Convergence and Quantum Advantage of Trotterized MERA for Strongly-Correlated Systems

Qiang Miao; Thomas Barthel

doi:10.22331/q-2025-02-11-1631

Convergence and Quantum Advantage of Trotterized MERA for Strongly-Correlated Systems

Qiang Miao¹ and Thomas Barthel^1,2,3

¹Duke Quantum Center, Duke University, Durham, North Carolina 27701, USA
²Department of Physics, Duke University, Durham, North Carolina 27708, USA
³Tensor Center, Auf dem Dresch 15, 52152 Simmerath, Germany

Published:	2025-02-11, volume 9, page 1631
Editor:	Remigiusz Augusiak
Eprint:	arXiv:2303.08910v3
Doi:	https://doi.org/10.22331/q-2025-02-11-1631
Citation:	Quantum 9, 1631 (2025).

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Abstract

Strongly-correlated quantum many-body systems are difficult to study and simulate classically. We recently proposed a variational quantum eigensolver (VQE) based on the multiscale entanglement renormalization ansatz (MERA) with tensors constrained to certain Trotter circuits. Here, we determine the scaling of computation costs for various critical spin chains which substantiates a polynomial quantum advantage in comparison to classical MERA simulations based on exact energy gradients or variational Monte Carlo. Algorithmic phase diagrams suggest an even greater separation for higher-dimensional systems. Hence, the Trotterized MERA VQE is a promising route for the efficient investigation of strongly-correlated quantum many-body systems on quantum computers. Furthermore, we show how the convergence can be substantially improved by building up the MERA layer by layer in the initialization stage and by scanning through the phase diagram during optimization. For the Trotter circuits being composed of single-qubit and two-qubit rotations, it is experimentally advantageous to have small rotation angles. We find that the average angle amplitude can be reduced considerably with negligible effect on the energy accuracy. Benchmark simulations suggest that the structure of the Trotter circuits for the TMERA tensors is not decisive; in particular, brick-wall circuits and parallel random-pair circuits yield very similar energy accuracies.

Featured image: The multi-scale entanglement renormalization ansatz (MERA) is a hierarchical tensor network and features a causal structure: The expectation value of any local observable $\hat{h}_i$ only depends on the the tensors of $\hat{h}_i$'s causal cone as indicated by the blue-shaded region on the left. This reduces computation costs from an exponential to a logarithmic scaling in the system size, and the number of required qubits from linear to system-size independent. For the implementation on a quantum computer, the tensors are Trotterized, i.e., built out of single-qubit and two-qubit gates (top right). Algorithmic phase diagrams and benchmark simulations substantiate a polynomial quantum advantage for the MERA variational quantum eigensolver (bottom right).

Popular summary

Strongly-correlated quantum matter like the insufficiently understood high-temperature superconductors is very difficult to simulate on classical computers. This work explores a hybrid quantum-classical simulation method for such systems, where the computationally intensive operations are executed on a quantum computer.

In contrast to classical systems, the state space for a quantum many-body system grows exponentially in the number of particles which often poses a challenge for the analysis by classical means. To resolve this, Richard Feynman suggested to employ quantum devices for the investigation of complex quantum systems. We describe how this can be achieved for models of quantum materials even though, for the time being, quantum computers have small numbers of quantum bits subject to noise. The decisive trick is to employ the so-called multiscale entanglement renormalization ansatz (MERA). Due to the causal structure of the MERA tensor networks, which are reminiscent of light cones, expectation values of local observables like energy densities can be evaluated efficiently with relatively small numbers of quantum bits. In this variational quantum algorithm (VQA), the parameters of the MERA are stored on a classical computer and the costly evaluations of energies and energy gradients, needed for the MERA optimization, are implemented on the quantum computer. To this purpose, the MERA tensors are Trotterized, which means that all tensors are chosen as circuits of two-qubit gates.

To substantiate a quantum advantage for the MERA VQA, we analyze the scaling of computation costs with respect to the desired groundstate approximation accuracy and find indeed a polynomial advantage compared to the traditional MERA techniques on classical computers as well as further quantum-inspired classical algorithms which employ tensor Trotterization (instead of unconstrained tensors) and stochastic sampling of energy gradients (instead of an exact gradient evaluation). We also explain methods to substantially improve the convergence properties of the MERA VQA and propose a way to reduce experimentally requirements by reducing the average gate rotation angles. These represent exciting steps towards viable quantum computing applications in condensed matter and particle physics.

► BibTeX data

@article{Miao2025convergencequantum,
  doi = {10.22331/q-2025-02-11-1631},
  url = {https://doi.org/10.22331/q-2025-02-11-1631},
  title = {Convergence and {Q}uantum {A}dvantage of {T}rotterized {MERA} for {S}trongly-{C}orrelated {S}ystems},
  author = {Miao, Qiang and Barthel, Thomas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1631},
  month = feb,
  year = {2025}
}

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