much more favorably with system size, leading to significant computational savings.
Scaling Advantages over pure AFQMC:

Sign Problem Mitigation: The hybrid approach can significantly reduce or even eliminate the sign problem plaguing AFQMC in certain regimes. This is achieved by treating the parts of the Hamiltonian prone to causing the sign problem (e.g., local repulsion in doped systems, frustration) with the numerically exact MPS method. This leads to a more benign scaling of statistical errors with system size and inverse temperature.
Potential Limitations:

Scaling with Transverse Coupling: The efficiency of the current update scheme, which updates all HS fields in a single imaginary time slice, deteriorates with increasing transverse coupling strength (V⊥). This is because the acceptance ratio of the Metropolis-Hastings algorithm decreases, requiring more computational effort to achieve the same statistical accuracy.

Overall Scaling: While the hybrid algorithm mitigates some scaling issues, it doesn't entirely escape the inherent complexities of correlated quantum systems. The computational cost still grows with system size, albeit at a potentially slower rate than pure MPS or AFQMC in favorable cases.
In summary: The hybrid algorithm shines when the transverse coupling is relatively weak and the sign problem in AFQMC is severe. In these scenarios, it can handle larger system sizes and lower temperatures than either pure method. However, as the transverse coupling strengthens, the efficiency gain might diminish, and alternative update schemes become crucial.

Could the sign problem observed in the effective hopping interaction model be a result of the specific update scheme used, and could alternative schemes mitigate this issue?

Yes, the sign problem observed in the effective hopping interaction model could be exacerbated by the global update scheme, and alternative schemes could potentially mitigate this issue. Here's why:

Global Updates and Frustration: The current scheme updates all HS fields in an entire imaginary time slice simultaneously. While this leverages the strength of MPS for 1D time evolution, it can also lead to a higher chance of proposing updates that increase frustration in the system, especially for strong transverse couplings. This is because the algorithm doesn't "see" the effect of individual HS field changes on the overall configuration energy until the entire slice is updated.

Local Update Schemes: Implementing local update schemes, where individual HS fields are updated sequentially, could help alleviate this problem. By evaluating the energy change associated with each local update, the algorithm can make more informed decisions, potentially leading to a higher acceptance ratio and a less severe sign problem.

Challenges and Potential Solutions:

MPS Efficiency: Efficiently implementing local updates within the MPS framework is crucial. The algorithm needs to quickly evaluate the energy change associated with flipping a single HS field, which requires careful consideration of the MPS structure and operations.
Differential Updates: One promising avenue is exploring techniques that efficiently compute the effect of differential operators on the MPS, representing the change due to a single HS field flip. This could involve calculating expectation values of operators like e^(√τ∆ητ,na(n)i ˆoi), where ∆ητ,n represents the change in the HS field.

Further Investigation: While local update schemes hold promise, their effectiveness in mitigating the sign problem for this specific model needs further investigation. It's crucial to carefully analyze the interplay between the update scheme, the model's Hamiltonian, and the chosen MPS algorithm to determine the optimal strategy.

Can this hybrid algorithm be generalized to study dynamical properties of correlated quantum systems, and what new insights could such simulations provide?

While the paper focuses on equilibrium properties, generalizing the hybrid MPS-AFQMC algorithm to study dynamical properties of correlated quantum systems is an exciting prospect, potentially offering valuable insights:
Generalization Strategies:

Real-Time Evolution: Instead of imaginary time evolution for the partition function, one could adapt the MPS component to perform real-time evolution under the influence of the HS fields. This would involve using a suitable real-time MPS algorithm (e.g., time-evolving block decimation - TEBD) to compute the time-dependent state.
Dynamical Correlators: By evolving the system in real-time, one could calculate dynamical correlation functions, providing information about the system's response to external perturbations or probes. This could involve measuring correlations between operators at different times, revealing information about excitations, relaxation processes, and transport phenomena.
Potential Insights:

Non-Equilibrium Dynamics: The hybrid algorithm could shed light on the non-equilibrium dynamics of correlated systems, a regime where traditional methods often struggle. This includes studying phenomena like quantum quenches, where a system is suddenly driven out of equilibrium, or driven-dissipative systems, where the interplay of coherent dynamics and dissipation leads to rich behavior.
Competition between Orders: For systems with competing orders, like the effective hopping interaction model, real-time simulations could provide insights into the dynamics of phase transitions and the formation of ordered states. This could involve analyzing how different order parameters evolve in time and how they are influenced by the interplay of interactions and quantum fluctuations.
Challenges and Opportunities:

Computational Cost: Real-time MPS simulations are generally more computationally demanding than imaginary time evolution. The entanglement growth in time poses a significant challenge, potentially limiting the accessible timescales.
Algorithm Development: Adapting the hybrid algorithm for real-time dynamics requires careful consideration of the interplay between the MPS and AFQMC components. Developing efficient algorithms for real-time evolution under HS field configurations is crucial.
In conclusion: Generalizing the hybrid MPS-AFQMC algorithm to study dynamical properties holds significant promise for advancing our understanding of correlated quantum systems. While challenges remain, the potential insights into non-equilibrium dynamics, phase transitions, and the behavior of complex materials make it a fruitful avenue for future research.