Researchers have developed a new variational time-evolution principle that optimizes simulations of quantum many-body systems, especially those that thermalize. Unlike conventional methods that address the exponential complexity of the global wavefunction, this approach focuses on the evolution of local observables, whose intrinsic complexity does not scale with the total system size. This distinction is crucial for thermalizing systems, where local observables lose memory of microscopic details and relax towards equilibrium values determined by only a few parameters.

The proposed algorithm is based on the local optimization of matrix-product states (MPS) and allows for closed-form equations of motion, analogous to the time-dependent variational principle. This method maintains coherent short-time dynamics while exploiting the simplification produced by thermalization at later times. The key lies in replacing global-state fidelity with a cost function defined on local reduced density matrices, significantly reducing the computational burden.

The same variational principle has a quantum-classical counterpart, combining quantum evaluation of the local cost with an optimization strategy robust to both shot and hardware noise. To demonstrate its feasibility, proof-of-concept implementations were carried out on quantum processors such as Quantinuum H2 and IBM Heron, which successfully recovered the characteristic local dynamics of these systems. This advance could enable more efficient simulations of complex quantum systems, opening new avenues for the study of condensed matter and other quantum phenomena.