The Integral Decimation Method for Quantum Dynamics and Statistical Mechanics

Abstract

The solutions to many problems in the mathematical, computational, and physical sciences often involve multidimensional integrals. A direct numerical evaluation of the integral incurs a computational cost that is exponential in the number of dimensions, a phenomenon called the curse of dimensionality. The problem is so substantial that one usually employs sampling methods, like Monte Carlo, to avoid integration altogether. Here, we derive and implement a quantum-inspired algorithm to decompose a multidimensional integrand into a product of matrix-valued functions -- a spectral tensor train -- changing the computational complexity of integration from exponential to polynomial. The algorithm constructs a spectral tensor train representation of the integrand by applying a sequence of quantum gates, where each gate corresponds to an interaction that involves increasingly more degrees of freedom in the action. Because it allows for the systematic elimination of small contributions to the integral through decimation, we call the method integral decimation. The functions in the spectral basis are analytically differentiable and integrable, and in applications to the partition function, integral decimation numerically factorizes an interacting system into a product of non-interacting ones. We illustrate integral decimation by evaluating the absolute free energy and entropy of a chiral XY model as a continuous function of temperature. We also compute the nonequilibrium time-dependent reduced density matrix of a quantum chain with between two and forty levels, each coupled to colored noise. When other methods provide numerical solutions to these models, they quantitatively agree with integral decimation. When conventional methods become intractable, integral decimation can be a powerful alternative.