We introduce a quantum optimal control algorithm for energy minimization that combines the diffeomorphic modulation under observable response preserving homotopy (D-MORPH) gradient and the Broyden Fletcher Goldfarb Shanno (BFGS) iterative scheme for nonlinear optimization. An extended set of controls defining the time-dependent mass, dipole moment, and external perturbational field are optimized to find an effective Hamiltonian that steers the dynamics of the system into the global minimum without getting trapped into local minima. The algorithm is illustrated as applied to energy minimization on rugged surfaces and golf potentials comparable to those previously explored for testing quantum annealing methodologies.

Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce the iterative power algorithm (IPA) for global optimization and a formal proof of convergence for both discrete and continuous global search problems, which is essential for applications in chemistry such as molecular geometry optimization. IPA implements the power iteration method in quantics tensor train (QTT) representations. Analogous to the imaginary time propagation method with infinite mass, IPA starts with an initial probability distribution ρ0(x) and iteratively applies the recurrence relation ρ k+1(x) = U(x) ρ k (x)/∥Uρ k ∥ L 1 , where U(x) = e–V(x) is defined in terms of the potential energy surface (PES) V(x) with global minimum at x = x*. Upon convergence, the probability distribution becomes a delta function δ­(x – x*), so the global minimum can be obtained as the position expectation value x* = Tr­[x δ­(x – x*)]. QTT representations of V(x) and ρ­(x) are generated by fast adaptive interpolation of multidimensional arrays to bypass the curse of dimensionality and the need to evaluate V(x) for all possible values of x. We illustrate the capabilities of IPA for global search optimization of two multidimensional PESs, including a differentiable model PES of a DNA chain with D = 50 adenine–thymine base pairs, and a discrete non-differentiable potential energy surface, V(p) = mod­(N,p), that resolves the prime factors of an integer N, with p in the space of prime numbers {2, 3,..., p max} folded as a d-dimensional 21 × 22 × ··· × 2 d tensor. We find that IPA resolves multiple degenerate global minima even when separated by large energy barriers in the highly rugged landscape of the potentials. Therefore, IPA should be of great interest for a wide range of other optimization problems ubiquitous in molecular and electronic structure calculations.

Numerically exact simulations of quantum reaction dynamics, including nonadiabatic effects in excited electronic states, are essential to gain fundamental insights into ultrafast chemical reactivity and rigorous interpretations of molecular spectroscopy. Here, we introduce the tensor-train split-operator KSL (TT-SOKSL) method for quantum simulations in tensor-train (TT)/matrix product state (MPS) representations. TT-SOKSL propagates the quantum state as a tensor train using the Trotter expansion of the time-evolution operator, as in the tensor-train split-operator Fourier transform (TT-SOFT) method. However, the exponential operators of the Trotter expansion are applied using a rank-adaptive TT-KSL scheme instead of using the scaling and squaring approach as in TT-SOFT. We demonstrate the accuracy and efficiency of TT-SOKSL as applied to simulations of the photoisomerization of the retinal chromophore in rhodopsin, including nonadiabatic dynamics at a conical intersection of potential energy surfaces. The quantum evolution is described in full dimensionality by a time-dependent wavepacket evolving according to a two-state 25-dimensional model Hamiltonian. We find that TT-SOKSL converges faster than TT-SOFT with respect to the maximally allowed memory requirement of the tensor-train representation and better preserves the norm of the time-evolving state. When compared to the corresponding simulations based on the TT-KSL method, TT-SOKSL has the advantage of avoiding the need to construct the matrix product state Laplacian by exploiting the linear scaling of multidimensional tensor-train Fourier transforms.