# Variational quantum algorithm for quantum matter using Trotterized entanglement renormalization

**Thomas Barthel, Duke University**

I will describe a variational quantum eigensolver for the simulation of strongly-correlated quantum matter based on a multi-scale entanglement renormalization ansatz (MERA) and gradient-based optimization. Due to its narrow causal cone, the algorithm can be implemented on noisy intermediate-scale (NISQ) devices and still describe large systems. The number of required qubits is system-size independent and increases only to a logarithmic scaling 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. For the practical implementation, the MERA disentanglers and isometries are Trotterized, i.e., implemented as brickwall circuits. With a few Trotter steps, one recovers the accuracy of the full MERA. Results of benchmark simulations for various critical spin models establish a quantum advantage, and I will report on first experimental tests on ion-trap devices. For systems with finite-range interactions, one can also show that the variational energy optimization of isometric tensor networks like MPS, TTNS, and MERA is free of barren plateaus.

**References:**

- arXiv:2108.13401 – Q. Miao and T. Barthel, “A quantum-classical eigensolver using multiscale entanglement renormalization”
- arXiv:2303.08910 – Q. Miao and T. Barthel, “Convergence and quantum advantage of Trotterized MERA for strongly-correlated systems”
- arXiv:2304.00161 – T. Barthel and Q. Miao, “Absence of barren plateaus and scaling of gradients in the energy optimization of isometric tensor network states”
- arXiv:2304.14320 – Q. Miao and T. Barthel, “Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus”