Quantum-inspired numerical analysis#
SeeMPS extends the MPS/TT formalism to function representation and numerical analysis. Multidimensional functions discretized on exponentially large grids can be encoded using quantized tensor trains (QTT), exploiting the fact that smooth functions often have bounded entanglement when represented in this binary encoding.
Consider a one-dimensional function \(f(x)\) defined over an interval \([a,b)\) sampled on a uniform grid of \(2^n\) points. The vector of function values \(v_i = f(x_i)\) can be tensorized using \(n\) binary indices (qubits) and compressed as an MPS. Provided the bond dimensions remain bounded, the storage cost scales as \(\mathcal{O}(n \chi^2)\) compared to \(\mathcal{O}(2^n)\) for the full grid representation.
The library provides tools for:
Function loading: Multiple strategies to construct MPS representations of functions, including direct analytic constructions, polynomial expansions, and tensor cross-interpolation.
Differentiation: Finite differences, Fourier differentiation, and HDAF methods implemented as MPO operators.
Integration: Newton-Cotes and Clenshaw-Curtis quadrature rules that reduce to inner products between MPS.
Interpolation: Finite differences and Fourier-based interpolation to estimate function values between grid points.
Optimization: Finding minima and maxima of functions encoded as MPS, as well as obtaining functions that solve optimization problems, such as eigenvalue equations.
These tools enable efficient solvers for ordinary and partial differential equations in high dimensions, both for eigenvalue problems and source problems with Dirichlet or periodic boundary conditions.