Quantum Systems: Simple Equations Unlock Exact Solutions for Complex Problems
A new equation determines when Matrix Product States accurately represent the eigenstates of local operators, covering scenarios from Hamiltonian eigenstates and driven quantum systems to steady states and symmetries. José Garre Rubio and colleagues at University of Vienna show that a concise, fixed-size equation, specifically, the action of an operator term on a block of tensors, provides both a necessary and sufficient condition for these exact solutions. This local characterisation enables thorough analysis of solutions across diverse physical settings, such as the recovery of quantum group symmetries within the XXZ model, and promises advancements in numerical algorithms using Matrix Product States and their generalisation to two dimensions with Projected Entangled Pair States A local Hamiltonian equation confirms Matrix Product State eigenstate accuracy Determining whether a Matrix Product State accurately represents a quantum system’s eigenstate is now achievable with a single, fixed-size equation, a strong improvement over previous indirect methods. These earlier methods lacked definitive confirmation of solutions. This local equation details how a Hamiltonian act on a block of tensors, providing both a necessary and sufficient condition for exact solutions. Previously, establishing such a condition necessitated complex analysis and was often limited to specific cases. This breakthrough transforms Matrix Product States from a primarily numerical technique into a powerful analytical framework, enabling thorough characterisation of solutions across diverse physical scenarios, including eigenstates, scar states, and symmetries. The local equation extends beyond simple ground states, also characterising exact MPS trajectories for driven quantum systems and steady states of local Lindbladians. These Lindbladians describe how quantum systems evolve with dissipation. The framework successfully recovers the quantum group symmetries present within the complex XXZ mode
