Certified algorithms for equilibrium states of local quantum Hamiltonians

Area of Science:

• Quantum Many-Body Physics
• Computational Quantum Physics
• Theoretical Computer Science

Background:

• Predicting equilibrium states in quantum many-body systems is computationally challenging.
• Existing methods struggle with convergence in the thermodynamic limit, leading to undecidability.
• Finite-size scaling often lacks certified bounds for reliable predictions.

Purpose of the Study:

• To develop certified algorithms for computing expectation values of observables in quantum systems.
• To provide rigorous lower and upper bounds for these expectation values.
• To demonstrate the feasibility of approximating observables in finite time.

Main Methods:

• Design of certified algorithms for expectation value computation.
• Application to local quantum Hamiltonians at zero and positive temperatures.
• Analysis of convergence for commuting Hamiltonians on 2D lattices.

Main Results:

• Algorithms provide rigorous bounds on expectation values.
• Demonstrated approximation of local observables in finite time, contrasting undecidability results.
• Proved fast convergence at high temperatures for commuting 2D lattice Hamiltonians.

Conclusions:

• The developed algorithms offer a certified approach to quantum many-body problem prediction.
• Local observables can be approximated efficiently using convex optimization for specific systems.
• This work bridges the gap between theoretical undecidability and practical computation.