Determining the N-Representability of a Reduced Density Matrix via Unitary Evolution and Stochastic Sampling

Area of Science:

• Computational Quantum Chemistry
• Many-Body Physics
• Quantum Information Theory

Background:

• The N-representability problem determines if a p-body matrix is derivable from an N-body density matrix.
• Solving this problem is crucial for finding exact ground states of quantum systems via constrained minimization.
• Existing N-representability conditions become computationally intractable for large systems due to exponential constraint growth.

Purpose of the Study:

• To introduce a novel hybrid quantum-stochastic algorithm to circumvent the computational limitations of traditional N-representability conditions.
• To develop a method that can determine if a given p-body matrix is N-representable, assess its quality, and correct it.
• To create a Hamiltonian-independent approach applicable to various quantum models.

Main Methods:

• A hybrid quantum-stochastic algorithm combining the Adaptive Derivative-Assembled Pseudo-Trotter (ADAPT) method with simulated annealing.
• The algorithm iteratively applies unitary evolution operators, guided by a stochastic process, to an initial N-body density matrix.
• The process aims to evolve the reduced density matrix of a p-body subsystem towards a target p-body matrix.

Main Results:

• The proposed hybrid ADAPT algorithm successfully approximates N-representability conditions.
• The method demonstrated expected behavior for 1- and 2-reduced density matrices (1-RDMs and 2-RDMs) across diverse models.
• The algorithm effectively evolved initial matrices towards specified target matrices, validating its applicability.

Conclusions:

• The hybrid quantum-stochastic algorithm offers an effective and computationally feasible alternative to traditional N-representability conditions.
• This approach provides a robust criterion for assessing and correcting the N-representability of p-RDMs.
• The Hamiltonian-independent nature of the algorithm broadens its applicability in quantum chemistry and condensed matter physics.