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 computation costs 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 proven the absence of barren plateaus [4-6]. I will also report on first experimental tests, probing a quantum phase transition and critical entanglement on ion-trap devices.

[1] "Scaling of contraction costs for entanglement renormalization algorithms including tensor Trotterization and variational Monte Carlo", arXiv:2407.21006
[2] "A quantum-classical eigensolver using multiscale entanglement renormalization", arXiv:2108.13401, PRR 5, 033141 (2023)
[3] "Convergence and quantum advantage of Trotterized MERA for strongly-correlated systems", arXiv:2303.08910
[4] "Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states", arXiv:2304.00161
[5] "Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus", arXiv:2304.14320, PRA 109, L050402 (2024)
[6] "Equivalence of cost concentration and gradient vanishing for quantum circuits: An elementary proof in the Riemannian formulation", arXiv:2402.07883, Quantum Sci. Technol. 9, 045039 (2024)