Matrix Product States for Quantum Stochastic Modeling.

Area of Science:

• Stochastic modeling
• Condensed matter physics
• Quantum information theory

Background:

• Predictive models for stochastic processes aim to forecast future behavior using limited past data.
• Matrix product states (MPS) provide an efficient representation for one-dimensional (1D) quantum spin chains.

Purpose of the Study:

• To establish a formal connection between optimal predictive models for stochastic processes and matrix product states (MPS).
• To leverage insights from quantum physics to improve stochastic modeling and quantify prediction memory requirements.

Main Methods:

• Associating each stochastic process with a corresponding quantum state of a spin chain.
• Demonstrating that optimal predictive models for these processes correspond to MPS representations of these quantum states.
• Utilizing MPS techniques to systematically construct optimal quantum predictive models.

Main Results:

• The optimal predictive model for a stochastic process is shown to be equivalent to the MPS representation of its associated quantum spin chain state.
• MPS methods provide a systematic framework for constructing the most effective quantum predictive models.
• A novel method is introduced for calculating the quantum memory required for optimal predictions.

Conclusions:

• A direct and powerful link exists between stochastic process prediction and the MPS formalism in quantum physics.
• The quantum memory required for optimal stochastic predictions is precisely quantified by the entanglement in the associated spin chain.
• This interdisciplinary connection offers new avenues for both improving predictive modeling and understanding quantum systems.