Title: Quantum simulation of condensed matter using entanglement renormalization
Abstract: Strongly-correlated quantum matter can be simulated with tensor network states. A very interesting approach, motivated by real-space renormalization group, is the multi-scale entanglement renormalization ansatz (MERA). While MERA has various advantages over alternative tensor network methods, it has relatively high classical computation costs, which limits the attainable approximation accuracy [1]. To avoid the classically expensive contractions of high-order tensors, we have developed a variational quantum eigensolver (VQE) based on MERA and tensor Trotterization [2]. Due to its causal structure and noise-resilience, the MERA VQE can be implemented on noisy intermediate-scale (NISQ) devices and still describe large physical systems. The number of required qubits is system-size independent and only grows logarithmically when using quantum amplitude estimation to speed up gradient evaluations. Translation invariance can be used to make the time complexity square-logarithmic in the system size and describe the thermodynamic limit. Results of benchmark simulations for various critical spin models and algorithmic phase diagrams substantiate a quantum advantage [3], and we have rigorously proven the absence of barren plateaus in the MERA VQE [4-6]. I will report on first experimental tests on ion-trap devices, which clearly demonstrate a continuous quantum phase transition with a sharp onset of the order parameter at the critical point. Using a new holographic tomography scheme, we were also able to resolve for the first time the transition from area-law to log-area law scaling of groundstate entanglement entropies when approaching criticality [7].
[1] "Scaling of contraction costs for entanglement renormalization algorithms including tensor Trotterization and variational Monte Carlo", PRB 111, 045104 (2025) [2] "A quantum-classical eigensolver using multiscale entanglement renormalization", PRR 5, 033141 (2023) [3] "Convergence and quantum advantage of Trotterized MERA for strongly-correlated systems", Quantum 9, 1631 (2025) [4] "Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states", Commun. Math. Phys. 406, 86 (2025) [5] "Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus", PRA 109, L050402 (2024) [6] "Equivalence of cost concentration and gradient vanishing for quantum circuits: An elementary proof in the Riemannian formulation", Quantum Sci. Technol. 9, 045039 (2024) [7] "Probing entanglement scaling across a quantum phase transition on a quantum computer", arXiv:2412.18602